uni wÜrzburg ,scan...uni wÜrzburg ,scan 16th gamm-imacs international symposium on scientific...
TRANSCRIPT
UNIWÜRZBURG SCAN,
16th GAMM-IMACS International Symposium onScientific Computing, Computer Arithmetic and Validated Numerics
September 21-26, 2014
Book of AbstractsDepartment of Computer Science
University of WürzburgGermany
SCA
N 2
014
Boo
k of
Abs
trac
ts
16th GAMM-IMACS International Symposium onScientific Computing, Computer Arithmetic and
Validated Numerics
SCAN 2014
Book of Abstracts
Department of Computer ScienceUniversity of Wurzburg
Germany
September 21-26, 2014
Editor: Marco NehmeierCover design: Marco NehmeierCover photo: Marco Nehmeier
Scientific Committee
• G. Alefeld (Karlsruhe, Germany)• J.-M. Chesneaux (Paris, France)• G.F. Corliss (Milwaukee, USA)• T. Csendes (Szeged, Hungary)• A. Frommer (Wuppertal, Germany)• R.B. Kearfott (Lafayette, USA)• W. Kraemer (Wuppertal, Germany)• V. Kreinovich (El Paso, USA)• U. Kulisch (Karlsruhe, Germany)• W. Luther (Duisburg, Germany)• G. Mayer (Rostock, Germany)• S. Markov (Sofia, Bulgaria)• J.-M. Muller (Lyon, France)• M. Nakao (Fukuoka, Japan)• M. Plum (Karlsruhe, Germany)• N. Revol (Lyon, France)• J. Rohn (Prague, Czech Republic)• S. Rump (Hamburg, Germany)• S. Shary (Novosibirsk, Russia)• Yu. Shokin (Novosibirsk, Russia)• W. Walter (Dresden, Germany)• J. Wol↵ von Gudenberg (Wurzburg, Germany)• N. Yamamoto (Tokyo, Japan)
Organizing Committee
• J. Wol↵ von Gudenberg (Chair)• Alexander Dallmann• Fritz Kleemann• Marco Nehmeier• Anika Schwind• Susanne Stenglin
3
Preface
SCAN 2014 will certainly become another flagship conference in theresearch areas of
• reliable computer arithmetic• enclosure methods• self validating algorithms
and will provide a lot of helpful and useful ideas for trustworthy ap-plications and further research. This booklet contains the abstracts ofthe invited or contributed talks.
The conference starts with awarding the Moore Prize for the bestapplication of interval methods to Prof. Kenta Kobayashi for hisComputer-Assisted Uniqueness Proof for Stokes’ Wave of ExtremeForm (p. 83).
Every morning and every afternoon starts with an invited keynote,and I think we can be proud that the following distinguished expertshave accepted our invitation:
• Ekaterina Auer (University of Duisburg-Essen, Germany)• Andrej Bauer (University of Ljubljana, Slovenia)• Sylvie Boldo (Inria, France)• Jack Dongarra (University of Tennessee and ORNL, USA; Univer-sity of Manchester, UK)
• John Gustafson (Ceranovo Inc., USA)• Bart lomiej Jacek Kubica (Warsaw University of Technology, Poland)• John Pryce (Cardi↵ University, UK)
SCAN 2014 in Wurzburg
The relations between Wurzburg and SCAN 2014 are surprisingly ob-vious. The University of Wurzburg was founded in 1402. Its logo isa left bracket symbol expressing the ongoing, open ended progress ofknowledge. If we permute the digits of the foundation year and closethe left bracket by a right one we have our particular interval for thelogo of the SCAN 2014 conference.
4
Organizing SCAN 2014 in Wurzburg
I want to thank all our sponsors, there donations made the conferencepossible.
I further want to thank the members of the scientific committee,the organizers of the last 3 meetings V. Kreinovich, N. Revol, andS. Shary in particular.
Organizing this conference was a pleasure for me, because of thetremendous assistance I got from the organizing committee:
Alexander Dallmann, Fritz Kleemann, Marco Nehmeier, AnikaSchwind and Susanne Stenglin.
Wurzburg, September 2014 Jurgen Wol↵ von Gudenberg
Last but not least I wish all of us a
SCAN
Successful Conference and Attractive Night program
5
Schedule
Sunday, September 21, 2014
18:00 – 20:00 Get-together and Registration (GHOTEL)
Monday, September 22, 2014
8:00 – 9:00 Registration (Conference O�ce)
9:00 – 9:30 Opening Session (Turing, Chair: J. Wol↵ von Gudenberg)
9:30 – 10:30 R. E. Moore Prize Awarding Ceremony (Turing, Chair:V. Kreinovich)
Kenta KobayashiComputer-Assisted Uniqueness Proof for Stokes’ Wave of ExtremeForm (p. 83)
10:30 – 11:00 Co↵ee Break
11:00 – 12:00 Plenary Talk (Turing, Chair: B. Kearfott)
John PryceThe architecture of the IEEE P1788 draft standard for intervalarithmetic (p. 135)
12:00 – 13:20 Lunch
13:20 – 14:20 Plenary Talk (Turing, Chair: W. Luther)
Ekaterina AuerResult Verification and Uncertainty Management in EngineeringApplications (p. 30)
14:20 – 16:00 Parallel Sessions
Session A1: Solution Sets (Turing, Chair: J. Garlo↵)
(1) Shinya MiyajimaVerified solutions of saddle point linear systems (p. 114)
6
(2) Takeshi OgitaIterative Refinement for Symmetric Eigenvalue Problems (p. 127)
(3) Sergey P. SharyMaximum Consistency Method for Data Fitting under Interval Un-certainty (p. 147)
(4) Gunter MayerA short description of the symmetric solution set (p. 107)
Session A2: Non Standard Interval Libraries (Zuse, Chair: C.P. Jean-nerod)
(1) Abdelrahman Elskhawy, Kareem Ismail and Maha ZohdyModal Interval Floating Point Unit with Decorations (p. 49)
(2) Jordan Ninin and Nathalie RevolAccurate and e�cient implementation of a�ne arithmetic usingfloating-point arithmetic (p. 125)
(3) Philippe ThevenyChoice of metrics in interval arithmetic (p. 157)
(4) Olga Kupriianova and Christoph LauterReplacing branches by polynomials in vectorizable elementary func-tions (p. 92)
Session A3: ODE – Orbits (Moore, Chair: A. Rauh)
(1) Tomohirio Hiwaki and Nobito YamamotoA numerical verification method for a basin of a limit cycle (p. 66)
(2) Christoph SpandlTrue orbit simulation of dynamical systems and its computationalcomplexity (p. 153)
(3) M. Konecny, W. Taha, J. Duracz and A. FarjudianImplementing the Interval Picard Operator (p. 87)
(4) Luc Jaulin, Jordan Ninin, Gilles Chabert, Stephane Le Menec,Mohamed Saad, Vincent Le Doze and Alexandru StancuComputing capture tubes (p. 72)
19:00 – 21:00 Reception (Town Hall Wurzburg)
7
Tuesday, September 23, 2014
9:00 – 10:00 Plenary Talk (Turing, Chair: U. Kulisch)
John GustafsonAn Energy-E�cient and Massively Parallel Approach to Valid Nu-merics (p. 62)
10:00 – 10:30 Co↵ee Break
10:30 – 11:45 Parallel Sessions
Session B1: Linear Systems (Turing, Chair: G. Mayer)
(1) Evgenija D. PopovaImproved Enclosure for Parametric Solution Sets with Linear Shape(p. 134)
(2) Irene A. Sharaya and Sergey P. SharyReserve as recognizing functional for AE-solutions to interval sys-tem of linear equations (p. 145)
(3) Katsuhisa Ozaki, Takeshi Ogita and Shin’ichi OishiAutomatic Verified Numerical Computations for Linear Systems(p. 129)
Session B2: Matrix Arithmetic (Zuse, Chair: L. Jaulin)
(1) Shinya MiyajimaFast inclusion for the matrix inverse square root (p. 111)
(2) Stef Graillat, Christoph Lauter, Ping Tak Peter Tang, Naoya Ya-manka and Shin’ichi OishiA method of calculating faithful rounding of l
2
-norm for n-vectors(p. 60)
(3) Lars BalzerSONIC – a nonlinear solver (p. 35)
Session B3: Dynamic Systems (Moore, Chair: W. Tucker)
(1) Balazs Banhelyi, Tibor Csendes, Tibor Krisztin and Arnold Neu-maierOn a conjecture of Wright (p. 31)
8
(2) S. I. KumkovApplied techniques of interval analysis for estimation of experimen-tal data (p. 90)
(3) Kenta Kobayashi and Takuya TsuchiyaError Estimations of Interpolations on Triangular Elements (p. 85)
11:45 – 13:00 Lunch
13:00 – 14:00 Plenary Talk (Turing, Chair: S. Shary)
Andrej BauerProgramming techniques for exact real arithmetic (p. 37)
14:00 – 15:15 Parallel Sessions
Session C1: Elliptic PDE (Turing, Chair: M. Nakao)
(1) Tomoki UdaNumerical Verification for Elliptic Boundary Value Problem withNonconforming P1 Finite Elements (p. 159)
(2) Henning BehnkeCurve Veering for the Parameter-Dependent Clamped Plate (p. 38)
(3) Takehiko Kinoshita, Yoshitaka Watanabe and Mitsuhiro T. NakaoSome remarks on the rigorous estimation of inverse linear ellipticoperators (p. 81)
Session C2: Modeling and Uncertainty (Zuse, Chair: C. Spandl)
(1) Rene Alt, Svetoslav Markov, Margarita Kambourova, Nadja Rad-chenkova and Spasen VassilevOn the mathematical modelling of a batch fermentation process us-ing interval data and verification methods (p. 26)
(2) Andreas Rauh, Ramona Westphal, Harald Aschemann and Ekate-rina AuerExponential Enclosure Techniques for Initial Value Problems withMultiple Conjugate Complex Eigenvalues (p. 141)
(3) Andreas Rauh, Luise Senkel and Harald AschemannComputation of Confidence Regions in Reliable, Variable-StructureState and Parameter Estimation (p. 139)
9
Session C3: Global Optimization (Moore, Chair: M. Konecny)
(1) Charlie Vanaret, Jean-Baptiste Gotteland, Nicolas Durand andJean-Marc AlliotCombining Interval Methods with Evolutionary Algorithms for GlobalOptimization (p. 162)
(2) Yao Zhao, Gang Xu and Mark StadtherrDynamic Load Balancing for Rigorous Global Optimization of Dy-namic Systems (p. 164)
(3) Ralph Baker KearfottSome Observations on Exclusion Regions in Interval Branch andBound Algorithms (p. 78)
15:15 – 15:45 Co↵ee Break
15:45 – 16:35 Parallel Sessions
Session D1: Interval Matrices (Turing, Chair: J. Horacek)
(1) Mihaly Csaba Markot and Zoltan HorvathFinding positively invariant sets of ordinary di↵erential equationsusing interval global optimization methods (p. 105)
(2) Jurgen Garlo↵ and Mohammad AdmSign Regular Matrices Having the Interval Property (p. 53)
Session D2: Non Linear Systems (Zuse, Chair: N. Yamamoto)
(1) Makoto Mizuguchi, Akitoshi Takayasu, Takayuki Kubo and Shin’ichiOishiA sharper error estimate of verified computations for nonlinear heatequations. (p. 119)
(2) Makoto Mizuguchi, Akitoshi Takayasu, Takayuki Kubo and Shin’ichiOishiA method of verified computations for nonlinear parabolic equations(p. 117)
10
Session D3: Tools and Workflows (Moore, Chair: E. Popova)
(1) Pacome Eberhart, Julien Brajard, Pierre Fortin and Fabienne JezequelTowards High Performance Stochastic Arithmetic (p. 47)
(2) Wolfram LutherA workflow for modeling, visualizing, and querying uncertain (GPS-)localization using interval arithmetic (p. 100)
16:45 – 17:30 Meeting of the Scientific Committee and EditorialBoard (Moore)
19:30 – 23:00 Conference Dinner (Festung Marienberg)
Wednesday, September 24, 2014
9:00 – 10:00 Plenary Talk (Turing, Chair: G. Alefeld)
Jack DongarraAlgorithmic and Software Challenges at Extreme Scales (p. 46)
10:00 – 10:30 Co↵ee Break
10:30 – 11:45 Parallel Sessions
Session E1: HPC (Turing, Chair: G. Bohlender)
(1) Sylvain Collange, David Defour, Stef Graillat and Roman Iakym-chukReproducible and Accurate Matrix Multiplication for High-PerformanceComputing (p. 42)
(2) Chemseddine Chohra, Philippe Langlois and David ParelloLevel 1 Parallel RTN-BLAS: Implementation and E�ciency Anal-ysis (p. 40)
(3) Hao Jiang, Feng Wang, Yunfei Du and Lin PengFast Implementation of Quad-Precision GEMM on ARMv8 64-bitMulti-Core Processor (p. 76)
11
Session E2: Parametric Linear Systems (Zuse, Chair: K. Ozaki)
(1) Milan HladıkOptimal preconditioning for the interval parametric Gauss–Seidelmethod (p. 68)
(2) Atsushi Minamihata, Kouta Sekine, Takeshi Ogita, Siegfried M.Rump and Shin’ichi OishiA Simple Modified Verification Method for Linear Systems (p. 109)
(3) Andreas Rauh, Luise Senkel and Harald AschemannVerified Parameter Identification for Dynamic Systems with Non-Smooth Right-Hand Sides (p. 137)
Session E3: Uncertainty (Moore, Chair: N. Louvet)
(1) Igor SokolovNon-arithmetic approach to dealing with uncertainty in fuzzy arith-metic (p. 151)
(2) Joe Lorkowski and Vladik KreinovichHow much for an interval? a set? a twin set? a p-box? a Kaucherinterval? An economics-motivated approach to decision making un-der uncertainty (p. 98)
(3) Boris S. Dobronets and Olga A. PopovaNumerical probabilistic approach for optimization problems (p. 44)
11:45 – 13:00 Lunch
13:00 – 22:00 Excursion to Bamberg
Thursday, September 25, 2014
9:00 – 10:00 Plenary Talk (Turing, Chair: N. Revol)
Sylvie BoldoFormal verification of tricky numerical computations (p. 39)
10:00 – 10:30 Co↵ee Break
12
10:30 – 11:45 Parallel Sessions
Session F1: Miscellaneous (Turing, Chair: B. Banhelyi)
(1) Roumen Anguelov and Svetoslav MarkovOn the sets of H- and D-continuous interval functions (p. 28)
(2) Luise Senkel, Andreas Rauh and Harald AschemannNumerical Validation of Sliding Mode Approaches with Uncertainty(p. 143)
(3) Amin Maher and Hossam A. H. FahmyUsing range arithmetic in evaluation of compact models (p. 103)
Session F2: Floating Point Operations (Zuse, Chair: P. Langlois)
(1) Hong Diep Nguyen and James DemmelToward hardware support for Reproducible Floating-Point Compu-tation (p. 123)
(2) Claude-Pierre Jeannerod and Siegfried M. RumpOn relative errors of floating-point operations: optimal bounds andapplications (p. 75)
(3) Stefan SiegelAn Implementation of Complete Arithmetic (p. 149)
Session F3: Solvability and Singularity (Moore, Chair: T. Ogita)
(1) Jaroslav Horacek and Milan HladıkOn Unsolvability of Overdetermined Interval Linear Systems (p. 70)
(2) Luc Longpre and Vladik KreinovichTowards the possibility of objective interval uncertainty in physics.II (p. 96)
(3) David Hartman and Milan HladıkTowards tight bounds on the radius of nonsingularity (p. 64)
11:45 – 13:00 Lunch
13:00 – 14:00 Plenary Talk (Turing, Chair: T. Csendes)
Bart lomiej Jacek KubicaInterval methods for solving various kinds of quantified nonlinearproblems (p. 89)
13
14:00 – 15:15 Parallel Sessions
Session G1: Verification Methods (Turing, Chair: H. Behnke)
(1) Balazs Banhelyi and Balazs Laszlo LevaiVerified localization of trajectories with prescribed behaviour in theforced damped pendulum (p. 33)
(2) Xuefeng Liu and Shin’ichi OishiVerified lower eigenvalue bounds for self-adjoint di↵erential opera-tors (p. 94)
(3) Kazuaki Tanaka and Shin’ichi OishiNumerical verification for periodic stationary solutions to the Allen-Cahn equation (p. 155)
Session G2: Bernstein Branch and Bound (Zuse, Chair: M. Stadtherr)
(1) Jurgen Garlo↵ and Tareq HamadnehConvergence of the Rational Bernstein Form (p. 56)
(2) Bhagyesh V. Patil and P. S. V. NatarajBernstein branch-and-bound algorithm for unconstrained global op-timization of multivariate polynomial MINLPs (p. 131)
Session G3: Stochastic Intervals (Moore, Chair: F. Jezequel)
(1) Ronald van Nooijen and Alla KolechkinaTwo applications of interval analysis to parameter estimation inhydrology. (p. 161)
(2) Tiago Montanher and Walter MascarenhasAn Interval arithmetic algorithm for global estimation of hiddenMarkov model parameters (p. 121)
(3) Valentin GolodovInterval regularization approach to the Firordt method of the spec-troscopic analysis of the nonseparated mixtures (p. 58)
15:20 – 15:35 Closing Session (Turing, Chair: J. Wol↵ von Guden-berg)
15:40 – 16:10 Co↵ee Break
19:00 – 22:30 Wine Tasting Party (Staatlicher Hofkeller)
14
Friday, September 26, 2014
10:00 – 13:00 IEEE P1788 Meeting (Moore)
15
Contents
On the mathematical modelling of a batch fermentation process us-ing interval data and verification methods 26
Rene Alt, Svetoslav Markov, Margarita Kambourova, Nadja Rad-chenkova and Spasen Vassilev
On the sets of H- and D-continuous interval functions 28
Roumen Anguelov and Svetoslav Markov
Result Verification and Uncertainty Management in EngineeringApplications 30
Ekaterina Auer
On a conjecture of Wright 31
Balazs Banhelyi, Tibor Csendes, Tibor Krisztin and Arnold Neu-maier
Verified localization of trajectories with prescribed behaviour in theforced damped pendulum 33
Balazs Banhelyi and Balazs Laszlo Levai
SONIC – a nonlinear solver 35
Lars Balzer
Programming techniques for exact real arithmetic 37
Andrej Bauer
Curve Veering for the Parameter-Dependent Clamped Plate 38
Henning Behnke
Formal verification of tricky numerical computations 39
Sylvie Boldo
16
Level 1 Parallel RTN-BLAS: Implementation and E�ciency Anal-ysis 40
Chemseddine Chohra, Philippe Langlois and David Parello
Reproducible and Accurate Matrix Multiplication for High-PerformanceComputing 42
Sylvain Collange, David Defour, Stef Graillat and Roman Iakym-chuk
Numerical probabilistic approach for optimization problems 44Boris S. Dobronets and Olga A. Popova
Algorithmic and Software Challenges at Extreme Scales 46Jack Dongarra
Towards High Performance Stochastic Arithmetic 47Pacome Eberhart, Julien Brajard, Pierre Fortin and FabienneJezequel
Modal Interval Floating Point Unit with Decorations 49Abdelrahman Elskhawy, Kareem Ismail and Maha Zohdy
Sign Regular Matrices Having the Interval Property 53Jurgen Garlo↵ and Mohammad Adm
Convergence of the Rational Bernstein Form 56Jurgen Garlo↵ and Tareq Hamadneh
Interval regularization approach to the Firordt method of the spec-troscopic analysis of the nonseparated mixtures 58
Valentin Golodov
A method of calculating faithful rounding of l2
-norm for n-vectors 60Stef Graillat, Christoph Lauter, Ping Tak Peter Tang, NaoyaYamanka and Shin’ichi Oishi
17
An Energy-E�cient and Massively Parallel Approach to Valid Nu-merics 62
John Gustafson
Towards tight bounds on the radius of nonsingularity 64David Hartman and Milan Hladık
A numerical verification method for a basin of a limit cycle 66Tomohirio Hiwaki and Nobito Yamamoto
Optimal preconditioning for the interval parametric Gauss–Seidelmethod 68
Milan Hladık
On Unsolvability of Overdetermined Interval Linear Systems 70Jaroslav Horacek and Milan Hladık
Computing capture tubes 72Luc Jaulin, Jordan Ninin, Gilles Chabert, Stephane Le Menec,Mohamed Saad, Vincent Le Doze and Alexandru Stancu
On relative errors of floating-point operations: optimal bounds andapplications 75
Claude-Pierre Jeannerod and Siegfried M. Rump
Fast Implementation of Quad-Precision GEMM on ARMv8 64-bitMulti-Core Processor 76
Hao Jiang, Feng Wang, Yunfei Du and Lin Peng
Some Observations on Exclusion Regions in Interval Branch andBound Algorithms 78
Ralph Baker Kearfott
Some remarks on the rigorous estimation of inverse linear ellipticoperators 81
Takehiko Kinoshita, YoshitakaWatanabe and Mitsuhiro T. Nakao
18
Computer-Assisted Uniqueness Proof for Stokes’ Wave of ExtremeForm 83
Kenta Kobayashi
Error Estimations of Interpolations on Triangular Elements 85Kenta Kobayashi and Takuya Tsuchiya
Implementing the Interval Picard Operator 87M. Konecny, W. Taha, J. Duracz and A. Farjudian
Interval methods for solving various kinds of quantified nonlinearproblems 89
Bart lomiej Jacek Kubica
Applied techniques of interval analysis for estimation of experimen-tal data 90
S. I. Kumkov
Replacing branches by polynomials in vectorizable elementary func-tions 92
Olga Kupriianova and Christoph Lauter
Verified lower eigenvalue bounds for self-adjoint di↵erential opera-tors 94
Xuefeng Liu and Shin’ichi Oishi
Towards the possibility of objective interval uncertainty in physics.II 96
Luc Longpre and Vladik Kreinovich
How much for an interval? a set? a twin set? a p-box? a Kaucherinterval? An economics-motivated approach to decision making un-der uncertainty 98
Joe Lorkowski and Vladik Kreinovich
19
A workflow for modeling, visualizing, and querying uncertain (GPS-)localization using interval arithmetic 100
Wolfram Luther
Using range arithmetic in evaluation of compact models 103Amin Maher and Hossam A. H. Fahmy
Finding positively invariant sets of ordinary di↵erential equationsusing interval global optimization methods 105
Mihaly Csaba Markot and Zoltan Horvath
A short description of the symmetric solution set 107Gunter Mayer
A Simple Modified Verification Method for Linear Systems 109Atsushi Minamihata, Kouta Sekine, Takeshi Ogita, Siegfried M.Rump and Shin’ichi Oishi
Fast inclusion for the matrix inverse square root 111Shinya Miyajima
Verified solutions of saddle point linear systems 114Shinya Miyajima
A method of verified computations for nonlinear parabolic equations117Makoto Mizuguchi, Akitoshi Takayasu, Takayuki Kubo and Shin’ichiOishi
A sharper error estimate of verified computations for nonlinear heatequations. 119
Makoto Mizuguchi, Akitoshi Takayasu, Takayuki Kubo and Shin’ichiOishi
An Interval arithmetic algorithm for global estimation of hiddenMarkov model parameters 121
Tiago Montanher and Walter Mascarenhas
20
Toward hardware support for Reproducible Floating-Point Compu-tation 123
Hong Diep Nguyen and James Demmel
Accurate and e�cient implementation of a�ne arithmetic usingfloating-point arithmetic 125
Jordan Ninin and Nathalie Revol
Iterative Refinement for Symmetric Eigenvalue Problems 127
Takeshi Ogita
Automatic Verified Numerical Computations for Linear Systems 129
Katsuhisa Ozaki, Takeshi Ogita and Shin’ichi Oishi
Bernstein branch-and-bound algorithm for unconstrained global op-timization of multivariate polynomial MINLPs 131
Bhagyesh V. Patil and P. S. V. Nataraj
Improved Enclosure for Parametric Solution Sets with Linear Shape134
Evgenija D. Popova
The architecture of the IEEE P1788 draft standard for intervalarithmetic 135
John Pryce
Verified Parameter Identification for Dynamic Systems with Non-Smooth Right-Hand Sides 137
Andreas Rauh, Luise Senkel and Harald Aschemann
Computation of Confidence Regions in Reliable, Variable-StructureState and Parameter Estimation 139
Andreas Rauh, Luise Senkel and Harald Aschemann
21
Exponential Enclosure Techniques for Initial Value Problems withMultiple Conjugate Complex Eigenvalues 141
Andreas Rauh, Ramona Westphal, Harald Aschemann and Eka-terina Auer
Numerical Validation of Sliding Mode Approaches with Uncertainty143
Luise Senkel, Andreas Rauh and Harald Aschemann
Reserve as recognizing functional for AE-solutions to interval sys-tem of linear equations 145
Irene A. Sharaya and Sergey P. Shary
Maximum Consistency Method for Data Fitting under Interval Un-certainty 147
Sergey P. Shary
An Implementation of Complete Arithmetic 149
Stefan Siegel
Non-arithmetic approach to dealing with uncertainty in fuzzy arith-metic 151
Igor Sokolov
True orbit simulation of dynamical systems and its computationalcomplexity 153
Christoph Spandl
Numerical verification for periodic stationary solutions to the Allen-Cahn equation 155
Kazuaki Tanaka and Shin’ichi Oishi
Choice of metrics in interval arithmetic 157
Philippe Theveny
22
Numerical Verification for Elliptic Boundary Value Problem withNonconforming P1 Finite Elements 159
Tomoki Uda
Two applications of interval analysis to parameter estimation inhydrology. 161
Ronald van Nooijen and Alla Kolechkina
Combining Interval Methods with Evolutionary Algorithms for GlobalOptimization 162
Charlie Vanaret, Jean-Baptiste Gotteland, Nicolas Durand andJean-Marc Alliot
Dynamic Load Balancing for Rigorous Global Optimization of Dy-namic Systems 164
Yao Zhao, Gang Xu and Mark Stadtherr
23
24
Abstracts
25
On the mathematical modelling of abatch fermentation process using interval
data and verification methods
Rene Alt1, Svetoslav Markov2, Margarita Kambourova3,Nadja Radchenkova3 and Spasen Vassilev3
1 Sorbonne Universites, LIP6, UPMC, CNRS UMR76062 Institute of Mathematics and Informatics, Bulgarian Academy of
Sciences3 Institute of Microbiology, Bulgarian Academy of Sciences1 Boite courrier 169 4 place Jussieu 75252 Paris Cedex 0,
2 “Akad. G. Bonchev” st., bl. 8, 1113 Sofia, Bulgaria3 “Akad. G. Bonchev” st., bl. 26, 1113 Sofia, Bulgaria
Keywords: batch fermentation processes, reaction schemes, dynamicmodels, numerical simulations, verification methods
An experiment in a batch laboratory bioreactor for the of EPS pro-duction by Aeribacillus pallidus 418 bacteria is performed and intervalexperimental data for the biomass-product dynamics are obtained [1].The dynamics of microbial growth and product synthesis is describedby means of several bio-chemical reaction schemes, aiming an under-standing of the underlying biochemical/metabolic mechanisms [2,3].The proposed reaction schemes lead to systems of ordinary di↵erentialequations whose solutions are fitted to the observed interval experi-mental data. Suitable parameter identification of the mathematicalmodels is performed aiming that the numerically computed results areincluded into the interval experimental data following a verificationapproach [4,5]. The proposed models reflect specific features of themechanism of the fermentation process, which may suggest further ex-perimental and theoretical work. We believe that using the proposedmethodology one can study the basic mechanisms underlying the dy-namics of cell growth, substrate uptake and product synthesis.
26
References:
[1] Radchenkova, N., M. Kambourova, S. Vassilev, R. Alt,S. Markov, On the mathematical modelling of EPS productionby a thermophilic bacterium, submitted to BIOMATH.
[2] Alt, R., S. Markov, Theoretical and computational studies ofsome bioreactor models,Computers and Mathematics with Appli-cations 64 (2012), 350–360.http://dx.doi.org/10.1016/J.Camwa.2012.02.046
[3] Markov, S., Cell Growth Models Using Reaction Schemes: BatchCultivation, Biomath 2/2 (2013), 1312301.http://dx.doi.org/10.11145/j.biomath.2013.12.301
[4] Wolff v. Gudenberg, J., Proceedings of the Conference Interval-96, Reliable Computing 3 (3).
[5] Kramer, W., J. Wolff v. Gudenberg, Scientific Computing,Validated Numerics, Interval Methods, Proceedings of the confer-ence Scan-2000/Interval-2000, Kluwer/Plenum, 2001.
27
On the sets of H- and D-continuousinterval functions
Roumen Anguelov1 and Svetoslav Markov21 Department of Mathematics and Applied Mathematics, University
of Pretoria, Pretoria 0002, South Africa2 Institute of Mathematics and Informatics, Bulgarian Academy of
Sciences, “Akad. G. Bonchev” st., bl. 8, 1113 Sofia, [email protected]
Keywords: real functions, interval functions, H-continuous functions,D-continuous functions, tight enclosures.
It has been shown that the space of Hausdor↵ continuous (H-continuous) functions is the largest linear space of interval functions[1]. This space has important applications in the Theory of PDE’s andReal Analysis [2]. Moreover, the space of H-continuous functions hasa very special place in Interval Analysis as well, more specifically inthe Analysis of Interval-valued Functions. It has been also shown thatthe practically relevant set, in terms of providing tight enclosures ofsets of real functions, is the set of the so-called Dilworth continuous (D-continuous) interval-valued functions [1]. Here we apply the concept ofquasivector space as defined in [3] which captures and preserves the es-sential properties of computations with interval-valued functions whilealso providing a relatively simple structure for computing. Indeed, aquasivector space is a direct sum of a vector (linear) space and a sym-metric quasivector space which makes the computations essentially aseasy as computations in a linear space. In the considered setting weprove that, the space of H-continuous functions is precisely the linearspace in the direct sum decomposition of the respective quasivectorspace.
References:
[1] R. Anguelov, S. Markov, B. Sendov, The set of Hausdor↵continuous functions — the largest linear space of interval func-
28
tions, Reliable Computing 12, 337–363 (2006).http://dx.doi.org/10.1007/s11155-006-9006-5
[2] J. H. van der Walt, The Linear Space of Hausdor↵ ContinuousInterval Functions, Biomath 2 (2013), 1311261.http://dx.doi.org/10.11145/j.biomath.2013.11.261
[3] S. Markov, On Quasilinear Spaces of Convex Bodies and Inter-vals, Journal of Computational and Applied Mathematics 162 (1),93–112, 2004.http://dx.doi.org/10.1016/j.cam.2003.08.016
29
Result Verification and UncertaintyManagement in Engineering Applications
Ekaterina Auer
University of Duisburg-Essen47048 Duisburg, [email protected]
Verified methods can have di↵erent uses in engineering applications.On the one hand, they are able to demonstrate the correctness of resultsobtained on a computer using a certain model of the considered systemor process. On the other hand, we can represent bounded parameteruncertainty in a natural way with their help. This allows us to studymodels and make statements about them over whole parameter rangesinstead of their (possibly incidental) point values. However, whole pro-cesses cannot be feasibly verified in all cases, the reasons ranging frominherent di�culties (e.g., for chaotic models) to problems caused bydependency and wrapping to drawbacks arising simply from choosingthe wrong verified technique. In this talk, we give a detailed overviewof how verified methods – sometimes in combinations with other tech-niques – improve the quality of simulations in engineering. We startby providing a general view on the role of verified methods in theverification/validation systematics and the modeling and simulationcycle for a given process. After that, we point out what concepts andtools are necessary to successfully apply verified methods, includingnot yet (fully) implemented ones that would be nonetheless advanta-geous. Finally, we consider several applications from (bio)mechanicsand systems control which exemplify the general approach and focuson di↵erent usage areas for verified methods.
Invited talk
30
On a conjecture of Wright
Balazs Banhelyi, Tibor Csendes, Tibor Krisztin, and ArnoldNeumaier
University of Szeged6720 Szeged, Hungary
Keywords: delayed logistic equation, Wright’s equation, Wright’sconjecture, slowly oscillating periodic solution, discrete Lyapunov func-tional, Poincare–Bendixson theorem, verified computational techniques,computer-assisted proof, interval arithmetic
In 1955 E.M. Wright proved that all solutions of the delay di↵er-ential equation x(t) = �↵
�ex(t�1) � 1
�converge to zero as t ! 1 for
↵ 2 (0, 3/2], and conjectured that this is even true for ↵ 2 (0, ⇡/2).The present paper proves the conjecture for ↵ 2 [1.5, 1.5706] (comparewith ⇡/2 = 1.570796...). The proof is based on the proven fact thatit is su�cient to guarantee the nonexistence of slowly oscillating peri-odic solutions, and that slowly oscillating periodic solutions with smallamplitudes cannot exist. The talk will give details on the a computer-assisted proof part that exclude slowly oscillating periodic solutionswith large amplitudes.
References:
[1] B. Banhelyi, Discussion of a delayed di↵erential equation withverified computing technique (in Hungarian), Alkalmazott Matem-atikai Lapok 24 (2007) 131–150.
[2] B. Banhelyi, T. Csendes, T. Krisztin, and A. Neumaier,Global attractivity of the zero solution for Wright’s equation. Ac-cepted for publication in the SIAM J. on Applied Dynamical Sys-tems.
31
[3] T. Csendes, B.M. Garay, and B. Banhelyi, A verified opti-mization technique to locate chaotic regions of a Henon system, J.of Global Optimization 35 (2006) 145–160.
[4] E.M. Wright, A non-linear di↵erence-di↵erential equation, J. furdie Reine und Angewandte Mathematik 194 (1955) 66–87.
32
Verified localization of trajectories withprescribed behaviour in the forced
damped pendulum
Balazs Banhelyi and Balazs Laszlo Levai
University of Szeged6720 Szeged, Hungary
Keywords: localization, forced damped pendulum, chaos
In mathematics, it is quite di�cult to define exactly what chaosreally means. In particular, it is easier to prepare a list of propertieswhich describe a so called chaotic system than give a precise definition.A dynamic system is generally classified as chaotic if it is sensitive toits initial conditions. Chaos can be also characterized by dense periodicorbits and topological transitivity.
While studying computational approximations of solutions of dif-ferential equations, it is an important question is whether the givenequation has chaotic solutions. The nature of chaos implies that thenumerical simulation must be carried out carefully, considering fittingmeasures against possible distraction due to accumulated rounding er-rors. Unfortunately except a few cases, the recognition of chaos hasremained a hard task that is usually handled by theoretical means[Hubbard1988].
In our present studies, we investigate a simple mechanical system,Hubbard’s sinusoidally forced damped pendulum [Hubbard1988]. Ap-plying rigorous computations, his 1999 conjecture on the existence ofchaos was proved in Banhelyi et al. [Banhelyi2008] in 2008 but theproblem of finding chaotic trajectories remained entirely open. Thistime, we present a fitting verified numerical technique capable to lo-cate finite trajectory segments theoretically with arbitrary prescribedqualitative behaviour and thus shadowing di↵erent types of chaotictrajectories with large precision. For example, we can achieve that our
33
pendulum goes through any specified finite sequence of gyrations bychoosing the initial conditions correctly.
To be able to provide solutions with mathematical precision, thecomputation of trajectories has to be executed rigorously. Keepingin mind this intention, we calculated the inclusion of a solution ofthe di↵erential equation with the VNODE algorithm [Nedialkov2001]and based on the PROFIL/BIAS interval environment [Knuppel1993].The search for a solution point is a global optimization problem towhich we applied the C version of the GLOBAL algorithm, a clusteringstochastic global optimization technique [Csendes1988]. This methodis capable to find the global optimizer points of moderate dimensionalglobal optimization problems, when the relative size of the region ofattraction of the global minimizer points are not very small.
References:
[1] Banhelyi, B., T. Csendes, B.M. Garay, and L. Hatvani,A computer–assisted proof for ⌃
3
–chaos in the forced damped pen-dulum equation, SIAM J. Appl. Dyn. Syst., 7, 843–867 (2008).
[2] Csendes, T., Nonlinear parameter estimation by global optimiza-tion – e�ciency and reliability, Acta Cybernetica, 8, 361-370 (1988).
[3] Hubbard, J.H., The forced damped pendulum: chaos, complica-tion and control, Amer. Math. Monthly, 8, 741–758 (1999).
[4] Knuppel, O., PROFIL – Programmer’s Runtime Optimized FastInterval Library, Bericht 93.4., Technische Universitat Hamburg-Harburg (1993).
[5] Nedialkov, N.S., VNODE – A validated solver for initial valueproblems for ordinary di↵erential equations, Available atwww.cas.mcmaster.ca/⇠nedialk/Software/VNODE/VNODE.shtml(2001).
34
SONIC – a nonlinear solver
Lars BalzerUniversity of Wuppertal
42097 Wuppertal, [email protected]
Keywords: verified computing, nonlinear systems, SONIC, C-XSC,filib++
This talk presents the program SONIC - a Solver and Optimizerfor Nonlinear Problems based on Interval Computation. It solves non-linear systems of equations and yields a list of boxes containing allsolutions within an initial box. The solver is written in C++ and useseither C-XSC or filib++ as an interval library. Parallelization of thecode is possible by the usage of MPI or OpenMP. Members of theApplied Computer Science Group of the University of Wuppertal areworking on the development of the solver.
SONIC uses a branch-and-bound approach to find and discard sub-boxes of the initial starting box that don’t contain a solution of thenonlinear system. To speed up the algorithm the branch-and-boundmethod is combined with further components. The goal is to reducethe computation time and the number of boxes that have to be con-sidered.
There is the constraint propagation whose idea is to spread theknown enclosures of the variables over the term net that represents thenonlinear system. After several sweeps over the term net the consideredbox is contracted.
Another key element of the algorithm is the interval Newton methodwith Gauss-Seidel iteration complemented by di↵erent choices for apreconditioning matrix.
The solver generates a list of small boxes that cover all solutionsof the system. To verify the existence of a solution in these boxes afinal verification step is applied. For this task SONIC has implementedseveral verification methods.
35
References:
[1] T. Beelitz, C.H. Bischof, B. Lang, P. Willems, SONIC – Aframework for the rigorous solution of nonlinear problems, reportBUW-SC 04/7, University of Wuppertal, 2004.
[2] W. Hofschuster, W. Kramer, C-XSC 2.0: A C++ Library forExtended Scientific Computing, Numerical Software with ResultVerification, Volume 2991/2004, Springer-Verlag, Heidelberg, pp.15 - 35, 2004.
[3] M. Lerch et al., filib++, a Fast Interval Library SupportingContainment Computations, ACM TOMS, volume 32, number 2,pp. 299-324, 2006.
[4] L. Balzer, Untersuchung des Einsatzes von Taylor-Modellen beider Losung nichtlinearer Gleichungssysteme, University of Wup-pertal, Master-Thesis, 2013.
36
Programming techniques for exact realarithmetic
Andrej Bauer
Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljana, Slovenia
There are several strategies for implementing exact computationwith real numbers. Two common ones are based on interval arithmeticwith forward or backward propagation of errors. A less common wayof computing with exact reals is to use Dedekind’s construction of realsas cuts. In such a setup a real number is defined by two predicatesthat describe its lower and upper bounds. We can extract e�cientevaluation strategies from such declarative descriptions by using aninterval Newton’s method. From the point of view of programminglanguage design it is desirable to express the mathematical content ofa computation in a direct and abstract way, while still retaining flexi-bility and control over evaluation strategy. We shall discuss how sucha goal might be achieved using techniques that modern programminglanguages have to o↵er.
Invited talk
37
Curve Veering for theParameter-Dependent Clamped Plate
Henning Behnke
TU Clausthal38678 Clausthal-Zellerfeld, [email protected]
Keywords: partial di↵erential equations, eigenvalue problem, eigen-value enclosures
The computation of vibrations of a thin rectangular clamped plateresults in an eigenvalue problem with a partial di↵erential equation offourth order.
@4
@ x4'+ P
@4
@ x2@ y2'+Q
@4
@ y4' = �' in ⌦,
' = 0 and@'
@n= 0 on @⌦,
for P ,Q 2 R, P > 0, Q > 0, and ⌦ = (�a2
, a2
)⇥ (� b2
, b2
) ✓ R2.If we change the geometry of the plate for fixed area, this results in a
parameter-dependent eigenvalue problem. For certain parameters, theeigenvalue curves seem to cross. We give a numerically rigorous proofof curve veering, which is based on the Lehmann-Goerisch inclusiontheorems and the Rayleigh-Ritz procedure.
References:
[1] H. Behnke, A Numerically Rigorous Proof of Curve Veering in anEigenvalue Problem for Di↵erential Equations, Z. Anal. Anwen-dungen, (1996), No. 15, pp. 181–200.
38
Formal verification of tricky numericalcomputations
Sylvie Boldo
InriaLRI, CNRS UMR 8623, Universite Paris-Sud
PCRI - Batiment 650Universite Paris-Sud91405 ORSAY Cedex
Keywords: Floating-point, formal proof, deductive verification
Computer arithmetic has applied formal methods and formal proofsfor years. As the systems may be critical and as the properties maybe complex to prove (many sub-cases, error-prone computations), aformal guarantee of correctness is a wish that can now be fulfilled.
This talk will present a chain of tools to formally verify numericalprograms. The idea is to precisely specify what the program requiresand ensures. Then, using deductive verification, the tools produceproof obligation that may be proved either automatically or interac-tively in order to guarantee the correctness of the specifications. Manyexamples of programs from the literature will be specified and formallyverified.
Invited talk
39
Level 1 Parallel RTN-BLAS:Implementation and E�ciency Analysis
Chemseddine Chohra,Philippe Langlois and David Parello
Univ. Perpignan Via Domitia, Digits, Architectures et LogicielsInformatiques, F-66860, Perpignan. Univ. Montpellier II, Laboratoired’Informatique Robotique et de Microelectronique de Montpellier,
UMR 5506, F-34095, Montpellier. CNRS, Laboratoire d’InformatiqueRobotique et de Microelectronique de Montpellier, UMR 5506,
F-34095, [email protected]
Keywords: Floating point arithmetic, numerical reproducibility,Round-To-Nearest BLAS, parallelism, summation algorithms.
Modern high performance computation (HPC) performs a hugeamount of floating point operations on massively multi-threaded sys-tems. Those systems interleave operations and include both dynamicscheduling and non-deterministic reductions that prevent numerical re-producibility, i.e. getting identical results from multiple runs, even onone given machine. Floating point addition is non-associative and theresults depend on the computation order. Of course, numerical repro-ducibility is important to debug, check the correctness of programs andvalidate the results. Some solutions have been proposed like paralleltree scheme [1] or new Demmel and Nguyen’s reproducible sums [2].Reproducibility is not equivalent to accuracy: a reproducible resultmay be far away from the exact result. Another way to guarantee thenumerical reproducibility is to calculate the correctly rounded value ofthe exact result, i.e. extending the IEEE-754 rounding properties tolarger computing sequences. When such computation is possible, it iscertainly more costly. But is it unacceptable in practice?We are motivated by round-to-nearest parallel BLAS. We can imple-ment such RTN-BLAS thanks to recent algorithms that compute cor-rectly rounded sums. This work is a first step for the level 1 of the
40
BLAS routines. We study the e�ciency of computing parallel RTN-sums compared to reproducible or classic ones – MKL for instance.We focus on HybridSum and OnlineExact, two algorithms that smooththe over-cost e↵ect of the condition number for large sums [3,4]. Westart with sequential implementations: we describe and analyze somehand-made optimizations to benefit from instruction level parallelism,pipelining and to reduce the memory latency. The optimized over-costis at least 25% reduced in the sequential case. Then we propose paral-lel RTN versions of these algorithms for shared memory systems. Weanalyze the e�ciency of OpenMP implementations. We exhibit bothgood scaling properties and less memory e↵ect limitations than exist-ing solutions. These preliminary results justify to continue towards thenext levels of parallel RTN-BLAS.
References:
[1] O. Villa, D. G. Chavarrıa-Miranda, V. Gurumoorthi,A. Marquez, and S. Krishnamoorthy. E↵ects of floating-point non-associativity on numerical computations on massivelymultithreaded systems. In CUG Proceedings, (2009), pp. 1–11.
[2] James Demmel and Hong Diep Nguyen, Fast ReproducibleFloating-Point Summation. In 21st IEEE Symposium on Com-puter Arithmetic, Austin, TX, USA, April 7-10, (2013), pp. 163—172.
[3] Yong-Kang Zhu and Wayne. B. Hayes, Correct roundingand a hybrid approach to exact floating-point summation. SIAMJ. Sci. Comput., (2009), Vol. 31, No. 4, pp. 2981–3001.
[4] Yong-Kang Zhu and Wayne. B. Hayes, Algorithm 908: On-line exact summation of floating-point streams. ACM Trans. Math.Software, (2010), 37:1–37:13.
41
Reproducible and AccurateMatrix Multiplication for
High-Performance Computing
Sylvain Collange, David Defour, Stef Graillat, andRoman Iakymchuk
INRIA – Centre de recherche Rennes – Bretagne AtlantiqueCampus de Beaulieu, F-35042 Rennes Cedex, France
[email protected]–LIRMM, Universite de Perpignan
52 avenue Paul Alduy, F-66860 Perpignan, [email protected]
Sorbonne Universites, UPMC Univ Paris 06, UMR 7606, LIP6F-75005 Paris, France
CNRS, UMR 7606, LIP6, F-75005 Paris, FranceSorbonne Universites, UPMC Univ Paris 06, ICS
F-75005 Paris, France{stef.graillat, roman.iakymchuk}@lip6.fr
Keywords: Matrix multiplication, reproducibility, accuracy, long ac-cumulator, multi-precision, multi- and many-core architectures.
The increasing power of current computers enables one to solvemore and more complex problems. This, therefore, requires to performa high number of floating-point operations, each one leading to a round-o↵ error. Because of round-o↵ error propagation, some problems mustbe solved with a longer floating-point format.
As Exascale computing (1018 operations per second) is likely tobe reached within a decade, getting accurate results in floating-pointarithmetic on such computers will be a challenge. However, anotherchallenge will be the reproducibility of the results – meaning gettinga bitwise identical floating-point result from multiple runs of the samecode – due to non-associativity of floating-point operations and dy-namic scheduling on parallel computers.
42
Reproducibility is becoming so important that Intel proposed a“Conditional Numerical Reproducibility” (CNR) in its MKL (MathKernel Library). However, CNR is slow and does not give any guar-antee concerning the accuracy of the result. Recently, Demmel andNguyen [1] proposed an algorithm for reproducible summation. Eventhough their algorithm is fast, no information is given on the accuracy.
More recently, we introduced [2] an approach to compute determin-istic sums of floating-point numbers e�ciently and with the best pos-sible accuracy. Our multi-level algorithm consists of two main stages:filtering that relies upon fast vectorized floating-point expansions; ac-cumulation which is based on superaccumulators in a high-radix carry-save representation. We presented implementations on recent Inteldesktop and server processors, on Intel Xeon Phi accelerator, and onboth AMD and NVIDIA GPUs. We showed that the numerical repro-ducibility and bit-perfect accuracy can be achieved at no additionalcost for large sums that have dynamic ranges of up to 90 orders ofmagnitude by leveraging arithmetic units that are left underused bystandard reduction algorithms.
In this talk, we will present a reproducible and accurate (roundingto the nearest) algorithm for the product of two floating-point matricesin parallel environments like GPU and Xeon Phi. This algorithm isbased on the DGEMM implementation. We will show that the perfor-mance of our algorithm is comparable with the classic DGEMM.
References:
[1] J. Demmel, H.D. Nguyen, Fast Reproducible Floating-PointSummation, Proceeding of the 21st IEEE Symposium on ComputerArithmetic, Austin, Texas, USA (2013), pp. 163-172.
[2] S. Collange, D. Defour, S. Graillat, R. Iakymchuk, Full-Speed Deterministic Bit-Accurate Parallel Floating-Point Summa-tion on Multi- and Many-Core Architectures, Research Report. HALID: hal-00949355. February 2014.
43
Numerical probabilistic approach foroptimization problems
Boris S. Dobronets and Olga A. PopovaSiberian Federal University79, Svobodny Prospect.
660041 Krasnoyarsk, [email protected], [email protected]
Keywords: Random programming, numerical probabilistic analysis,mathematical programming.
Currently being developed methods and approaches to solving op-timization problems under di↵erent types of uncertainty [1]. In most ofuncertain programming algorithms are used the expectation operatorand are held the averaging procedure. We consider a new approach tooptimization problems with uncertain input data. This approach usesa numerical probabilistic analysis and allows us to construct the jointprobability density function of optimal solutions.
Methods to construct the solution set for the optimization problemwith random input parameters, we determine as the Random Program-ming [2,3].
Let us formulate problem of random programming as follows:
f(x, ⇠) ! min, (1)
gi(x, ⇠) 0, i = 1, ...,m. (2)
where x is the solution vector, ⇠ is random vector of parameters, f(x, ⇠)is objective function, gi(x, ⇠) are constraint functions.
Vector x⇤ is the solution of problem (1)–(2), if
f(x⇤, ⇠) = infU
f(x, ⇠),
whereU = {x|gi(x, ⇠) 0, i = 1, ...,m.}
44
The solution set of (1)–(2) is defined as follows
X = {x|f(x, ⇠) ! min, gi(x, ⇠) 0, i = 1, ...,m, ⇠ 2 ⇠}
Note that x⇤ is random vector. So in contrast to the deterministic prob-lem, for x⇤ is necessary to determine the probability density functionfor each component of x⇤i as the joint probability density function.
Unlike most methods of stochastic programming in Random Pro-gramming we can to construct a joint probability density function Px
of random vector x⇤.To construct the joint probability density function Px for problem
(1)–(2) we use probabilistic extensions and quasi Monte Carlo method[2]. This allows us to construct procedures for solving systems of linearalgebraic equations and nonlinear equations with random coe�cients.
Relying on numerical examples, we showed that the random pro-gramming procedures is e↵ective method for linear and non linear op-timization problems.
References:
[1] B.Liu, Theory and Practice of Uncertain Programming (2nd Edi-tion), Springer-Verlag, Berlin, 2009.
[2] O.A. Popova, Optimization Problems with Random Data Jour-nal of Siberian Federal University. Mathematics & Physics, 6,(2013), No. 4, pp. 506–515
[3] B. Dobronets, O. Popova, Linear optimization problems withrandom data. VII Moscow International Conference on OperationResearch (ORM 2013): Moscow, October 15–19, 2013. Proceed-ings Vol. 1 / Eds. P.S. Krasnoschekov , A.A. Vasin , A.F. Izmailov.— M., MAKS Press, 2013. pp. 15–18
45
Algorithmic and Software Challenges atExtreme Scales
Jack DongarraUniversity of Tennessee, USA
Oak Ridge National Laboratory, USAUniversity of Manchester, UK
In this talk we examine how high performance computing haschanged over the last 10-year and look toward the future in termsof trends. These changes have had and will continue to have a majorimpact on our software. Some of the software and algorithm challengeshave already been encountered, such as management of communicationand memory hierarchies through a combination of compile–time andrun–time techniques, but the increased scale of computation, depth ofmemory hierarchies, range of latencies, and increased run–time envi-ronment variability will make these problems much harder. We willlook at five areas of research that will have an importance impact inthe development of software and algorithms.
We will focus on following themes:
• Redesign of software to fit multicore and hybrid architectures
• Automatically tuned application software
• Exploiting mixed precision for performance
• The importance of fault tolerance
• Communication avoiding algorithms
Invited talk
46
Towards High Performance StochasticArithmetic
Pacome Eberhart1, Julien Brajard2,Pierre Fortin1 and Fabienne Jezequel3
1 Sorbonne Universites, UPMC Univ Paris 06, CNRS, UMR 7606,LIP6, F-75005, Paris, France
2 Sorbonne Universites (UPMC, Univ Paris 06), CNRS, IRD, MNHN,LOCEAN Laboratory, 4 place Jussieu, F-75005, Paris, France
3 Sorbonne Universites, UPMC Univ Paris 06, CNRS, UMR 7606,LIP6, F-75005, Paris, France and Universite Paris 2, France
Keywords: numerical validation, stochastic arithmetic, high perfor-mance computing, SIMD processing
Because of the finite representation of floating-point numbers incomputers, the results of arithmetic operations need to be rounded.The CADNA library [1],based on discrete stochastic arithmetic [2],can be used to estimate the propagation of rounding errors in scientificcodes. By synchronously computing each operation three times witha randomly chosen rounding mode, CADNA estimates the number ofexact significant digits of the result within a 95% confidence interval.To ensure the validity of the method and allow a better analysis of theprogram, several types of anomalies are checked at execution time.
However, the overhead on computation time can be of up to 80times depending on the program and on the level of anomaly detec-tion [3]. There are two main factors that can explain this: the cost ofanomaly detection and that of stochastic operations. Firstly, cancella-tion (sudden loss of accuracy in a single operation) detection is basedon the computation of the number of exact significant digits that re-lies on a logarithmic evaluation. This mathematical function is muchmore costly than floating-point arithmetic operations. Secondly, the
47
stochastic operators are currently implemented through the overload-ing of arithmetic operators and the change of the rounding mode ofthe FPU (Floating Point Unit). However, this method makes vector-ization impossible, as each vector lane would need a di↵erent roundingmode. Moreover, it causes performance overhead due to function callsand to the flushing of the FPU pipelines, respectively. This impliesan even greater performance drop for HPC applications that rely onSIMD (Single Instruction Multiple Data) processing and on pipelinefilling for better e�ciency.
To bypass these overheads and allow the use of vector instructionsfor SIMD parallelism, we propose several improvements in the CADNAlibrary. Since only the integer part of the number of exact significantdigits is required, we can use the exponent of a floating-point valueas an approximation of the logarithm evaluation, which removes thelogarithm function call. To avoid the cost of function calls, we proposeto inline the stochastic operators. Finally, rather than depending onthe rounding modes of the FPU, we compute the randomly roundedarithmetic operations by handling the sign bit of the operands throughmasks. These contributions provide a speedup factor of up to 2.5 ona scalar code. They also enable the use of CADNA with vectorizedcode: SIMD performance results on high-end CPUs and on an IntelXeon Phi are presented.
References:
[1] J. Vignes, Discrete stochastic arithmetic for validating results ofnumerical software Num. Algo., 37(1–4):377–390, Dec. 2004.
[2] Universite Pierre et Marie Curie, Paris, CADNA: Controlof Accuracy and Debugging for Numerical Applications http://www.lip6.fr/cadna
[3] F. Jezequel, J.-L. Lamotte, and O. Chubach., Paralleliza-tion of discrete stochastic arithmetic on multicore architectures2013 Tenth International Conference on Information Technology:New Generations (ITNG), pp. 160–166, Apr. 2013.
48
Modal Interval Floating Point Unit withDecorations
Abdelrahman Elskhawy, Kareem Ismail and Maha Zohdy
Electronics and Communications department,Faculty of Engineering,Cairo University, [email protected]
1 Introduction
Rounding errors in digital computations using floating point numbersmay result in totally inaccurate results. One of the mathematical so-lutions to monitor and control rounding errors is the interval com-putations which was popularized as classical interval arithmetic byRamon E.Moore in 1966 [1]. Results obtained via interval arithmeticoperations are mathematically proven to bind the correct result of thecomputation.
2 Modal intervals
A generalized extension of the classical intervals was presented in 1980which is the modal intervals. Modal Intervals Arithmetic (MIA) hassome advantages over classical intervals as it managed to solve someof the problems in the latter like existence of Additive inverse andMultiplicative inverse, a stronger sub-distributive law, and ability tosolve the interval equations that classical intervals failed to solve andobtaining meaningful interval result when solving these equations [2].This leads to solving serious problems in applications like control andcomputer graphics. Due to the bad performance of software implemen-tation of intervals basic operations, researches were lead to hardwareimplementation of MIA.
49
3 Previous Work
There is only one hardware implementation published for Modal inter-val Adder/Subtractor in [3]. It provides a hardware implementationof the Modal Interval Double Floating Point Adder/Subtractor andMultiplier units. It proposes two di↵erent hardware implementationapproaches(serial and parallel) for each of these units.
4 Decorations
A decoration, mechanism to handle the exceptions, is information at-tached to an interval; the combination is called a decorated interval.It’s used to describe a property not of the interval it is attached to butof the function evaluated on the input.Worth mentioning that this is a part of the draft standard P1788 forInterval floating point arithmetic, and that there is NO previous hard-ware implementation for it[4].This standard’s decoration model, in contrast with IEEE-754’s, has nostatus flags [5].The set D of decorations has five members shown in Table 1.A decorated interval may take one of five values, thus it is imple-
mented as 3 –bits giving di↵erent decorated output intervals accordingto the input intervals and the operation evaluated.
5 Proposed Implementation
The work presented in here adopts the double path floating pointadders which are based on performing speculatively addition on twodistinct low latency paths (CLOSE and FAR path) depending on theexponent di↵erence and e↵ective operation [6].The correct result isthen selected at the end of the computation. The double path unit isbuilt on the following assumptions:1) FAR path: If exponent di↵erence >1, for e↵ective subtraction themaximum number of leading zeros is one, and only one-bit left shift
50
Decoration Logicalvalue
Shortdescription
Definition
com 000 Common x is a bounded, nonempty sub-set of Dom(f); f is continuousat each point of x and the com-puted interval f(x) is bounded.
dac 001 Defined &continuous
x is a nonempty subset ofDom(f) and the restriction of fto x is continuous.
def 010 Defined x is a nonempty subset ofDom(f).
trv 011 Trivial Always true (so gives no infor-mation).
ill 100 Ill-Formed Not an Interval; formallyDom(f)= ø.
Table 1: Decorations values
might be required for normalization and no need for the leading zerodetection. For e↵ective addition, no possibility of leading zeros ap-pearance, however a large full length right shifter is required for thetwo mantissas’ alignment.2) Near Path, used for e↵ective subtraction only, if exponent di↵erenceis 0 or 1, then only one bit right shift might be needed. Counting thepossible leading zeros is performed in parallel with the operation.
In the NEAR path, a compound adder is used to calculate all thepossible result and reduce the conversion step to only a simple selection[7].While in the FAR path the mantissas are swapped based on theexponent di↵erence to produce only positive results [7]. Thus, theconversion step (2’s complementing) is not any more needed.
The proposed design supports all addition/subtraction, special casesresulting from infinities, de-normalized numbers, and Nan input, aswell as Decorations according to IEEE P1788 standard.
51
6 Results & Testing
After implementation using Verilog and simulation using Quartus &DC Compiler, we obtained a maximum frequency of 283 MHZ usingdevice number EP3SL50F780C2 of Stratix III Family, and 1.126 GHZin 65nm technology for ASIC simulation.The design was tested using a C++ algorithm to generate testing vec-tors, and “File Compare Tool” to compare the outputs of the designwith the testing vectors. Due to the impossibility to cover all the pos-sible combinations of the input, numbers were divided into di↵erentranges that were covered independently.
References:
[1] R.E. Moore, Interval Analysis, Prentice Hall Inc., EnglewoodCli↵s, New Jersey, 1966.
[2] M. A. E. Gardenes, Modal Intervals, Reliable Computing., Au-gust 2008.
[3] A.A.B. Omar, Hardware Implementation of Modal Interval Adder/Subtractor and Multiplier, MSc thesis, Electronics and Commu-nications Department, Faculty of Engineering, Cairo University,2012.
[4] IEEE P1788 draft standard for interval floating point arithmetic .
[5] 754-2008 IEEE Standard for Binary Floating Point Arithmetic.,August 2008.
[6] G. Even and P.M. Seidel, Delay-Optimized Implementation ofIEEE Floating Point Addition, IEEE Transaction on Computers,Vol. 53, No.2 , pp. 97-113, 2004.
[7] S. Oberman, Design Issues in High Performance Floating PointArithmetic Units, PhD. Thesis, Stanford University,” 1996.
52
Sign Regular Matrices Having theInterval Property
Jurgen Garlo↵1) and Mohammad Adm2)
1)University of Applied Sciences / HTWG KonstanzFaculty for Computer ScienceD-78405 Konstanz, Germany
P. O. Box [email protected]
and2)University of Konstanz
Department of Mathematics and StatisticsD-78464 Konstanz, Germany
We say that a class C of n-by-nmatrices possesses the interval prop-erty if for any n-by-n interval matrix [A] = [A,A] = ([aij, aij])i,j=1,...,n
the membership [A] ⇢ C can be inferred from the membership to Cof a specified set of its vertex matrices, where a vertex matrix of [A] isa matrix A = (aij) with aij 2
�aij, aij
, i, j = 1, . . . , n. Examples of
such classes include the
• M -matrices or, more generally, inverse-nonnegative matrices [8],where only the bound matrices A and A are required to be in theclass;
• inverse M -matrices [7], where all vertex matrices are needed;
• positive definite matrices [3], [11], where a subset of cardinality2n�1 is required (here only symmetric matrices in [A] are consid-ered).
A class of matrices which in the nonsingular case are somewhat re-lated to the inverse nonnegative matrices are the totally nonnegative
53
matrices. A real matrix is called totally nonnegative if all its minorsare nonnegative. Such matrices arise in a variety of ways in mathe-matics and its applications, e.g., in di↵erential and integral equations,numerical mathematics, combinatorics, statistics, and computer aidedgeometric design. For background information we refer to the recentlypublished monographs [4], [10]. The speaker posed in 1982 the con-jecture that the set of the nonsingular totally nonnegative matricespossesses the interval property, where only two vertex matrices are in-volved [5], see also [4, Section 3.2] and [10, Section 3.2]. The two vertexmatrices are the bound matrices with respect to the checkerboard or-dering which is obtained from the usual entry-wise ordering in the setof the square matrices of fixed order by reversing the inequality signfor each entry in a checkerboard fashion. This conjecture originatedin the interpolation of interval-valued data by using B-splines. Dur-ing the last three decades many attempts have been made to settlethe conjecture. Some subclasses of the totally nonnegative matriceshave been identified for which the interval property holds, however,the general problem remained open. In our talk we apply the Cauchonalgorithm (also called deleting derivation algorithm [6] and Cauchonreduction algorithm [9]) to settle the conjecture. We report further onsome other recent results, viz. we
• give for each entry of a nonsingular totally nonnegative matrixthe largest amount by which this entry can be perturbed withoutlosing the property of total nonnegativity,
• identify other subclasses exhibiting the interval property of thesign regular matrices, i.e., of matrices with the property that alltheir minors of fixed order have one specified sign or are allowedalso to vanish. This leads us to a new open problem.
References:
[1] M. Adm and J. Garloff, Invariance of total nonnegativity of atridiagonal matrix under element-wise perturbation, Oper. Matri-ces, in press.
54
[2] M. Adm and J. Garloff, Intervals of totally nonnegative ma-trices, Linear Algebra Appl., 439 (2013), No. 12, pp. 3796-3806.
[3] S. Bialas and J. Garloff, Intervals of P-matrices and relatedmatrices, Linear Algebra Appl., 58 (1984), pp. 33-41.
[4] S. M. Fallat and C. R. Johnson, Totally Nonnegative Matri-ces, Princeton Series in Applied Mathematics, Princeton Univer-sity Press, Princeton and Oxford, 2011.
[5] J. Garloff, Criteria for sign regularity of sets of matrices, LinearAlgebra Appl., 44 (1982), pp. 153-160.
[6] K. R. Goodearl, S. Launois and T. H. Lenagan, Totallynonnegative cells and matrix Poisson varieties, Adv. Math., 226(2011), pp. 779-826.
[7] C. R. Johnson and R. S. Smith, Intervals of inverseM -matrices,Reliab. Comput., 8 (2002), pp. 239-243.
[8] J.R. Kuttler, A fourth-order finite-di↵erence approximation forthe fixed membrane eigenproblem,Math. Comp., 25 (1971), pp. 237-256.
[9] Launois and T. H. Lenagan, E�cient recognition of totallynonnegative matrix cells, Found. Comput. Mat., in press.
[10] A. Pinkus, Totally Positive Matrices, Cambridge Tracts in Math-ematics 181, Cambridge Univ. Press, Cambridge, UK, 2010.
[11] J. Rohn, Positive definiteness and stability of interval matrices,SIAM J. Matrix Anal. Appl., 15 (1994), pp. 175-184.
55
Convergence of the Rational BernsteinForm
Jurgen Garlo↵1) and Tareq Hamadneh2)
1)University of Applied Sciences / HTWG KonstanzFaculty for Computer ScienceD-78405 Konstanz, Germany
P. O. Box [email protected]
and2)University of Konstanz
Department of Mathematics and StatisticsD-78464 Konstanz, Germany
A well-established tool for finding tight bounds on the range of amultivariate polynomial
p(x) =lX
i=0
aixi, x = (x
1
, . . . , xn), i = (i1
, . . . , in),
over a box X is the (polynomial) Bernstein form [1,2,4-6]. This isobtained by expanding p by Bernstein polynomials. Then the mini-mum and maximum of the coe�cients of this expansion, the so-calledBernstein coe�cients
bi(p) =iX
j=0
�ij
��lj
�aj, i = 0, . . . , l,
provide lower and upper bounds for the range of p over X. It is knownthat the bounds converge to the range
• linearly if the degree of the Bernstein polynomials is elevated,• quadratically with respect to the width of X,• quadratically with respect to the width of subboxes if subdivisionis applied.
56
In [3] the rational Bernstein form for bounding the range of therational function f = p/q over X is presented, viz.
lmini=0
bi(p)
bi(q) f(x) l
maxi=0
bi(p)
bi(q), x 2 X.
It turned out that some important properties of the polynomialBernstein form do not carry over to the rational Bernstein form, e.g.,the convex hull property and the monotonic convergence of the bounds.In our talk we show that, however, the convergence properties listedabove remain in force for the rational Bernstein form. Similar resultshold for the rational Bernstein form over triangles.
References:
[1] G. T. Cargo and O. Shisha, The Bernstein form of a polyno-mial, J. Res. Nat. Bur. Stand., 70B (1966), pp.79-81.
[2] ] J. Garloff, Convergent bounds for the range of multivari-ate polynomials, in Interval Mathematics 1985, K. NICKEL, Ed.,Lecture Notes in Computer Science, 212 (1986), Springer-Verlag,Berlin, Heidelberg, New York, pp. 37-56.
[3] A. Narkawicz, J. Garloff, A. P. Smith and C. Munoz,Bounding the range of a multivariate rational function over a box,Reliab. Comput., 17 (2012), pp. 34-39.
[4] T. J. Rivlin, Bounds on a polynomial, J. Res. Nat. Bur. Stand.,74B (1970), pp. 47-54.
[5] V. Stahl, Interval methods for bounding the range of polynomialsand solving systems of nonlinear equations, dissertation, JohannesKepler Universitat Linz (1995).
[6] M. Zettler and J. Garloff, Robustness analysis of polyno-mials with polynomial parameter dependency, IEEE Trans. Au-tomat. Contr., 43 (1998), pp. 425-431.
57
Interval regularization approach to theFirordt method of the spectroscopic
analysis of the nonseparated mixtures
Valentin Golodov
South Ural State University454080 Chelyabinsk, Russia
Keywords: system of linear equations, interval uncertainty, intervalregularization, Firordt method, exact computations
Firordt method is one of the methods of the analysis of the non-separated mixtures [1]. According Firordt’s method we can determineconcentration cj of the components in the m-component mixture assolving system of the equations of the form
bi =mX
j=1
aij · l · cj, (1)
where bi is an absorbancy of the analized mixture on the i-th analyt-ical wave length(AWL), aij is an molar coe�cient of the absorbtionof the j-th component on i-th AWL, l is constant. Number of theAWL(k) (number of the equations) usually is equal to the numberof the components(m) in the mixture. Overdetermined systems withk > m are used for the enhanced accuracy.
Results of the spectroscopy may be imprecise so we have some im-precise system of linear algebraic equations for analysis with equationsof the form (1).
bi =mX
j=1
aij · l · cj, (2)
We consider interval linear algebraic systems of equations Ax = b,with an interval matrix A and interval right-hand side vector b, as a
58
model of imprecise systems of linear algebraic equations of the sameform.
We use a new regularization procedure proposed in [3] that reducesthe solution of the imprecise linear system to computing a point fromthe tolerable solution set for the interval linear system with a widenedright-hand side. Tolerable solution set is least sensitive, among allthe solution sets [2], to the change in the interval matrix of the systemAx = b. We exploit this idea that may be called interval regularizationfor the system of equations of the Firordt method of the form (2).
With regards to system of equations of the Firordt method (es-pecially overdetermined) such interval regularization technique pro-vides the enhanced accuracy. Our computing technique uses exactrational computations, it allows to solve sensitive and ill-conditionedproblems[4].
References:
[1] Vlasova I.V., Vershinin V.I., Determination of binary mix-ture components by the Firordt method with errors below thespecified limit, Journal of Analytical Chemistry, vol. 64(2009),No. 6, pp. 553–558.
[2] S.P. Shary, A new technique in systems analysis under inter-val uncertainty and ambiguity, Reliable Computing, vol. 8 (2002),No. 5, pp. 321–418.
[3] Anatoly V. Panyukov, Valentin A. Golodov, Computing Best Pos-sible Pseudo-Solutions to Interval Linear Systems of Equations, .Reliable Computing, Volume 19(2013), Issue 2, pp. 215-228.
[4] V.A. Golodov and A.V. Panyukov, Library of classes “ExactComputation 2.0”. State. reg. 201361818, March 14, 2013. O�cialBulletin of Russian Agency for Patents and Trademarks, FederalService for Intellectual Property, 2013, No. 2. Series “Programsfor Computers, Databases, Topology of VLSI”. (in Russian)
59
A method of calculating faithful roundingof l2-norm for n-vectors
Stef Graillat1, Christoph Lauter1, Ping Tak Peter Tang2,Naoya Yamanka3 and Shin’ichi Oishi3
1 Sorbonne Universites 2 Intel Corporation 3 Faculty of ScienceUPMC Univ Paris 06 2200 Mission College Blvd and EngineeringUMR 7606, LIP6 Santa Clara, CA 95054 Waseda University4, place Jussieu USA 3-4-1 OkuboF - 75005 Paris Tokyo 169-8555France Japan
[email protected], [email protected], [email protected],
[email protected], [email protected]
Keywords: Floating-point arithmetic, error-free transformations, faith-ful rounding, 2-norm, underflow, overflow
In this paper, we present an e�cient algorithm to compute the
faithful rounding of the l2
-norm,qPn
j x2
j , of a floating-point vector
[x1
, x2
, . . . , xn]T . This means that the result is accurate to within onebit of the underlying floating-point type. The algorithm is also faithfulin exception generations: an overflow or underflow exception is gen-erated if and only if the input data calls for this event. This newalgorithm is also well suited for parallel and vectorized implementa-tions. In contrast to other algorithms, the expensive floating-pointdivision operation is not used. We demonstrate our algorithm withan implementation that runs about 4.5 times faster then the netlibversion [1].
There are three novel aspects to our algorithm for l2
-norms:First, for an arbitrary real value �, we establish an accuracy con-
dition for a floating-point approximation S to � that guarantees thecorrect rounding of the square root �(
pS) to be a faithful rounding ofp
�.Second, we propose a way of computing an approximation S to the
sum � =P
j x2
j that satisfies the accuracy condition. This summation
60
algorithm makes use of error-free transformations [4] at crucial steps.Our error-free transformation is custom designed for l
2
-norm computa-tion and thus requires fewer renormalization steps than a more generalerror-free transformation needs. We show that the approximation S isaccurate up to a relative error bound of �`(3"2), where " is the ma-chine epsilon and �`(�) = `�/(1 � `�) bounds the accumulated errorover ` summation steps [3] for an underlying addition operation with arelative error bound of �. Our derivation of � = 3"2 is an enhancement;the standard bounds on � in the literature are strictly greater than 3"2.
Third, in order to avoid spurious overflow and underflow in the in-termediate computations, our algorithm extends the previous work byBlue [2]: the input data xj are appropriately scaled into “bins” suchthat computing and accumulating their squares x2j are guaranteed ex-ception free. While Blue uses three bins and the division operation,our algorithm uses only two and is division free. These properties econ-omize registers usage and improve performance. The claim of faith-ful rounding and exception generation is supported by mathematicalproofs. The proof of faithful overflow generation is relatively straight-forward, but that for faithful underflow generation requires consider-ably greater care.
References:
[1] Anderson, Bai, Bischof, Blackford, Demmel, Dongarra,Croz, Hammarling, Greenbaum, McKenney, andSorensen, LAPACK Users’ guide (third ed.), Society for Indus-trial and Applied Mathematics, Philadelphia, PA, USA, 1999.
[2] Blue, A portable Fortran program to find the Euclidean norm ofa vector, ACM Trans. Math. Softw., 4 (1978), No. 1, pp. 15–23.
[3] Higham, Accuracy and stability of numerical algorithms (seconded.), Society for Industrial and Applied Mathematics, Philadel-phia, PA, USA, 2002.
[4] Ogita, Rump, and Oishi, Accurate Sum And Dot Product,SIAM J. Sci. Comput., 26 (2005), No. 6, pp. 1955–1988.
61
An Energy-E�cient and MassivelyParallel Approach to Valid Numerics
John Gustafson
Ceranovo Inc.Palo Alto, CA USA
Computer hardware manufacturers have shown little interest in im-proving the validity of their numerics, and have accepted the hazards offloating point arithmetic. However, they have a very strong and grow-ing interest in energy e�ciency as they compete for the battery life ofmobile devices as well as the amount of capability they can achieve ina large data center with strict megawatt-level power budgets. Theyare also concerned that multicore parallelism is growing much fasterthan algorithms can exploit. It may be possible to persuade manufac-turers to embrace valid numerics not because of validity concerns butbecause having valid numerics can solve energy/power and parallelismconcerns.
A new universal number format, the “unum”, allows valid (prov-ably bounded) arithmetic with about half the bits of conventional IEEEfloating point on average; the bit reduction saves energy and power byreducing the bandwidth and storage demands on the processor. Italso relieves the programmer from being an expert in numerical anal-ysis, by automatically tracking the exact or ULP-wide inexact state ofeach value and by promoting and demoting dynamic range and frac-tion precision automatically. Unums pass the di�cult validity testspublished by Kahan, Rump, and Bailey. When used to solve physicsproblems, such as nonlinear ordinary di↵erential equations, they alsoexpose a new source of massive parallelism in what were thought to behighly serial time-dependent problems. Furthermore, because unumarithmetic obeys associative and distributive laws, parallelization of
Invited talk
62
algorithms does not produce changes in the answer from rounding er-rors that unsophisticated programmers mistake for logic errors; thisfurther facilitates the use of parallel architectures.
The new format and the new algorithms that go with it have thepotential to completely disrupt the way computers are designed andused for technical computing.
63
Towards tight bounds on the radius ofnonsingularity
David Hartman, Milan HladıkDepartment of Applied Mathematics, Charles University
11800 Prague, Czech Republic{hartman,hladik}@kam.mff.cuni.cz
Institute of Computer Science Academy of Sciences18207 Prague 8, Czech Republic
Keywords: radius of nonsingularity, semidefinite programming, ap-proximation algorithm
Radius of nonsingularity of a square matrix is the minimal distanceto a singular matrix in the Chebyshev norm. More formally, for amatrix A 2 Rn⇥n, the radius of nonsingularity [1,2] is defined by
d(A) := inf {" > 0; 9 singular B : |aij � bij| " 8i, j}.
It has been shown [2,3] that this characteristic can be computed as
d(A) =1
kA�1k1,1, (1)
where k · k1,1 is a matrix norm defined as
kMk1,1 := max {kMxk1
; kxk1 = 1} = max {kMzk1
; z 2 {±1}n}.
Unfortunately, computing k·k1,1 is an NP-hard problem [2]. In fact,provided P 6= NP no polynomial time algorithm for approximatingd(A) with a relative error at most 1
4n2 exists [3]. That is why therewere investigated various lower and upper bounds. Rohn [3] providedthe following bounds
1
⇢(|A�1|E) d(A) 1
maxi=1,...,n
(E|A�1|)ii
64
On the other side, Rump [4, 5] developed other estimations1
⇢(|A�1|E) d(A) 6n
⇢(|A�1|E).
We provide better bounds based on approximation of k ·k1,1. Moreconcretely, we propose a randomized approximation method with ex-pected error 0.7834. The mentioned algorithm is based on a semidef-inite relaxation of original problem [6]. This relaxation gives the bestknown approximation algorithm for Max-Cut problem, and we utilizesimilar principle to derive tight bounds on the radius of nonsingularity.
Supported by grants 13-17187S and 13-10660S of the Czech Science Foundation.
References:
[1] S. Poljak, J. Rohn, Radius of Nonsingularity, Technical reportKAM Series, Department of Applied Mathematics, Charles Uni-versity, 117 (1988), pp. 1–11.
[2] S. Poljak, J. Rohn, Checking robust nonsingularity is NP-hard,Math. Control Signals Syst., 6 (1993), No. 1, pp. 1–9.
[3] J. Rohn, Checking properties of interval matrices, Technical Re-port, Institute of Computer Science, Academy of Sciences of theCzech Republic, Prague, 686 (1996).
[3] V. Kreinovich and A. Lakeyev and J. Rohn and P. Kahl,Computational Complexity and Feasibility of Data Processing andInterval Computations, Kluwer, 1998.
[4] S. M. Rump, Almost sharp bounds for the componentwise dis-tance to the nearest singular matrix, Linear Multilinear Algebra,42 (1997), No. 2, pp.93–107.
[5] S. M. Rump, Bounds for the componentwise distance to the near-est singular matrix, SIAM J. Matrix Anal. Appl., 18 (1997), No. 1,pp.83–103.
[6] B. Gartner and J. Matousek, Approximation Algorithms andSemidefinite Programming, Springer, 2012.
65
A numerical verification method for abasin of a limit cycle
Tomohirio Hiwaki and Nobito YamamotoThe University of Electro-CommunicationsChofugaoka 1-5-1, Chofu, Tokyo, Japan
Keywords: numerical verification, dynamical system, basin of limitcycle
1 Introduction
We propose a method of validated computation to verify a domainwithin a basin of a closed orbit, which is asymptotic stable in a dynam-ical system described by ODEs. This method proves a contractibilityof Poincare map.
2 Problem
We treat ordinary di↵erential equations
du
dt= f(u), 0 < t < 1, (1)
u 2 D ⇢ Rn,f : D 7! Rn,
where f(u) is continuously di↵erentiable with respect to u.In order to specify a time period T explicitly, we apply variable
transformation by s =t
Tand v(s) = u(Ts), then get an expression
(dv
ds= Tf(v), 0 < s < 1,
v(0) = v(1),(2)
66
for a closed orbit. We define a unit vector n
�
together with � as aPoincare section, which is a plain perpendicular to n
�
. Additionally'(T,w) is defined as the point of the trajectory at s = 1 with an initialvalue w and a period T .
3 Our idea for verification of a basin
Using a projection P�
of (T,v(1)) to (T,�) which is defined by
P�
= I � 1
n
�
Tn
�
✓0n
�
◆�0,n
�
T�,
we prove the contractibility of the Poincare map in [W ] which is a setof points on �. In verification process we compute 2-norm of a certainmatrix which comes from an operator P
�
'(T,w) and verify the normis less than 1 by validated computation. Consequently we prove that[W ] is included by a basin of a limit cycle.
In actual calculation, we use numerical verification technique, e.g.Lohner method, mean value form and so on. Especially, an e�cientmethod is adopted which is developed by P.Zgliczynski, so calloed C1-Lohner method.
We will present numerical examples in our talk.
References:
[1] RihmR.Rihm, Interval methods for initial value problems in ODEs,Elsevier(North-Holland), Topics in validated computation (ed. byJ.Herzberger), 1994
[2] Zgli P.Zgliczynski, C1-Lohner algorithm, Found.Comput.Math.,2 (2002), 429-465
67
Optimal preconditioning for the intervalparametric Gauss–Seidel method
Milan HladıkCharles University, Faculty of Mathematics and Physics, Department
of Applied MathematicsMalostranske nam. 25, 11800 Prague, Czech Republic
Keywords: interval computation, interval parametric system, precon-ditioner, linear programming
Consider an interval parametric system of linear equations
A(p)x = b(p), p 2 p,
where the constraint matrix and the right-hand side vector linearlydepends on parameters p
1
, . . . , pK as follows
A(p) =KX
k=1
Akpk, b(p) =KX
k=1
bkpk.
Herein, A1, . . . , AK 2 Rn⇥n are given matrices, b1, . . . , bK 2 Rn aregiven vectors, and p = (p
1
, . . . ,pK) is a given interval vector. Thecorresponding solution set is defined as
{x 2 Rn; 9p 2 p : A(p)x = b(p)}.
Various methods for computing an enclosure to the solution set exist[1]. In our contribution, we focus on the interval Gauss–Seidel iteration[4]. In particular, we will be concerned with the problem of determin-ing an optimal preconditioner that minimizes either the widths of theresulting intervals, or their upper/lower bounds. A preconditioner is amatrix C 2 Rn⇥n, by which the system is pre-multiplied
CA(p)x = Cb(p), p 2 p.
68
Usually, the midpoint inverse A(pc)�1 is chosen since it performs wellpractically, but it needn’t be the optimal choice.
The problem of computing an optimal preconditioner for linearnon-parametric interval systems was studied, e.g., in [2, 3]. In ourpaper, we will extend some of their results for the parametric systems.We will show that optimal preconditioners can be computed by solvingsuitable linear programs using approximately Kn variables and similarnumber of constraints, which means that the problem is polynomiallysolvable.
We also show by several examples that, in some cases, such optimalpreconditioners are able to significantly decrease overestimation of theenclosures computed by common methods.
References:
[1] M. Hladık. Enclosures for the solution set of parametric intervallinear systems. Int. J. Appl. Math. Comput. Sci., 22(3):561–574,2012.
[2] R. B. Kearfott. Preconditioners for the interval Gauss–Seidelmethod. SIAM J. Numer. Anal., 27(3):804–822, 1990.
[3] R. B. Kearfott, C. Hu, and M. Novoa III. A review of precondi-tioners for the interval Gauss–Seidel method. Interval Comput.,1991(1):59–85, 1991.
[4] E. D. Popova. On the solution of parametrised linear systems. InW. Kramer and J. W. von Gudenberg, editors, Scientific Comput-ing, Validated Numerics, Interval Methods, pages 127–138. Kluwer,2001.
69
On Unsolvability of OverdeterminedInterval Linear Systems
Jaroslav Horacek and Milan HladıkCharles University, Faculty of Mathematics and Physics, Department
of Applied MathematicsMalostranske nam. 25, 118 00, Prague, Czech [email protected], [email protected]
Keywords: interval linear systems, overdetermined systems, unsolva-bility conditions
By an overdetermined interval linear system (OILS) we mean an in-terval linear system with more equations than variables. By a solutionset of an interval linear system Ax = b we mean
⌃ = {x | Ax = b for some A 2 A, b 2 b},
where A is an interval matrix and b is an interval vector. If ⌃ is anempty set, we call the system unsolvable. It is appropriate to pointout that this approach is di↵erent from the least squares method.
The set ⌃ is usually hard to be described. That is why it is of-ten enclosed (among other possibilities) by some n-dimensional box.Computing the tightest possible box (interval hull) containing ⌃ isNP-hard. Therefore, we usually compute in polynomial time a slightlybigger box containing the interval hull (interval enclosure). For moresee [2].
There exist many methods for computing interval enclosures ofOILS see e.g, [1]. Nevertheless, many of them return nonempty so-lution set even if the OILS has no solution (e.g., if we use the intervalleast squares as an enclosure method). In some applications we docare whether systems are solvable or unsolvable (e.g. system valida-tion, technical computing).
Unfortunatelly, deciding whether an interval system is solvable isan NP-hard problem. There exist some results for square systems (i.e,
70
systems where A in Ax = b is a square matrix) like [3]. In our talk wewould like to address the solvability and unsolvability of OILS. Thereis a lack of necessary and su�cient conditions for detecting solvabilityand unsolvability of OILS. We would like to present some newly de-veloped conditions and algorithms concerning these problems. We willtest the strength of various conditions numerically and nicely visualizethe results.
References:
[1] J. Horacek, M. Hladık, Computing enclosures of overdeter-mined interval linear systems, Reliable Computing, 19 (2013), No. 2,pp. 142–155.
[2] A. Neumaier, Interval methods for systems of equations, Cam-bridge University Press, Cambridge, 1990.
[3] J. Rohn, Solvability of systems of interval linear equations and in-equalities, Linear optimization problems with inexact data, (2006),pp. 35–77.
71
Computing capture tubes
Luc Jaulin1, Jordan Ninin1, Gilles Chabert3,Stephane Le Menec2, Mohamed Saad1, Vincent Le Doze2,
Alexandru Stancu4
1 Labsticc, IHSEV, OSM, ENSTA-Bretagne2 EADS/MBDA, Paris, France3 Ecole des Mines de Nantes
4 Aerospace Research Institute, University of Manchester, UK
Keywords: capture tube, contractors, interval arithmetic, robotics,stability.
1 Introduction
A dynamic system can often be described by a state equation x =h(x,u, t) where x 2 Rn is the state vector, u 2 Rm is the controlvector and h : Rn ⇥ Rp ⇥ R ! Rn is the evolution function. As-sume that the control low u = g (x, t) is known (this can be obtainedusing control theory), the system becomes autonomous. If we definef (x, t) = h (x,g (x, t) , t), we get the following equation.
x = f (x, t) .
The validation of some stability properties of this system is an impor-tant and di�cult problem [2] which can be transformed into provingthe inconsistency of a constraint satisfaction problem. For some par-ticular properties and for invariant system (i.e., f does not depend ont), it has been shown [1] that the V-stability approach combined inter-val analysis [3] can solve the problem e�ciently. Here, we extend thiswork to systems where f depends on time.
72
2 Problem statement
Consider an autonomous system described by a state equation x =f (x, t). A tube G(t) is a function which associates to each t 2 R asubset of Rn. A tube G(t) is said to be a capture tube if the fact thatx(t) 2 G(t) implies that x(t + t
1
) 2 G(t + t1
) for all t1
> 0. Considerthe tube
G (t) = {x,g (x, t) 0} (1)
where g : Rn ⇥ R ! Rm. The following theorem, introduced recenltly[4], shows that the problem of proving that G (t) is a capture tube canbe cast into solving a set of inequalities.
Theorem. If the system of constraints8<
:
(i) @gi@x (x, t) .f(x, t) + @gi
@t (x, t) � 0(ii) gi (x, t) = 0(iii) g (x, t) 0
(2)
is inconsistent for all x, all t � 0 and all i 2 {1, . . . ,m} then G (t) ={x,g (x, t) 0} is a capture tube.
3 Computing capture tubes
If a candidate G (t) for a capture tube is available, we can check thatG (t) is a capture tube by checking the inconsistency of a set of nonlin-ear equations (see the previous section). This inconsistency can theneasily be checked using interval analysis [3]. Now, for many systemssuch as for non holonomous systems, we rarely have a candidate fora capture tube and we need to find one. Our main contribution is toprovide a method that can help us to find such a capture tube. Theidea if to start from a non-capture tube G(t) and to try to characterizethe smallest capture tube G+(t) which encloses G(t). To do this, wepredict for all (x, t), that are solutions of (2), a guaranteed envelope fortrajectory within finite time-horizon window [t, t+ t
2
] (where t2
> 0 isfixed). If all corresponding x(t+t
2
) belongs to G(t+t2
), then the union
73
of all trajectories and the initial G (t) (in the (x, t) space) correspondsto the smallest capture tube enclosing G (t).
References:
[1] L. jaulin, F. Le Bars, An interval approach for stability analy-sis; Application to sailboat robotics, IEEE Transaction on Robotics,2012.
[2] S. Le Menec, Linear Di↵erential Game with Two Pursuers andOne Evader, Advances in Dynamic Games, 2011.
[3] R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction toInterval Analysis, SIAM, Philadelphia, 2009.
[4] A. Stancu, L. Jaulin, A. Bethencourt, Set-membership track-ing using capture tubes, to be submitted.
74
On relative errors of floating-pointoperations: optimal bounds and
applications
Claude-Pierre Jeannerod1 and Siegfried M. Rump2,3
1Inria, laboratoire LIP (CNRS, ENS de Lyon, Inria, UCBL),Universite de Lyon, France.
2Institute for Reliable Computing, Hamburg University ofTechnology, Germany.
3Faculty of Science and Engineering, Waseda University, Tokyo,Japan.
[email protected] — [email protected]
Keywords: floating-point arithmetic, rounding error analysis
Rounding error analyses of numerical algorithms are most oftencarried out via repeated applications of the so-called standard modelsof floating-point arithmetic. Given a round-to-nearest function fl andbarring underflow and overflow, such models bound the relative errorsE
1
(t) = |t�fl(t)|/|t| and E2
(t) = |t�fl(t)|/|fl(t)| by the unit roundo↵ u.This talk will investigate the possibility of refining these bounds,
both in the case of an arbitrary real t and in the case where t is theexact result of an arithmetic operation on some floating-point numbers.Specifically, we shall provide explicit and attainable bounds on E
1
(t),which are all less than or equal to u/(1 + u) and, therefore, smallerthan u. For E
2
(t) we will see that the situation is di↵erent and thatoptimal bounds can or cannot equal u, depending on the operation andthe floating-point radix.
Then we will show how to apply this set of sharp bounds to therounding error analysis of various numerical algorithms, including sum-mation, dot products, matrix factorizations, and complex arithmetic:in all cases, we obtain much shorter proofs of the best-known errorbounds for such algorithms and/or improvements on these boundsthemselves.
75
Fast Implementation of Quad-PrecisionGEMM on ARMv8 64-bit Multi-Core
ProcessorHao Jiang, Feng Wang, Yunfei Du and Lin Peng
National University of Defence Technology410072 Changsha, China
[email protected],[email protected]
Keywords: quad-precision GEMM, error-free transformation, ARMv864-bit multi-core processor,
In recent years, ARM-based SoCs have a rapid evolution. Thepromising qualities, such as competitive performance and energy e�-ciency, make ARM-based SoCs the candidates for the next generationHigh Performance Computing (HPC) system [1]. For instance, sup-ported by the Mont-Blanc project, Barcelona Supercomputing Centerbuilds the world’s first ARM-based HPC cluster–Tibidabo. Recently,the new 64-bit ARMv8 instruction set architecture (ISA) improvessome important features including using 64-bit addresses, introducingdouble-precision floating point in the NEON vector unit, increasing thenumber of registers, supporting fused multiply-add (FMA) instruction,etc. Hence, it shows increasing interest to build the HPC system withARMv8-based SoC.
In HPC, large-scale and long-time numerical calculations often pro-duce inaccurate and invalidated results owing to cancellation fromround-o↵ errors. In the cases above, double precision accuracy is notsu�cient, then higher precision is required. Some high precision em-ulation softwares, such as MPFR, GMP and QD library, perform wellfor some applications. BLAS is the fundamental math library. Toimprove it’s accuracy, M. Nakata designed MBLAS [2] based on thethree high precision libraries above, and some other researchers didthe similar researches on GPUs. All the high precision BLAS librariesabove are independent of the computer architectures. Hence, in someplatforms, they can not achieve the optimal performance of processors.
76
Matrix-matrix multiplication (GEMM) is the basic function in thelevel-3 BLAS. In this paper, we present the first implementation of thequad precision GEMM (QGEMM) on ARMv8 64-bit multi-core pro-cessor. We utilize double-double type format to store the quadrupleprecision floating point value. We choose the blocking and packing al-gorithms and parallelism method from GotoBLAS [3]. We propose theoptimization model with the purpose of maximizing the compute-to-memory access ratio to construct the inner kernel of QGEMM. Con-sidering the double-double format and the 128-bit vector register inARMv8 64-bit processor, we let one vector register store one double-double floating point value to save the memory space. As each ARMv864-bit processor core contains 32 vector registers, we choose the 4x2register blocking. The basic segment of the inner kernel is the prod-uct of two double-double values adding a double-double value. Witherror-free transformation, we implement this segment in the assem-bly language, using ARM 64-bit memory accessing instruction, cachepre-fetching instruction and FMA instruction. Considering the datadependency, we reorder the instruction and unroll the loop to performcalculation in parallel. The numerical results show that our imple-mentation shows better performance than MBLAS on ARMv8 64-bitprocessor.
References:
[1] N. Rajovic, P.M. Carpenter, I. Gelado, N. Puzovic, A.Ramirez, M. Valero, Supercomputing with Commodity CPUs:Are Mobile SoCs Ready for HPC?, SC’13.
[2] M. Nakata, The MPACK(MBLAS/MLAPACK):A multiple pre-cision arithmetic version of BLAS and LAPACK, version 0.8.0,2012, http://mplapack.sourceforge.net.
[3] K. Goto, R. A. v. d. Geijn, Anatomy of high-performance ma-trix multiplication, ACM Transactions on Mathematics Software,34 (2008), No. 12, pp. 1–25.
77
Some Observations on Exclusion Regionsin Interval Branch and Bound Algorithms
Ralph Baker Kearfott
University of Louisiana at LafayetteDepartment of Mathematics, U.L. Box 4-1010
Lafayette, Louisiana 70504-1010 [email protected]
Keywords: cluster problem, backboxing, epsilon-inflation, completesearch, branch and bound, interval computations
In branch and bound algorithms for constrained global optimiza-tion, an acceleration technique is to construct regions x around localoptimizing points x, then delete these regions from further search. Theresult of the algorithm is then a list these constructed small regionsin which all globally optimizing points must lie. If the constructedregions are too small, the algorithm will not be able to easily rejectadjacent regions in the search, while, if the constructed regions are toolarge, the set of optimizing points is not known accurately. We brieflyreview previous methods of constructing boxes about approximate op-timizing points. We then derive a formula for determining the size ofa constructed solution-containing region, proportional to the smallestradius ✏ of any box generated in the branch and bound algorithm. Weprove that, if a box of this size is constructed, adjacent regions of ra-dius ✏ on qualifying faces will necessarily be rejected, without the needto actually process them in the branch and bound algorithm. Based onthis, we propose a class of algorithms that construct exclusion boxesfrom concentric shells of small boxes of increasing size surrounding theinitial exclusion box x. The behavior of such algorithms would be morepredictable and controllable than use of branch and bound algorithmswithout such auxiliary constructions.
78
References:
[1] Ferenc Domes and Arnold Neumaier. Rigorous verification of fea-sibility, 2014. submitted, preprint at http://www.mat.univie.ac.at/~neum/ms/feas.pdf.
[2] Jurgen Herzberger, editor. Topics in Validated Computations: pro-ceedings of IMACS-GAMM International Workshop on ValidatedComputation, Oldenburg, Germany, 30 August–3 September 1993,volume 5 of Studies in Computational Mathematics, Amsterdam,The Netherlands, 1994. Elsevier.
[3] Ralph Baker Kearfott. Abstract generalized bisection and a costbound. Mathematics of Computation, 49(179):187–202, July 1987.
[4] Ralph Baker Kearfott. Rigorous Global Search: Continuous Prob-lems. Number 13 in Nonconvex Optimization and its Applications.Kluwer Academic Publishers, Dordrecht, Netherlands, 1996.
[5] Ralph Baker Kearfott and Kaisheng Du. The cluster problem inmultivariate global optimization. Journal of Global Optimization,5:253–265, 1994.
[6] Gunter Mayer. Epsilon-inflation in verification algorithms. J.Comput. Appl. Math., 60(1-2):147–169, 1995.
[7] Siegfried M. Rump. Verification methods for dense and sparsesystems of equations. In Herzberger [2], pages 63–136.
[8] Hermann Schichl, Mikaly Csaba Markot, and Arnold Neumaier.Exclusion regions for optimization problems, 2013. accepted forpublication; preprint available at http://www.mat.univie.ac.at/~neum/ms/exclopt.pdf.
[9] Hermann Schichl and Arnold Neumaier. Exclusion regions forsystems of equations. SIAM Journal on Numerical Analysis,42(1):383–408, 2004.
79
[10] R. J. Van Iwaarden. An Improved Unconstrained Global Optimiza-tion Algorithm. PhD thesis, University of Colorado at Denver,1996.
80
Some remarks on the rigorous estimationof inverse linear elliptic operators
Takehiko Kinoshita1,2, Yoshitaka Watanabe3, Mitsuhiro T.Nakao4
1Center for the Promotion of Interdisciplinary Education andResearch, Kyoto University, Kyoto 606-8501, Japan
2Research Institute for Mathematical Sciences, Kyoto University,Kyoto 606-8502, Japan
3Research Institute for Information Technology, Kyushu University,Fukuoka 812-8581, Japan
4Sasebo National College of Technology, Nagasaki 857-1193, [email protected]
Keywords: Linear elliptic PDEs, inverse operators, validated compu-tations
In this talk, we consider several kinds of constructive a posterioriestimates for the inverse linear elliptic operator. We show that thecomputational costs depend on the concerned elliptic problems as wellas the approximation properties of used finite element subspaces, e.g.,mesh size or so. Also, we propose a new estimate which is e↵ective foran intermediate mesh size. Moreover, we describe some results on theasymptotic behaviour of the approximate inverse estimates. Numericalexamples which confirm us these facts are presented.
References:
[1] T. Kinoshita, Y. Watanabe, and M. T. Nakao, An improve-ment of the theorem of a posteriori estimates for inverse ellipticoperators, Nonlinear Theory and Its Applications, 5 (2014), no. 1,pp. 47–52.
[2] M. T. Nakao, K. Hashimoto, and Y. Watanabe, A numer-ical method to verify the invertibility of linear elliptic operators
81
with applications to nonlinear problems, Computing, 75 (2005),pp. 1–14.
[3] Y. Watanabe, T. Kinoshita, and M. T. Nakao, A posteri-ori estimates of inverse operators for boundary value problems inlinear elliptic partial di↵erential equations, Mathematics of Com-putation, 82 (2013), pp. 1543–1557.
82
Computer-Assisted Uniqueness Prooffor Stokes’ Wave of Extreme Form
Kenta KobayashiHitotsubashi University
2-1 Naka, Kunitachi-City, Tokyo 186-8601, [email protected]
Keywords: Stokes’ wave, global uniqueness, Stokes conjecture
We present computer-assisted proof for the global uniqueness ofStokes’ wave of extreme form. The gravity and the surface tensionhave much influence on the form of water waves. Assuming that theflow is infinitely deep, the gravitational acceleration is a unique exter-nal force of the system and the wave profile is stationary, we obtainNekrasov’s equation[1]. In particular, a positive solution of Nekrasov’sequation corresponds to a water wave which has just one peak andone trough per period. Stokes’ wave of extreme form has a sharp crestand is considered to be the limit of the positive solution of Nekrasov’sequation with respect to parameters such as gravity, wave length, andwave velocity.
Stokes’ wave of extreme form is obtained by the following nonlinearintegral equation for the unknown ✓ : (0, ⇡] ! R:
8>>>><
>>>>:
✓(s) =1
3⇡
Z ⇡
0
log
����sin s+t
2
sin s�t2
���� ·sin ✓(t)
R t
0
sin ✓(w)dwdt,
0 < ✓(s) <⇡
2s 2 (0, ⇡),
✓(⇡) = 0.
The wave profile of Stokes’ wave of extreme form is represented as(x(s), y(s)) (0 < s < 2⇡), where x and y are determined by
dx
ds= � L
2⇡e�H✓(s) cos ✓(s),
dy
ds= � L
2⇡e�H✓(s) sin ✓(s).
Invited talk
83
Here, L is the wavelength and H is the Hilbert transform.Although the existence of Stokes’ wave of extreme form has been
proved[2], the global uniqueness had not been proved for long time. Wesuppose it is almost 30 years during which a lot of e↵orts were devotedin order to solve the problem. Finally we proved the global uniquenessin 2010 [3] and published the summarized version in 2013 [4].
The uniqueness of Stokes’ wave concerns the second Stokes’ conjec-ture[5] which has been a longtime open problem for 130 years. The sec-ond conjecture supposed that the profile of Stokes’ wave between twoconsecutive crests should be downward convex. Existence of Stokes’wave of extreme form which has the profile of downward convex wasproved in 2004 [6]. Therefore, the complete settlement of the secondStokes’ conjecture was brought by our result.
References:
[1] A. I. Nekrasov, On waves of permanent type I, Izv. Ivanovo-Voznesensk. Polite. Inst., 3 (1921), pp. 52–56. (in Russian)
[2] J. F. Toland, On the existence of waves of greatest height andStokes’ conjecture, Proc. R. Soc. Lond., A 363 (1978), pp. 469–485.
[3] K. Kobayashi, On the global uniqueness of Stokes’ wave of ex-treme form, IMA J. Appl. Math., 75 (2010), pp. 647–675.
[4] K. Kobayashi, Computer-assisted uniqueness proof for Stokes’wave of extreme form, Nankai Series in Pure, Applied Mathematicsand Theoretical Physics, 10 (2013), pp. 54–67.
[5] G. G. Stokes, On the theory of oscillatory waves, Appendix B: Consideration relative to the greatest height of oscillatory irro-tational waves which can be propagated without change of form,Math. Phys. Paper, 1 (1880), pp. 225–228.
[6] J. F. Toland, & P. I. Plotnikov Convexity of Stokes wavesof extreme form, Arch. Ration. Mech. Anal., 171 (2004), pp.349–416.
84
Error Estimations of Interpolationson Triangular Elements
Kenta Kobayashi and Takuya Tsuchiya
Graduate School ofCommerce and Management,Hitotsubashi University, Japan
Graduate School ofScience and Engineering,Ehime University, Japan
Keywords: the circumradius condition, interpolation, finite elementmethods
The interpolations and their error estimations are important fun-damentals for, in particular, the finite element error analysis. LetK ⇢ R2 be a triangle with apices xi, i = 1, 2, 3. Let P
1
be the set ofall polynomials whose degree is at most 1. For a continuous functionv 2 C0(K), the linear interpolation I1
Kv 2 P1
is defined by
(I1
Kv)(xi) = v(xi), i = 1, 2, 3.
It has been known that we need to impose a geometric conditionto K to obtain an error estimation of ku� I1
Kuk1,2,K . One of the well-known such conditions is the following. Let hK be the diameter ofK.
• The maximum angle condition, Babuska-Aziz [1], Jamet [2](1976). Let ✓
1
, 2⇡/3 ✓1
< ⇡, be a constant. If any angle ✓ of Ksatisfies ✓ ✓
1
and hK 1, then there exists a constant C = C(✓1
)independent of hK such that
kv � I1
Kvk1,2,K ChK |v|2,2,K , 8v 2 H2(K).
85
Since its discovery, the maximum angle condition was believed to bethe most essential condition for convergence of solutions of the finiteelement method.
Recently, we obtained the following error estimate which is moreessential than the maximum angle condition. Let RK be the circum-radius of K.
• The circumradius condition, Kobayashi-Tsuchiya [3] (2014).For an arbitrary triangle K with RK 1, there exists a constantCp independent of K such that the following estimate holds :
kv � I1
Kvk1,p,K CpRK |v|2,p,K , 8v 2 W 2,p(K), 1 p 1.
We have also pointed out that the circumradius condition is closelyrelated to the definition of surface area [4]. In this talk we will explainthe circumradius condition and the related topics. We will also mentionrecent developments on the subject.
References:
[1] I. Babuska, A.K. Aziz, On the angle condition in the finiteelement method, SIAM J. Numer. Anal., 13 (1976) 214–226
[2] P. Jamet, Estimations d’erreur pour des elements finis droitspresque degeneres, R.A.I.R.O. Anal. Numer., 10 (1976) 43–61
[3] K. Kobayashi, T. Tsuchiya, A Babuska-Aziz type proof of thecircumradius condition, Japan J. Indus. Appl. Math., 31 (2014)193–210
[4] K. Kobayashi, T. Tsuchiya, On the circumradius condition forpiecewise linear triangular elements, submitted, arXiv:1308.2113
[5] X. Liu, F. Kikuchi, Analysis and estimation of error constants forP0
and P1
interpolations over triangular finite elements, J. Math.Sci. Univ. Tokyo, 17 (2010) 27–78
86
Implementingthe Interval Picard Operator
M. Konecny, W. Taha, J. Duracz and A. FarjudianAston University (first author only)
Aston TriangleB4 7ET, Birmingham, [email protected]
Keywords: ODE, interval Picard operator, function arithmetic
Edalat & Pattinson give an elegant constructive description of theexact solutions of Lipschitz ODE IVPs based on an interval Picardoperator [1]. We build on this theoretical work and propose a verifiableand practically useful method for validated ODE solving. In particular,this method
• is very simple;
• is correct by construction in a strong sense;
• produces arbitrarily precise results;
• works for problems with uncertain initial values;
• produces tight enclosures for non-trivial problems.
Simplicity, correctness by construction and arbitrary precision are prop-erties that our method inherits from Edalat & Pattinson’s work.
We employ a number of ideas from established validated ODE solv-ing approaches. Most importantly, we employ a function arithmeticsimilar to Taylor Models (TMs) [2]. In our arithmetic, an enclosureis formed by two independent polynomials, which makes it possibleto closely approximate interval functions of non-constant width. Suchfunctions arise naturally from the interval Picard operator.
To support problems with uncertain initial value, we borrow twofurther techniques from TM methods. First, we enclose the (n+1)-aryODE flow instead of enclosing the union of the graphs of the unary
87
ODE solutions over all initial values. Second, we use a version ofshrink-wrapping [3] to minimize the loss of information between steps.
We demonstrate that our implementation of the method is capableof enclosing solutions of non-smooth ODEs and classical examples ofnon-linear systems, including the Van der Pol system and the Lorenzsystem with uncertain initial conditions. While our method is not at-tempting to compete with mature systems such as COSY and VNODEin terms of speed and power, we believe it is a theoretically pleasing andeasily verifiable alternative worth exploring and testing to its limits.
References:
[1] A. Edalat, D. Pattinson, A Domain-Theoretic Account of Pi-card’s Theorem, LMS Journal of Computation and Mathematics,10 (2007), pp. 83–118.
[2] K. Makino, M. Berz, Taylor Models and Other Validated Func-tional Inclusion Methods, International Journal of Pure and Ap-plied Mathematics, 4 (2003), No. 4, pp. 379–456.
[3] M. Berz, K. Makino, Taylor Models and Other Validated Func-tional Inclusion Methods, International Journal of Di↵erential Equa-tions and Applications, 10 (2005), No. 4, pp. 385–403.
88
Interval methods for solving various kindsof quantified nonlinear problems
Bart lomiej Jacek Kubica
Warsaw University of TechnologyWarsaw, Poland
Interval branch-and-bound type methods can be used to sovle vari-ous problems, in particular: equations systems, constraint satisfactionproblems, global optimization, Pareto sets seeking, Nash points andother game equilibria seeking and other problems, e.g., seeking all lo-cal (but non-global) optima of a function.
We show that each of these problems can be expressed by a specifickind of first-order logic formula and investigate, how this a↵ects thestructure of the algorithm and used tools. In particular, we discussseveral aspects of parallelization of these algorithms.
The focus is on seeking game equilibria, that is a relatively novelapplication of interval methods.
Invited talk
89
Applied techniques of intervalanalysis for estimationof experimental data
S. I. Kumkov1
Institute of Mathematics and Mechanics UrB RAS16, S.Kovalevskaya str., 620990, Ekaterinburg, Russia
Ural Federal University, Ekaterinburg, [email protected]
Keywords: interval analysis, estimation, experimental data
In practice of processing experimental data with bounded measur-ing errors and unknown probabilistic characteristics, interval analysismethods [1,2] are successfully applied in contrast to statistical ones.Moreover, using the concrete properties of the process, it becomes pos-sible to elaborate more e↵ective procedures that ones based the box-techniques [1,2]. The paper deals with adjusting the interval analysismethods to practical processing the experimental chemical processes[3,4]. Here, two version of description the processes are considered.In the first version, an analytical function with parameters to be esti-mated is used, In the second one, a system of the process kinetic systemof ordinary di↵erential equations with parameters is used. Also, shortsamples of measurements with bounded errors are given.
The problem of estimation is formulated as follows: it is neces-sary to estimate the set of admissible values of the process parametersconsistent with the given process description and input data.
In the first version, the experimental process is described by thefunction S(t, V,↵, BG) = V exp(↵t) + BG, where, t is the processtime, ↵, V, BG are parameters. Here, the sought-for information setI(↵, lnV,BG) is constructed (Fig.1a) as a collection of two-dimensionalcross-sections {I(↵, lnV,Bn)}, n = 1, 101 on the grid {Bn}. The cross-sections are convex polygons with exact linear boundaries. The exam-ple of a middle cross-section is marked by the thick boundary.
1The work was supported by the RFBR Grants, nos. 12-01-00537 and 13-01-96055
90
lnV
-2900 -2800 -2700 -2600 -2500 -2400 -2300
-1.355
-1.360
-1.365
-1.370
-1.375
-1.380
-1.385
-1.390
I BG( )!,ln ,V
with�101�cross-sections
I( )n!,lnV�BG,
I BG( )min!,lnV,
I ,B( )!,lnV Gmax
!
a) K3
K2
K1=0
K1=�0.00133�(maximal�value)
point-wisesection�formaximalvalue)
0.00057500
0.00195
0.04
cross-sectionfor�minimal
outer�minimalbox-approximation
b)
In the second practical example, the process is described by thekinetic system of ordinary di↵erential equations:x1
= �K1
x1
x2
�K2
x1
, x2
= �K1
x1
x2
�K3
x2
x3
, x3
= �K2
x1
�K3
x2
x3
,where x
1
, x2
, x3
is the phase vector, K1
, K2
, K3
are parameters to beestimated. To construct the informational set, the three–dimensional(on parameters K
1
, K2
, K3
) grid-wise approach was used. The infor-mation set I(K
1
, K2
, K3
) is represented (Fig.1b) by a collection of two-dimensional cross-sections {I(K
1,k, K2,m, K3,n)}, k = 1, 51, m = 1, 101,n = 1, 101 on the tree-dimensional grid. The cross-sections are non-convex polygons with approximate grid-wise boundary. Note that thegrid techniques allows to construct this collection to be a “practicallymaximal” internal approximation of the set I(K
1
, K2
, K3
).Note that both used approaches works significantly faster and gives
more accurate results than ones based on application of usual box-parallelotopes [1,2].
References:[1] E. Hansen, G.W. Walster, Global Optimization Using Interval Analysis,
Marcel Dekker Inc., New York, USA, 2004.
[2] Shary S.P., Finite–Dimensional Interval Analysis, Electronic Book, 2013,http://www.nsc.ru/interval/Library/InteBooks
[3] S.I. Kumkov, Yu.V. Mikushina, Interval Approach to Identification of Cat-alytic Process Parameters, Reliable Computing 2014; 19, issue 2: 197-214. Re-liable Computing, 19 (2014), No. 2, pp. 197–214.
[4] S.I. Kumkov, Procession of experimental data on ionic conductivity of moltenelectrolyte by the interval analysis methods, Rasplavy, (2010), No. 3, pp. 86–96.
91
Replacing branches by polynomials invectorizable elementary functions
Olga Kupriianova, Christoph Lauter
Sorbonne Universites, UPMC Univ Paris 06, UMR 7606, LIP6,F-75005 Paris, France
4 place Jussieu 75252 PARIS CEDEX 05{olga.kupriianova, christoph.lauter}@lip6.fr
Keywords: vectorizable code, interpolation polynomial, elementaryfunctions, linear tolerance problem
The collection of codes that give the value of the elementary mathe-matical function (e.g. sin, exp) is called a mathematical library. Thereare several existing examples of such libraries, libms, but they all con-tain only manually implemented codes, i.e. written long time ago andnon-adapted for particular tasks [1]. The existing implementations area compromise between speed, accuracy and portability [2]. In order toproduce di↵erent flavors of implementation for each elementary func-tion (e.g. fast or precise), we use Metalibm1, an academic prototypefor a parametrized code generator of mathematical functions.
Metalibm splits the specified domain I for the function implemen-tation in order to reduce argument range [3], hence we get the splitting{Ik}
0kn�1
. Then on each of the subdomains Metalibm approximatesthe given function with a minimax polynomial. Thus, in order to getthe value f(x) for a particular input x, one has to get the correspond-ing polynomial coe�cients. This means first determinating the indexk 2 Z of the subinterval: Ik 3 x, 8x 2 I. Typically this is done withif -statements. To avoid branches on this step we propose to build amapping function P (x) that returns the needed interval index k.
We propose to build a continuous function p(x) such that bp(x)c =P (x). The splitting intervals can be represented as Ik = [ak, ak+1
]. As
1http://lipforge.ens-lyon.fr/www/metalibm/
92
we want the function to verify bp(x)c = P (x), we get the followingconditions for its values:
p(x) 2 [k, k + 1) when ak x ak+1
, 0 k n� 1. (1)
Among the continuous functions we choose the interpolation polynomi-als which pass through the split points and take the values p(ak) = k.However, classical interpolation theory does not allow us to take intoaccount the conditions (1), so they are checked a posteriori. This veri-fication can reliably be done using interval arithmetic implemented inSollya2.
First testing results of the proposed method show that it is possibleto build a mentioned mapping function with an interpolation polyno-mial and a posteriori condition check. However, sometimes the valuesof the polynomial exceed the required bounds.
In order to take into account the conditions (1) a priori we couldalso find the tolerable solution set [3] of the interval system of linearalgebraic equations Ac = k, where A is Vandermonde matrix com-posed of splitting intervals, k is an interval vector of allowable valuesfor the function. The system solution c will give the set of the possiblepolynomial coe�cients. Constructing polynomials by solving lineartolerance problem [3] is left for future work.
References:
[1] Jean-Michel Muller, Elementary Functions: Algorithms andImplementation, Birkhauser, Boston, 1997.
[2] Christoph Lauter, Arrondi correct de fonctions mathematiques.Fonctions univaries et bivaries, certification et automatisation, phDthesis, ENS de Lyon, 2008.
[3] Sergey P. Shary, Solving the linear interval tolerance problem,Mathematics and Computers in Simulation. – 1995. – Vol. 39. –P. 53-85.
2http://sollya.gforge.inria.fr/
93
Verified lower eigenvalue bounds forself-adjoint di↵erential operators
Xuefeng Liu and Shin’ichi OishiResearch Institue of Science and Engineering, Waseda University
3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, [email protected]
Keywords: Eigenvalue bounds, finite element method, self-adjointdi↵erential operators
By using finite element methods(FEM), we develop a new theoremto give verified eigenvalue bounds for generally defined self-adjoint dif-ferential operators, which includes the Laplace operator, the Bihar-monic operators and so on. The explicit a priori error estimationsfor conforming and non-conforming FEMs play an import role in con-structing explicit lower eigenvalue bounds. As a feature of proposedtheorem, it can even give bounds for eigenvalues that the correspondingeigenfunctions may have a singularity.
We consider the eigenvalue problem in an abstract form. Let Vbe a Hilbert function space and V h be a finite dimensional space,Dim(V h) = n. Here, V h may not be a subspace of V . Suppose M(·, ·),N(·, ·) are semi-positive symmetric bilinear forms on both V and V h.Moreover, for any u 2 V or V h, N(u, u) � 0 implies u = 0. Definenorm | · |N and semi-norm | · |M by, | · |M :=
pM(·, ·), | · |N :=
pN(·, ·).
We consider an eigenvalue problem defined by the bilinear formsM(·, ·)and N(·, ·): Find u 2 V and � 2 R such that,
M(u, v) = �N(u, v) 8v 2 V. (1)
With proper setting of M and N , the main theorem to provide lowereigenvalue bounds is given as below.
Theorem 1 Let Ph : V ! V h be a projection such that,
M(u� Phu, vh) = 0 8vh 2 V h , (2)
94
along with an error estimation as
|u� Phu|N Ch|u� Phu|M . (3)
Then we have lower bounds for �k’s,
�h,k1 + �h,kC2
h
�k (k = 1, 2, · · · , n) . (4)
The concrete form of projection Ph and the value of constant Ch,which tends to 0 as mesh size h ! 0, are depending on the FEM spacesin use. In case of the eigenvalue problems of Laplacian, a new methodbased on hyper-circle equation for conforming FEM spaces is devel-oped to give an explicit bound for the constant Ch; see [1]. Generally,by using proper non-conforming FEMs, the projection Ph is just aninterpolation operator and the constant Ch can be easily obtained byconsidering the interpolation error estimation on local elements.
To impove the precision of eigenvalue bounds, we combine lowereigenvalue bounds of Theorem 1 and Lehmann-Goerisch’s theorem togive sharp bounds; see [2]. Also, the explicit a priori error estimationfor FEMs has been successfully applied to verify the solution existencefor semi-linear elliptic partial di↵erential equations [3].
References:
Xuefeng Liu and Shin’ichi Oishi, Verified eigenvalue evaluationfor Laplacian over polygonal domain of arbitrary shape, SIAM J.Numer. Anal., 51(3), 1634-1654, 2013.
Xuefeng Liu and Shin’ichi Oishi, Guaranteed high-precisionestimation for P
0
interpolation constants on triangular finite el-ements, Japan Journal of Industrial and Applied Mathematics,30(3), 635-652, 2013.
Akitoshi Takayasu, Xuefeng Liu and Shin’ichi Oishi, Verifiedcomputations to semilinear elliptic boundary value problems onarbitrary polygonal domains, NOLTA, IEICE, Vol.E96-N, No.1,pp.34-61, Jan. 2013.
95
Towards the possibility of objectiveinterval uncertainty in physics. II
Luc Longpre and Vladik KreinovichUniversity of Texas at El Paso
El Paso, TX 79968, [email protected], [email protected]
Keywords: algorithmic randomness, interval uncertainty, quantumphysics
Applications of interval computations usually assume that whilewe only know an interval [x, x] containing the actual (unknown) valueof a physical quantity x, there is the exact value x of this quantity –and that in principle, we can get more and more accurate estimates ofthis value. This assumption is in line with the usual formulations ofphysical theories – as partial di↵erential equations relating exact valuesof di↵erent physical quantities, fields, etc., at di↵erent spatial locationsand moments of time; see, e.g., [2]. Physicists know, however, thatdue, e.g., to Heisenberg’s uncertainty principle, there are fundamentallimitations on how accurately we can determine the values of physicalquantities [2, 5].
One of the important principles of modern physics is operationalism– that a physical theory should only use observable quantities. Thisprinciple is behind most successes of the 20 century physics, startingwith relativity theory (vs. un-observable aether) and quantum mechan-ics. From this viewpoint, it is desirable to avoid using un-measurableexact values and modify physical theories so that they explicitly takeobjective uncertainty into account.
According to quantum physics, we can only predict probabilities ofdi↵erent events. Thus, uncertainty means that instead of exact valuesof these probabilities, we can only determine intervals; see, e.g., [3].
From the observational viewpoint, a probability measure meansthat we observe a sequence which is random (in Kolmogorov-Martin-Lof (KML) sense) relative to this measure. What we thus need is
96
the ability to describe a sequence which is random relative to a set ofpossible probability measures. This is not easy: in [1, 4], we have shownthat in seemingly reasonable formalizations, every random sequence isactually random relative to one of the original measures. Now we knowhow to overcome this problem: for example, for a sequence of events!1
!2
. . . occurring with the interval probability [p, p], we require thatthis sequence is random relative to a product measure correspondingto some sequence of values pi 2 [p, p] – and that it is not random inthis sense for any narrower interval. We show that this can be achievedwhen lim inf pi = p and lim sup pi = p.
We also analyze what will happen if we take into account that inphysics, not only events with probability 0 are physically impossible(this is the basis of KML definition), but also events with very smallprobability are impossible (e.g., it is not possible that all gas moleculeswould concentrate, by themselves, in one side of a vessel).
References:
[1] D. Cheu, L. Longpre, Towards the possibility of objective in-terval uncertainty in physics, Reliable Computing, 15(1) (2011),pp. 43–49.
[2] R. Feynman, R. Leighton, M. Sands, Feynman Lectures onPhysics, Basic Books, New York, 2005.
[3] I.I. Gorban, Theory of Hyper-Random Phenomena, UkrainianNational Academy of Sciences Publ., Kyiv, 2007 (in Russian).
[4] V. Kreinovich, L. Longpre, Pure quantum states are funda-mental, mixtures (composite states) are mathematical construc-tions: an argument using algorithmic information theory, Interna-tional Journal on Theoretical Physics, 36(1) (1997) pp. 167–176.
[5] L. Longpre, V. Kreinovich, When are two wave functions dis-tinguishable: a new answer to Pauli’s question, with potential ap-plication to quantum cosmology, International Journal of Theoret-ical Physics, 47(3) (2008), pp. 814–831.
97
How much for an interval? a set?a twin set? a p-box? a Kaucher interval?An economics-motivated approach todecision making under uncertainty
Joe Lorkowski and Vladik Kreinovich
University of Texas at El PasoEl Paso, TX 79968, USA
[email protected], [email protected]
Keywords: decision making, interval uncertainty, set uncertainty,p-boxes
There are two main reasons why decision making is di�cult. First,we need to take into account many di↵erent factors, there is usually atrade-o↵. For example, shall we stay in a slightly better hotel or in areasonably good cheaper one?
But even when we know how to combine di↵erent factors into asingle objective function, decision making is still di�cult because ofuncertainty. For example, when deciding on the best way to investmoney, the problem is that we are not certain which financial instru-ment will lead to higher returns.
Let us use economic ideas to solve such economic problems: namely,let us assign a fair price to each case of uncertainty.
What does “fair price” mean? One of the reasonable properties isthat if v is a pair price for an instrument x and v0 is a fair price for aninstrument x0, then the fair price for a combination x+x0 of these twoinstruments should be equal to the sum of the prices.
In [3], this idea was applied to interval uncertainty [4], for whichthis requirement takes the form v([x, x]+[x0, x0]) = v([x, x])+v([x0, x0]).Under reasonable monotonicity conditions, all such functions have theform v([x, x]) = ↵·x+(1�↵)·x for some ↵ 2 [0, 1]; this is a well-knownHurwicz criterion.
98
In this talk, we show that for sets S, we similarly get v(S) =↵ · supS + (1� ↵) · inf S.
For probabilistic uncertainty, for large N , buying N copies of thisrandom instrument is equivalent to buying a sample ofN values comingfrom the corresponding probability distribution. One can show thatfor this type of uncertainty, additivity implies that the fair price shouldbe equal to the expected value µ.
A similar idea can be applied to finding the price of a p-box (see,e.g., [1, 2]), a situation when, for each x, we only know an interval[F (x), F (x)] containing the actual (unknown) value
F (x) = Prob(⌘ x)
of the cumulative distribution function. In this case, additivity leadsto the fair price ↵ · µ+ (1� ↵) · µ, where [µ, µ] is the range of possiblevalues of the mean µ.
We also come up with formulas describing fair price of twins (in-tervals whose bounds are only known with interval uncertainty) andof Kaucher (improper) intervals [x, x] for which x > x.
References:
[1] S. Ferson, Risk Assessment with Uncertainty Numbers: RiskCalc,CRC Press, Boca Raton, Florida, 2002.
[2] S. Ferson et al., Experimental Uncertainty Estimation and Statis-tics for Data Having Interval Uncertainty, Sandia National Labo-ratories, Report SAND2007-0939, May 2007; available ashttp://www.ramas.com/intstats.pdf
[3] J. McKee, J. Lorkowski, T. Ngamsantivong, Note on FairPrice under Interval Uncertainty, Journal of Uncertain Systems,(8) 2014, to appear.
[4] R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction toInterval Analysis, SIAM, Philadelphia, 2009.
99
A workflow for modeling, visualizing, andquerying uncertain (GPS-)localization
using interval arithmetic
Wolfram Luther
University of Duisburg-EssenLotharstrasse 63, 47057 Duisburg, Germany
Keywords: GPS-localization, Dempster-Shafer theory, query language,3D visualization
A number of applications use GPS-based localization for a varietyof purposes, such as navigation of cars, robots or to localize images.Global Positioning System receivers usually report an error magni-fication factor as a ratio of output variables and input parametersas Geometric/Positional/Horizontal/Vertical/ Time Dilution of Preci-sion (XDOP) using one- to four-dimensional coordinate systems. Overtime, various algorithmic approaches have been developed to compen-sate for errors due to environmental disturbances and uncertain pa-rameters occurring in GPS signal measurement, such as an adaptiveKalman filter-based approach that was implemented using a fuzzy logicinference system, or by combining GPS measurements with furthersensory data. In terms of querying GPS-based data, less work can befound that takes the uncertain characteristic of GPS data into account,especially if semantic querying mechanisms are involved.
In this talk we highlight new verified method for uncertain (GPS-)localization based on Dempster-Shafer theory (DST) [1], with multidi-mensional and interval-valued basic probability assignments (nDIBPA)and masses estimated via statistical observations or/and expert knowl-edge. In order to define and work with these focal elements, we ex-tended the Dempster-Shafer with Intervals (DSI) toolbox by addingfunctions that provide the capability to compute imprecise(GPS-)localization.
100
Besides aggregation and normalization methods on 2DIBPAs aswell as plausibility (PL) and belief (BEL) interval functions, the DSItoolbox provides discrete interval generalizations of the normal andWeibull distribution with compact support and interval-valued param-eters to model the error of a GPS measurement(http://udue.de/DSIexamples). Thus, an adequate error distributionmeasured in radians, (x, y)- or (x, y, z)-coordinates can be assumedvia nDIBPAs, n = 1, 2, 3, using expert knowledge based on long-timemeasurements or results reported in the literature, i.e., the Weibulldistribution for radial GPS position error.
In [2], we provide an extended example concerning localization andalignment of a truck equipped with two GPS sensors at a given dis-tance. Roughly speaking, the grid approach using 2DIBPAs is similarto the Riemann integral concept using upper and lower sums. We askfor inner and outer domains I and O, which contain all possible lo-calizations of the object with a given high probability, and alignmentsenclosed by a contour C situated in the shape S := O\I. The thicknessof the shape S depends on the sample size and the computing errorswhen rounding to ±1. By summing up the lower mass bounds of allfocal elements in I and the upper bounds for focal elements constitut-ing O, we acquire an enclosure for the BEL(Y ) and PL(Y ).
To dynamically generate X3D scenes, we use the Replicave frame-work [3] –a Java based X3D and X3DOM toolkit for modeling of 3Dscenes–which can be interfaced by C and C++ via the Java NativeInterface (JNI) together with a layered visualization approach for vi-sualizing map and terrain data and multiple 2D/3D overlays and gridsfor visualizing information as demonstrated in [4].
To support queries of the type Are objects A and B in space C attime T with a plausibility of 50 percent, we extended the GeoSPARQLstandard based on the ability of introducing DST models into the on-tology as features of spatial objects and the set of custom SPARQLand/or extensions of GeoSPARQL functions that should be able toevaluate queries with uncertainty [5].
The main benefit our approach o↵ers for GIS applications is aworkflow concept using DST-based models that are embedded into
101
an ontology-based semantic querying mechanism accompanied by 3Dvisualization techniques. This workflow provides an interactive wayof semantically querying uncertain GIS models and providing visualfeedback.
Our future work consists in the final implementation and extensionof the di↵erent components of the workflow. Thanks to Nelson Baloian,Gabor Rebner, Daniel Sacher, and Benjamin Weyers for preparing thematerial during a stay at the University of Chile and to the DFG andthe DAAD for funding this cooperation within the SADUE13 and thePRASEDEC projects.
References:
[1] G. Rebner, D. Sacher, W. Luther, Verified stochastic meth-ods: The evolution of the Dempster-Shafer with intervals (DSI)toolbox, Taylor & Francis, London, 2013, pp. 541–548.
[2] G. Rebner, D. Sacher, B. Weyers, W. Luther, Verifiedstochastic methods in geographic information system applicationswith uncertainty, to appear in Structural Safety.
[3] D. Biella, W. Luther, D. Sacher, Schema migration intoa web-based framework for generating virtual museums and lab-oratories, 18th International Conference on Virtual Systems andMultimedia (VSMM) (2012), pp. 307–314.
[4] F. Calabrese, C. Ratti, Real Time Rome, Networks and Com-munication Studies, (2006), No. 3 & 4, pp. 247–258.
[5] Open Geospatial Consortium, OGC GeoSPARQL – A Geo-graphic Query Language for RDF Data,http://www.opengis.net/doc/IS/geosparql/1.0
102
Using range arithmetic in evaluation ofcompact models
Amin Maher1 and Hossam A. H. Fahmy2
1 Deep Submicron Division, Mentor Graphics CorporationCairo, Egypt, 11361
amin [email protected] and Communications Engineering, Cairo University
Giza, Egypt, [email protected]
Keywords: range arithmetic, interval arithmetic, a�ne arithmetic,compact models, circuit simulation, Monte Carlo simulation, designvariability
At nowadays semiconductor technologies, electronics circuits per-formance is a↵ected by process variation. To accommodate for thesevariation at design stage, it is normal to simulate the design severaltimes for several process corners. As well doing Monte Carlo simula-tion to take random parameters variation into consideration. Largenumber of runs are needed to have good results with Monte Carlo sim-ulation. As an alternative to these simulations, range arithmetic maybe used to simulate variations in parameters.
Approaches use range arithmetic in circuit simulation, show goodresults[1][2]. To complete the simulation flow, a set of device mod-els should be available. These models should be able to work withrange arithmetic based simulator. For high level compact models, likeBSIM4, replacing the floating point calculation with interval ones isnot enough. Re-writing parts of the model is necessary to providecorrect results with the interval calculations.
In this work we evaluate compact models using range arithmetic.We compare the results for accuracy, e�ciency and reliability whenusing di↵erent representation of range arithmetic, interval and a�nearithmetic. Results are tested against point intervals data and Monte-Carlo simulations.
103
References:
[1] Grabowski, Darius, Markus Olbrich, and Erich Barke.”Analog circuit simulation using range arithmetics.” In Design Au-tomation Conference, 2008. ASPDAC 2008. Asia and South Pa-cific, pp. 762-767. IEEE, 2008.
[2] Tang, Qian Ying, and Costas J. Spanos. ”Interval-valuebased circuit simulation for statistical circuit design.” In SPIE Ad-vanced Lithography, pp. 72750J-72750J. International Society forOptics and Photonics, 2009.
104
Finding positively invariant sets ofordinary di↵erential equations usinginterval global optimization methods
Mihaly Csaba Markot and Zoltan Horvath
University of ViennaOskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
Szechenyi Istvan UniversityEgyetem ter 1., H-9026 Gyor, Hungary
[email protected], [email protected]
Keywords: global optimization, interval branch-and-bound, ODE,positive invariance
Let us consider the initial value problem y0(t) = f(y(t)), t 0,y(0) = y
0
, where f : V ✓ RN ! RN is continuously di↵erentiable, andassume that a unique solution exists for all u
0
2 V . A set C ✓ V iscalled positively invariant w.r.t. f , if for all u
0
2 C the solution u(t)stays in C for all t � 0. If C is convex and closed, then a su�cient con-dition of C being positively invariant is the existence of a real positive "constant such that for all v 2 C the containment relation v+"f(v) 2 Cholds.
For the discretized case, one can introduce the concept of discretepositive invariance: Let us given f , y
0
, and a stepsize ⌧ > 0, anddenote a numerical integration scheme (e.g., one from the family ofRunga-Kutta methods) by �, i.e., y(i + 1) = �(y(i), ⌧, f), i = 0, . . .A set C ✓ V is discrete positively invariant w.r.t. � with stepsizeconstant ⌧ ⇤ > 0, if for all ⌧ 2 (0, ⌧ ⇤] and for all y(0) 2 C the relationy(i+ 1) 2 C holds for i = 0, . . . Finding discrete positively invariantsets and large stepsize constants may be essential for carrying out long-term integration in an e�cient way, especially for sti↵ problems.
In both the continuous and the discrete cases, the related basicproblem is the following: given F : V ✓ RN ! RN , C ✓ V , and aconstant " > 0, decide whether v + "F (v) 2 C for all v 2 C. That is,
105
we have to verify a mathematical property in all points of a given set,hence, for tackling the problem on a computer we need interval-basedreliable numerical algorithms. In the talk we show how to translate theabove decision problem into a set of global optimization problems whenC is box-shaped, and solve them with an interval branch-and-boundalgorithm. Furthermore, we introduce a reliable method for findingboxes that are positively invariant for given F and ", and an algorithmto find the maximal " for which a given C set is positively invariant.The applicability and e�ciency of the methods are demonstrated on(low-dimensional) sti↵ chemical reaction models.
106
A short description of the symmetricsolution set
Gunter Mayer
University of Rostock18051 Rostock, Germany
Keywords: interval linear systems, symmetric solution set, Oettli–Prager–like theorem
Given a regular real n⇥n interval matrix [A] and an interval vector[b] with n components the well–known Oettli–Prager theorem describesthe solution set S = {x 2 Rn| Ax = b, A 2 [A], b 2 [b] } by means ofthe vector inequality
|b� Ax| rad([A]) · |x|+ rad([b]), (1)
i.e., the statements ‘x 2 S ’ and ‘ (1) holds for x 2 Rn ’ are equiva-lent. Here, A, b denote the midpoints of [A], and [b], respectively, andrad([A]), rad([b]) denote their radii. Restricting A and [A] in S to besymmetric leads to the symmetric solution set
Ssym
= {x 2 Rn| Ax = b, A = AT 2 [A] = [A]T , b 2 [b] } ✓ S
which is more complicated to be described. Starting with 1995 sev-eral attempts were made to find such a description. They essentiallyrepresented a way how to end up with a set of inequalities extendingthat in (1); cf. for instance [1], [2], and [6]. It lasted up to 2008 untilHladık presented in [3] an extension comparable with (1). In [4] thefollowing more compact reformulation of [3] was stated:
x 2 Ssym
if and only if
|b� Ax| rad([A]) · |x|+ rad([b]) as in (1)
107
and
|xT (Dp �Dq)(b� Ax)| |x|T · |Dp rad([A])� rad([A])Dq| · |x|+ |x|T · |Dp �Dq| · rad([b]) (2)
for all vectors p, q 2 {0, 1}n\{0, (1, . . . , 1)T} such that p �lex
q andpT q = 0. Here Dv denotes a diagonal matrix Dv = diag(v) forv = (vi) 2 Rn and ‘�
lex
’ denotes the strict lexicographic orderingof vectors, i.e., u �
lex
v if for some index k we have ui = vi, i < k, anduk < vk. At most (3n � 2n+1 + 1)/2 inequalities are needed in (2).
In our talk we review these inequalities, outline a proof in [4] dif-ferent from that in [3] and indicate some ways from various authorshow to enclose S
sym
; cf. the survey [5].
References:
[1] G. Alefeld, G. Mayer, On the symmetric and unsymmetricsolution set of interval systems SIAM J. Matrix Anal. Appl., 16(1995), pp. 1223–1240.
[2] G. Alefeld, V. Kreinovich, G. Mayer, On the shape of thesymmetric, persymmetric and skew–symmetric solution set SIAMJ. Matrix Anal. Appl., 18 (1997), pp. 693–705.
[3] M. Hladık, Description of symmetric and skew–symmetric solu-tion set, SIAM J. Matrix Anal. Appl., 30 (2008), No. 2, pp. 509–521.
[4] G. Mayer, An Oettli–Prager–like theorem for the symmetric solu-tion set and for related solution sets, SIAM J. Matrix Anal. Appl.,33 (2012), No. 3, pp. 979–999.
[5] G. Mayer, A survey on properties and algorithms for the symmet-ric solution set, Preprint 12/2, Universitat Rostock, Preprints ausdem Institut fur Mathematik, ISSN 0948–1028, Rostock, 2012.
[6] E. Popova, Explicit description of 2D Parametric solution sets,BIT Numerical Mathematics, 52 (2012), No. 1, pp. 179–200.
108
A Simple Modified Verification Methodfor Linear Systems
Atsushi Minamihata, Kouta Sekine, Takeshi Ogita, SiegfriedM. Rump and Shin’ichi Oishi
Graduate School of Fundamental Science and Engineering,Waseda University
3–4–1 Okubo, Shinjuku-ku, Tokyo 169–8555, [email protected]
Keywords: verified numerical computations, componentwise errorbound, INTLAB,
This talk is concerned with the problem of verifying the accuracy ofapproximate solutions of linear systems. We propose a simple modifiedmethod of calculating a componentwise error bound of the computedsolution, which is based on the following Rump’s theorem:
Theorem 1 (Rump [1, Theorem 2.1]) Let A 2 Rn⇥n and b, x 2Rn be given. Assume v 2 Rn with v > 0 satisfies u := hAiv > 0. LethAi = D � E denote the splitting of hAi into the diagonal part D andthe o↵-diagonal part �E, and define w 2 Rn by
wk := max1in
Gik
uifor 1 k n,
where G := I � hAiD�1 = ED�1 � O. Then A is nonsingular, and
|A�1b� x| (D�1 + vwT )|b� Ax|. (1)
In particular, the method based on Theorem 1 is implemented in theroutine verifylss in INTLAB Version 7 [2].
We modify Theorem 1 as follows:
Theorem 2 Let A, b, x, u, v, w be defined as in Theorem 1. Definec := |b� Ax| and Ds := diag(s) where s 2 Rn with
sk := ukwk for 1 k n.
109
Then,
|A�1b� x| (D�1 + vwT )(I +Ds)�1c. (2)
Moreover,
|A�1b� x| �v + (D�1 + vwT )(I +Ds)�1(c� �u), (3)
where � := min1in
ciui.
Theorem 3 Let A, b, x, u, v, w be defined as in Theorem 1. Define� := uwT � ED�1 and c := |b� Ax|. Then,
|A�1b� x| (D�1 + vwT )(c� �(I � �)c). (4)
Moreover,
|A�1b� x| �v + (D�1 + vwT )
min�c� �u, (c� �u� �(I � �)(c� �u))
�, (5)
where � := min1in
ciui.
(3) and (5) always give better bounds than Theorem 1. Detailedproofs will be presented. Numerical results will be shown to illustratethe e�ciency of the proposed theorems.
References:
[1] S. M. Rump, Accurate solution of dense linear systems, Part II:Algorithms using directed rounding, J. Comp. Appl. Math., 242(2013), 185–212.
[2] S. M. Rump, INTLAB - INTerval LABoratory, Developments in Re-liable Computing, T. Csendes, ed., 77–104, Kluwer, Dordrecht,1999. http://www.ti3.tuhh.de/rump/
110
Fast inclusion for the matrix inversesquare root
Shinya Miyajima
Faculty of Engineering, Gifu University1-1 Yanagido, Gifu-shi, Gifu 501-1193, Japan
Keywords: numerical inclusion, matrix inverse square root, Newtonoperator
Given a nonsingular matrix A 2 Cn⇥n, a matrix X such that
AX2 = I,
where I is the n⇥n identity matrix, is called an inverse square root ofA.The matrix inverse square root always exists for A being nonsingular,and appears in important problems in science and technology, e.g., theoptimal symmetric orthogonalization of a set of vectors [1] and thegeneralized eigenvalue problem [2]. Several numerical algorithms forcomputing the matrix inverse square root have been proposed (see [1–5], e.g.). It is known that the inverse square root is not unique (see [6],in which the matrix square root is treated, but all considerations thereimmediately carry over to the matrix inverse square root). If A has nononpositive real eigenvalues, the principal inverse square root (see [1])can be defined by requiring that all the eigenvalues of X have positivereal parts. The principal inverse square root is of particular interest,since this has important applications such as the matrix sign function,the unitary polar factor and the geometric mean of two positive definitematrices (see [7], e.g.).
In this talk, we consider enclosing the matrix inverse square root,specifically, computing an interval matrix containing the inverse squareroot using floating point computations. Frommer, Hashemi and Sab-lik [8] have firstly proposed two such algorithms. In these algorithms,numerical spectral decomposition of A is e↵ectively utilized, so that
111
they require only O(n3) operations. The first algorithm computes theinterval matrix containing the inverse square root by enclosing a solu-tion of the matrix equation F (X) = 0, where F (X) := XAX � I, viathe Krawczyk operator, and guarantees the uniqueness of the inversesquare root contained in the computed interval matrix. In the secondalgorithm, an a�ne transformation of F (X) = 0 making use of thenumerical results for the spectral decomposition is adopted. Althoughthis algorithm does not verify the uniqueness of the contained inversesquare root, the numerical results in [8] show that this algorithm usu-ally computes narrower interval matrices and is successful for largerdimensions than the first algorithm. As an application of these twoalgorithms, the algorithms for enclosing the matrix sign function havealso been developed in [8].
The purpose of this talk is to propose an algorithm for enclosing thematrix inverse square root. The proposed algorithm also utilizes thespectral decomposition of A and requires only O(n3) operations. Inthis algorithm, the a�ne transformation of F (X) = 0 and the Newtonoperator is adopted in order to verify the existence of the inverse squareroot in a candidate interval matrix, and the uniqueness is also verifiedusing the nontransformed equation. This algorithm moreover verifiesthe principal property of the inverse square root uniquely containedin the computed interval matrix by utilizing the theory in [9] whichenables us to enclose all eigenvalues of a matrix. We finally reportnumerical results to observe the properties of the proposed algorithm.
References:
[1] N. Sherif, On the computation of a matrix inverse square root,Computing, 46 (1991), pp. 295–305.
[2] P. Laasonen, On the iterative solution of the matrix equationAX2 � I = 0, Math. Tables Other Aids Comput., 12 (1958),pp. 109–116.
[3] A. Borici, A Lanczos approach to the inverse square root of alarge and sparse matrix, J. Comput. Phys., 162 (2000), pp. 123–131.
112
[4] C.H. Guo, N.J. Higham, A Schur-Newton method for the matrixpth root and its inverse, SIAM J. Matrix Anal. Appl., 28 (2006),pp. 788–804.
[5] S. Lakic, A one parameter method for the matrix inverse squareroot, Appl. Math., 42 (1997), No. 6, pp. 401–410.
[6] N.J. Higham, Functions of Matrices: Theory and Computation,SIAM, Philadelphia, 2008.
[7] B. Iannazzo, B. Meini, Palindromic matrix polynomials, matrixfunctions and integral representations, Linear Algebra Appl., 434(2011), pp. 174–184.
[8] A. Frommer, B. Hashemi, T. Sablik, Computing enclosuresfor the inverse square root and the sign function of a matrix, Lin-ear Algebra Appl., (2014), http://dx.doi.org/10.1016/j.laa.2013.11.047
[9] S. Miyajima, Numerical enclosure for each eigenvalue in gener-alized eigenvalue problem, J. Comp. Appl. Math., 236 (2012),pp. 2545–2552.
113
Verified solutions of saddle point linearsystems
Shinya MiyajimaFaculty of Engineering, Gifu University
1-1 Yanagido, Gifu-shi, Gifu 501-1193, [email protected]
Keywords: verified computation, saddle point linear systems, errorestimation
In this talk, we are concerned with the accuracy of numericallycomputed solutions of saddle point linear systems
Hu = b, H :=
✓A BT
B �C
◆, u :=
✓xy
◆, b :=
✓fg
◆, (1)
where A 2 Rn⇥n, B 2 Rm⇥n, f 2 Rn and g 2 Rm are given, x 2 Rn
and y 2 Rm are to be solved, n � m, A is symmetric positive definite(SPD), B has full rank, and C is symmetric positive semi-definite,which implies that H is nonsingular. The systems (1) arise a varietyof science and engineering applications, including partial di↵erentialequations and optimization problems.
Let u⇤ = (x⇤T, y⇤
T)T and u = (xT , yT )T denote the exact and nu-
merical solution of (1), respectively. We consider in this talk verifiedcomputation of u⇤, specifically, computing rigorous upper bounds forku� u⇤k1 using floating point operations.
The pioneering work has been given by Chen and Hashimoto [1].They skillfully exploited the special structure of (1). Their result en-ables us to avoid computing an approximation of H�1. Let
✓rfrg
◆:=
✓A BT
B �C
◆✓xy
◆�✓
fg
◆
be residual vectors. They have been presented the error estimation
kx� x⇤k2
kA�1k2
(krfk2 + kBTk2
ky � y⇤k2
), (2)
ky � y⇤k2
⇣(kBA�1k2
krfk2 + krgk2), (3)
114
where
⇣ :=kAk
2
k(BBT )�1k2
1 + kAk2
k(BBT )�1k2
�min
(C)
and �min
(C) denotes the smallest singular value of C. From (2) and(3), we can obtain the upper bound for ku� u⇤k1, since ku� u⇤k1 =max(kx� x⇤k1, ky� y⇤k1) max(kx� x⇤k
2
, ky� y⇤k2
). Substituting(3) into (2), we obtain
kx� x⇤k2
kA�1k2
((1+ ⇣kBTk2
kBA�1k2
)krfk2+ ⇣kBTk2
krgk2). (4)
The important special case is when C = 0. We then have �min
(C) = 0,so that (3) and (4) give
kx� x⇤k2
kA�1k2
(1 + kAk2
kBTk2
k(BBT )�1k2
kBA�1k2
)krfk2+(A)kBTk
2
k(BBT )�1k2
krgk2, (5)
ky � y⇤k2
kAk2
k(BBT )�1k2
(kBA�1k2
krfk2 + krgk2), (6)
where (A) := kAk2
kA�1k2
. They have been proposed the verificationalgorithms based on (5) and (6).
Hashimoto [2] has also treated the case when C = 0 and improved(5) and (6). Since B has full rank, there exists a nonsingular m ⇥mmatrix LB such that LBLT
B = BBT . He has been presented the errorestimation
kx� x⇤k2
kA�1k2
krfk2 + (A)kL�1
B rgk2, (7)
kLB(y � y⇤)k2
(A)krfk2 + kAk2
kL�1
B rgk2. (8)
From (8), we have
ky�y⇤k2
kL�1
B k2
kLB(y�y⇤)k2
kL�1
B k2
((A)krfk2+kAk2
kL�1
B rgk2).(9)
The purpose of this talk is to present and propose error estima-tions and verification algorithms in (1), respectively. Since A is SPD,there exists a nonsingular n⇥ n matrix LA such that LALT
A = A. LetL�1
A BT = QR be thin QR factorization of L�1
A BT , where Q 2 Rn⇥m
is column-orthogonal and R 2 Rm⇥m is upper triangular. Since LA is
115
nonsingular and B has full rank, R is also nonsingular. We then derivethe error estimation
kx� x⇤k2
kL�1
A k2
kL�1
A rfk2 + ⌘kA�1BTk2
krgk2, (10)
ky � y⇤k2
⌘(kBA�1rfk2 + krgk2), (11)
where
⌘ :=kR�1k2
2
1 + kR�1k22
�min
(C).
Let R�T := (R�1)T . When C = 0, we present the error estimation
kx� x⇤k2
kL�1
A k2
(kL�1
A rfk2 + kR�Trgk2), (12)
ky � y⇤k2
kR�Tk2
kL�1
A rfk2 + kR�1R�Trgk2. (13)
Let "CH , �CH , �CH0, "CH0, �H , "H , �M , "M , �M0 and "M0 be the righthand sides of (3), (4), (5), (6), (7), (9), (10), (11), (12) and (13),respectively. We prove �M �CH , "M "CH , �M0 �H �CH0 and"M0 "H "CH0, and propose the verification algorithms based on(12) and (13). These algorithms do not assume but prove that A isSPD and B has full rank. Numerical results are finally reported toshow the properties of the proposed algorithms.
References:
[1] X. Chen, K. Hashimoto, Numerical validation of solutions ofsaddle point matrix equations, Numer. Linear Algebra Appl., 10(2003), pp. 661–672.
[2] K. Hashimoto, A preconditioned method for saddle point prob-lems, MHF Preprint Series, MHF 2007-6 (2007), http://www2.math.kyushu-u.ac.jp/coe/report/pdf/2007-6.pdf
116
A method of verified computations fornonlinear parabolic equations
Makoto Mizuguchi1, Akitoshi Takayasu1, Takayuki Kubo2,and Shin’ichi Oishi1,3
1Waseda University, 2University of Tsukuba, 3CREST JST
1698555 Tokyo, Japan,3050006 Ibaraki, Japan
Keywords: parabolic initial-boundary value problems, verified com-putations, existence, error bounds
Let ⌦ be a bounded polygonal or polyhedral domain in Rd (d =1, 2, 3). Let V := H1
0
(⌦), X := L2(⌦) and V ⇤ := H�1(⌦). A dualproduct between V and V ⇤ is defined by h·, ·i. The inner product inX is denoted by (·, ·)X . In this talk, we consider the initial-boundaryvalue problem of heat equations:
8<
:
@tu� �u = f(u) in (0,1)⇥ ⌦,u(t, x) = 0 on (0,1)⇥ @⌦,u(0, x) = u
0
(x) in ⌦,(1)
where @tu := dudt , f : V ! X is a Frechet di↵erentiable nonlinear
function, and u0
2 X is a given initial function. A : V ! V ⇤ isdefined by hAu, vi := (ru,rv)X for all v 2 V . Here, �A generatesan analytic semigroup {e�tA}t�0
. Letting n 2 N be a fixed naturalnumber, we divide the time: 0 = t
0
< t1
< · · · < tn < 1. Fork = 1, 2, ..., n, we define Tk = (tk�1
, tk] and T =S
Tk. The main aimof this paper is to present a computer-assisted method of verifying localexistence and uniqueness of exact solution of (1) in the function space:
L1(T ;V ) :=
⇢u : ess sup
t2Tku(t)kV < 1
�.
117
Let uk ⇡ u(tk) be the approximate solution by the finite elementmethod and the backward Euler method [1]. We construct an ap-proximate solution ! denoted by
!(t) :=nX
k=1
uk�k(t), t 2 T,
where �k(t) is a piecewise linear Lagrange basis in Tk. We have thefollowing result with the semigroup thory and Banach’s fixed-pointtheorem.
Theorem 1 (Verification principle) For t 2 Tk and v(t) 2 V , letBk(v, ⇠) be a ball centered at v with radius ⇠ in the norm k · kL1
(Tk;V )
.Suppose that we have constants Lk(v, ⇠), �k and "k, defined by
Lk(v, ⇠) := supy2Bk(v,⇠),
w2V, kwkV =1
kf 0[y]wkL1(Tk;X)
,
�k :=
����uk � uk�1
⌧k+Auk � f(uk)
����V ⇤
, and "k := kuk � uk�1
kV .
Also let us assume that �min
> 0 is the minimal eigenvalue of A in V ⇤.
We denote ⌘ > 0 by ⌘ := 2p
⌧ke Lk(uk, "k)"k + �k +
⇣1 + 1�e�⌧k�min
⌧k�min
⌘"k +
⇢k�1
, where e is Napier’s constant, ⇢0
= 0. If ⇢k > 0 satisfies2p
⌧ke Lk(!, ⇢k)⇢k + ⌘ < ⇢k, then the solution u(t) of (1) uniquely exists
in the ball Bk(!, ⇢k).
References:
[1] V. Thomee, Galerkin Finite Element Methods for Parabolic Prob-lems, Springer, Berlin, 1997.
[2] A. Pazy, Semigroups of linear operators and applications to partialdi↵erential equations, Springer, New York, 1983.
118
A sharper error estimate of verifiedcomputations for nonlinear heat
equations.
Makoto Mizuguchi1, Akitoshi Takayasu1, Takayuki Kubo2,and Shin’ichi Oishi1,3
1Waseda University, 2University of Tsukuba, 3CREST JST
1698555 Tokyo, Japan,3050006 Ibaraki, Japan
Keywords: parabolic initial-boundary value problems, computer-assisted proof, rigorous error estimate
Let ⌦ be a bounded polygonal or polyhedral domain in Rd (d =1, 2, 3). Let V := H1
0
(⌦), X := L2(⌦) and V ⇤ := H�1(⌦). A dualproduct between V and V ⇤ is defined by h·, ·i. The inner product inX is denoted by (·, ·)X . In this talk, we consider the initial-boundaryvalue problem of heat equations:
8<
:
@tu� �u = f(u) in (0,1)⇥ ⌦,u(t, x) = 0 on (0,1)⇥ @⌦,u(0, x) = u
0
(x) in ⌦,(1)
where @tu := dudt , f : V ! X is a Frechet di↵erentiable nonlinear
function, and u0
2 X is a given initial function. A : V ! V ⇤ is definedby hAu, vi := (ru,rv)X for all v 2 V . Letting n 2 N be a fixednatural number, we divide the time: 0 = t
0
< t1
< · · · < tn < 1.For k = 1, 2, ..., n, we define Tk = (tk�1
, tk] and T =STk. Let uk ⇡
u(tk) be a fully discretized approximate solution obtained by the finiteelement method and the backward Euler method [1]. We define anapproximate solution ! by
!(t) :=nX
k=1
uk�k(t), t 2 T,
119
where �k(t) is a piecewise linear Lagrange basis in Tk. By using !(t),we have established a computer-assisted proof of local existence anduniqueness of u(t). In this method, the precision of the error estimateis slightly rough.
The topic of this talk is to derive a shaper error estimate by addingsome assumptions that the initial data u
0
2 V and f 0, which denotesby a Frechet derivative of f , is a local continuous function. The keypoint of this talk is to use an ideal approximation: u(t) defined by
u(t) :=nX
k=1
uk�k(t), t 2 T,
where uk 2 V satisfies an elliptic equation:✓uk � uk�1
⌧k, v
◆
X
+ (ruk,rv) = (f(uk), v)X , 8v 2 V.
Then, we divide the error estimate into the following two parts:
ku� !kL1(Tk;V )
ku� ukL1(Tk;V )
+ ku� !kL1(Tk;V )
.
First, we rigorously construct the ideal approximation u(t) using theframework of verified computations for elliptic equations, e.g. M. Plum[3]. Next, by using u, a local existence and uniqueness of u(t) is val-idated by a computer-assisted method depending on Banach’s fixed-point theorem and the semigroup theory [2]. Then, the sharper errorestimate is provided.
References:
[1] V. Thomee, Galerkin Finite Element Methods for Parabolic Prob-lems, Springer, Berlin, 1997.
[2] A. Pazy, Semigroups of linear operators and applications to partialdi↵erential equations, Springer, New York, 1983.
[3] M. Plum, Existence and multiplicity proofs for semilinear ellip-tic boundary value problems by computer assistance, Jahresber.Dtsch. Math.-Ver., 110 (2008), pp. 19–54.
120
An Interval arithmetic algorithm forglobal estimation of hidden Markov
model parameters
Tiago Montanher and Walter Mascarenhas
University of Sao Paulo1010 Matao St. Sao Paulo, SP. Brazil
Keywords: Hidden Markov models, interval arithmetic, linear pro-gramming bounds
Hidden Markov Models are important tools in statistics and appliedmathematics, with applications in speech recognition, physics, mathe-matical finance and biology. The Hidden Markov Models we considerhere are formed by two discrete time and finite state stochastic pro-cess. The first process is a Markov chain (A, ⇡) and is not observabledirectly. Instead, we observe a second process B which is driven bythe hidden process. For instance, a Markov chain is a simple HiddenMarkov Model in which the observed process and the hidden processare the same. These models have received much attention in the lit-erature in the past forty years, and the book by Cappe[2] presents agood didactic overview on the topic. From a historical perspective,the seminal paper by Rabiner[1] provides a good motivation for thissubject.
In order to extract conclusions from a Hidden Markov Models wemust estimate the parameters defining the hidden process (A, ⇡) andthe observed process B. In this article we present e�cient global op-timization techniques to estimate these parameters by maximum like-lihood and compare our estimates with the ones obtained by the locallikelihood maximization methods already described in the literature.Usually, this estimation problem is solved by local methods, like theBaum-Welch algorithm. These methods are e�cient, however theyonly find local maximizers and do not estimate the distance from the
121
resulting parameters to global optima. Our work aims to improve thissituation in practice.
We develop a global optimization algorithm based on the classicalinterval branch and bound framework described by Kearfott[3]. In asuccessful execution, the algorithm is able to find a box with prescribedwidth which rigorously contains at least one feasible point x⇤ for theproblem and such that x⇤ is a ✏�-global maximum. The objectivefunction and its derivatives are evaluated by the so called backwardrecursion presented on Rabiner’s work. In order to obtain sharper esti-mates of functions we do not evaluate them using the natural intervalextension. Instead, at each evaluation we solve a set of small linearprograms given by the backward recursion. We also try to improve thelower bound for the maximum implementing a multi-start Baum-Welchprocedure. To handle the underflow problems which arise frequentlyin the estimation problem for Hidden Markov models we derive a newscaling scheme based on C + + functions scalbln and frexp. Thisapproach is significantly di↵erent from the literature where authorssuggests to take log of the objective function. We present numericalexperiments illustrating the e↵ectiveness of our method.
References:
[1] Rabiner, Lawrence R., A tutorial on hidden markov modelsand selected applications in speech recognition, Proceedings of theIEEE, 1989, pp. 257–286
[2] Cappe, Olivier and Moulines, Eric and Ryden, Tobias,Inference in Hidden Markov Models (Springer Series in Statistics),Springer-Verlag New York, Inc., 2005
[3] Kearfott, R. B., Rigorous Global Search: Continuous Problems,Kluwer Academic Dordrecht, 1996
122
Toward hardware support forReproducible Floating-Point
ComputationHong Diep Nguyen, James Demmel
University of California, BerkeleyBerkeley, CA 94720, USA
{hdnguyen,demmel}@eecs.berkeley.edu
Keywords: Reproducibility, Floating-point arithmetic, Hardware im-plementation, Parallel Computation
Reproducibility is the ability to compute bit-wise identical resultsfrom di↵erent runs of the same program on the same input data. Thisis very important for debugging and for understanding the reliability ofthe program. In a parallel computing environment, especially on verylarge-scale systems, it is usually not possible to control the availablecomputing resources such as the processor count and the reduction treeshape. Therefore the order of evaluation di↵ers from one run to anotherrun of the same program, which leads to di↵erent computed results dueto the non-associativity of floating-point addition and multiplication.
In a previous paper [1] we proposed the pre-rounding techniquefor reproducible summation regardless of the available computing re-sources such as processor count, reduction tree shape, data partition,multimedia instruction set (SSE, AVX), etc. This technique has beenimplemented in a production library, ReproBLAS [2], which currentlysupports SSE instructions and MPI computing environment. Exper-imental results showed that on large-scale systems, for example ona CRAY XC30 machine with more than 1024 processors, the repro-ducible sum runs only 20% slower than the performance-optimizednonreproducible sum. On a single processor, however, the reproduciblesum can be up to 8 times slower than the performance-optimized non-reproducible sum. That is because the pre-rounding technique is im-plemented using extra floating-point operations for multiple passes oferror-free vector transformation.
123
For the purpose of performance enhancement, having hardware sup-port will help to reduce the additional cost of extra floating-point op-erations needed. In this paper, we propose a new instruction for re-producible add, which ideally can be issued at every clock cycle, whichwill reduce the cost of reproducible summation to almost as small asa normal nonreproducible sum on a single processor. In comparisonwith using a long accumulator, which can also provide reproducibilityby computing exact dot product and summation, our new instructionexhibits the following advantages:
• the new instruction operates on the existing register file instead ofhaving to implement a special accumulator unit,
• it does not require drastic changes to the existing scheduling sys-tem,
• it requires less memory space, hardware area as well as energyconsumption,
• the new instruction can be well pipelined and multi-threaded.
In this talk, I will present some preliminary results of this on-goingwork of hardware implementation. First, I will present the sketchof the hardware layout in order to implement the reproducible addinstruction. Then I will show some experimental results to demonstratethat the chosen hardware configuration is su�cient to obtain goodaccuracy. Finally I will discuss some possible future work.
References:
[1] J. Demmel, H.D. Nguyen, Fast Reproducible Floating-PointSummation, ARITH 21, Austin, Texas, April 7-10, 2013.
[2] ReproBLAS, Reproducible Basic Linear Algebra Subprograms,http://bebop.cs.berkeley.edu/reproblas.
124
Accurate and e�cient implementationof a�ne arithmetic
using floating-point arithmetic
Jordan Ninin and Nathalie Revol
J. Ninin : IHSEV team, LAB-STICC, ENSTA-Bretagne2 rue Francois Verny, 29806 Brest, France
N. Revol: INRIA - Universite de Lyon - AriC teamLIP (UMR 5668 CNRS - ENS de Lyon - INRIA - UCBL)
ENS de Lyon, 46 allee d’Italie, 69007 Lyon, [email protected]
Keywords: interval arithmetic, a�ne arithmetic, floating-point arith-metic, roundo↵ error
A�ne arithmetic is one of the extensions of interval arithmeticthat aim at counteracting the variable dependency problem. Witha�ne arithmetic, defined in [5] by Stolfi and Figueiredo, variables arerepresented as a�ne combination of symbolic noises. It di↵ers fromthe generalized interval arithmetic, defined by Hansen in [1], wherevariables are represented as a�ne combination of intervals. Non-a�neoperations are realized through the introduction of a new noise, thataccounts for nonlinear terms. Variants of a�ne arithmetic have beenproposed, they aim at limiting the number of noise symbols. Let usmention [4] by Messine and [6] by Vu, Sam-Haroud and Faltings toquote only a few.
The focus here is on the implementation of a�ne arithmetic usingfloating-point arithmetic, specified in [2]. With floating-point arith-metic, an issue is to handle roundo↵ errors and to incorporate them inthe final result, so as to satisfy the inclusion property, which is the fun-damental property of interval arithmetic. In [4], [5] and [6], roundo↵errors are accounted for in a manner that implies frequent switches ofthe rounding mode; this incurs a severe time penalty. Implementationsof these variants are available in YalAA, developed by Kiel [2].
125
We propose an implementation that uses one dedicated noise sym-bol for accumulated roundo↵ errors. For accuracy purposes, the round-o↵ error ✏ of each arithmetic operation is computed exactly via EFT(Error Free Transforms). For e�ciency purposes, the rounding modeis never switched. Instead, a brute-force bound on the roundo↵ errorincurred by the accumulation of the ✏s mentioned above is used.
Experimental results are presented. The proposed implementationis one of the most accurate and its execution time is significantly re-duced; it can be up to 50% faster than other implementations. Fur-thermore, the use of a FMA (Fused Multiply-and-Add) reduces thecost of the EFT and the overall performance is even better.
References:
[1] E.R. Hansen, A generalized interval arithmetic, Lecture Notes inComputer Science, No. 29, pp. 7–18, 1975.
[2] American National Standards Institute and Instituteof Electrical and Electronic Engineers, IEEE standardfor binary floating-point arithmetic. Std 754-2008. ANSI/IEEEStandard, 2008.
[3] S. Kiel, YalAA: yet another library for a�ne arithmetic, ReliableComputing, Vol. 16, pp. 114–129, 2012.
[4] F. Messine, Extensions of a�ne arithmetic: Application to un-constrained global optimization, Journal of Universal ComputerScience, Vol. 8, No. 11, pp. 992–1015, 2002.
[5] J. Stolfi and L. de Figueiredo, Self-Validated Numerical Meth-ods and Applications, Monograph for 21st Brazilian MathematicsColloquium, Rio de Janeiro, Brazil, 1997.
[6] X.-H. Vu, D. Sam-Haroud, and B. Faltings, Combining mul-tiple inclusion representations in numerical constraint propagation,in IEEE Int. Conf. on Tools with Artificial Intelligence, pp. 458–467, IEEE Computer Society, 2004.
126
Iterative Refinement for SymmetricEigenvalue Problems
Takeshi OgitaTokyo Woman’s Christian University
Tokyo 167-8585, [email protected]
Keywords: eigenvalue problem, iterative refinement, accurate numer-ical algorithms
Let us consider a standard eigenvalue problem
Ax = �x (1)
where A = AT 2 Rn⇥n. To solve (1) is ubiquitous since it is one of thesignificant tasks in scientific computing. The purpose of this talk is tocompute an arbitrarily accurate eigenvalue decomposition:
A = bX bD bX�1 = bX bD bXT ,
where bX 2 Rn⇥n is orthogonal and bD 2 Rn⇥n is diagonal.Most of the existing refinement algorithms are based on Newton’s
method for nonlinear equations, e.g. [1,2]. These methods can improveeigenpairs one-by-one. On the other hand, we develop a method ofimproving all eigenvalues and eigenvectors at the same time.
Let bX 2 Rn⇥n be an orthogonal matrix consisting of all exact eigen-vectors of A. Let X
0
2 Rn⇥n be an initial approximation of bX. Herewe assume that X
0
satisfies
k bX �X0
k =: "0
<1
2.
In this talk we propose a simple iterative refinement algorithm forcalculating Xk, k = 1, 2, . . . such that
• Xk := Xk�1
+Wk for k = 1, 2, . . .
127
• k bX �Xkk =: "k ⇡ "2k�1
⇡ "2k0
(quadratic convergence)
which implies
max |�i � e�(k)i | ⇡ "k max |�i| = "kkAk, e�(k)i := (XTk AXk)ii.
The idea is as follows: for an approximation X 2 Rn⇥n of bX, defineE 2 Rn⇥n such that bX = X(I + E). Then we try to compute a goodapproximation eE of E by utilizing the following two relations:
⇢ bX�1 bX = bXT bX = I (orthogonality)bX�1A bX = bXTA bX = bD (diagonalizability)
After obtaining eE, we can update X by X(I + eE). In general, we caniteratively update Xk by
Xk+1
:= Xk(I + eEk) = Xk +XkeEk.
Detailed discussions and numerical results will be presented in thetalk.
References:
[1] J.J. Dongarra, C.B. Moler, J.H. Wilkinson, Improving theaccuracy of computed eigenvalues and eigenvectors, SIAM J. Nu-mer. Anal., 20:1 (1983), 23–45.
[2] F. Tisseur, Newton’s method in floating point arithmetic anditerative refinement of generalized eigenvalue problems, SIAM. J.Matrix Anal. Appl., 22:4 (2001), 1038–1057.
128
Automatic Verified NumericalComputations for Linear Systems
Katsuhisa Ozaki, Takeshi Ogita and Shin’ichi OishiShibaura Institute of Technology
307 Fukasaku, Minuma-ku, Saitama-shi, Saitama 337-8570, [email protected]
Keywords: Verified Numerical Computations, Floating-Point Arith-metic, Numerical Linear Algebra
This talk is concerned with verified numerical computations forlinear systems. Our aim is to improve an automatic verified methodfor linear systems which is discussed in [2]. Let F be a set of floating-point numbers as defined by IEEE 754. Let I be the identity matrixwith suitable size. For A 2 Fn⇥n, if R exists such that
kRA� Ik ↵ < 1, (1)
then A is non-singular. This is dominant computations in verifiednumerical computations of linear systems. The discussion is how toobtain ↵ in (1) as fast as possible. The notation fl(·) and fl4(·) meansthat each operation in the parenthesis is evaluated by floating-pointarithmetic as defined by IEEE 754 with rounding to nearest and round-ing upwards, respectively. Let e = (1, 1, . . . , 1)T 2 Fn. A constant udenotes the roundo↵ unit, for example, u = 2�53 for binary64. Assumethat neither overflow nor underflow occurs in fl(·). If we apply a priorierror analysis (for example [1]) for fl(RA� I), then an upper bound ofkRA� Ik1 can be computed by
kRA� Ik1 fl4(kfl(|RA� I|e) + du(|R|(|A|e) + e)k1) (2)
where d is a constant with log2
n . d . n depending on the order ofthe evaluation. For (2), the following relation often holds [2]:
fl4(fl(|RA� I|e)) ⌧ fl4(du(|R|(|A|e) + e))
129
Our idea is as follows: After obtaining R ⇡ A�1, we first evaluatefl4(u(|R|(|A|e) + e)). Then, the constant d can be controlled by thefollowing block computations in order to prove kRA � Ik1 < 1. As-sume that s = dn/n0e for block size n0. We use block notations forC = AB (A,B,C 2 Fn⇥n) as follows:
0
@C
11
· · · C1s
... . . . ...Cs1 · · · Css
1
A =
0
@A
11
· · · A1s
... . . . ...As1 · · · Ass
1
A
0
@B
11
· · · B1s
... . . . ...Bs1 · · · Bss
1
A
We introduce a variant of block matrix computations with ↵ 2 N,w = ds/↵e and 1 q w � 1 as follows:
Cij = fl(wX
k=1
Tk), Tq = fl(↵qX
l=↵(q�1)+1
AilBlj), Tw = fl(sX
l=↵(w�1)+1
AilBlj).
Then, |C � AB| ��|A||B|, � = n0 + ↵ + w � 2. The minimal � forn = r3 (r 2 N) is identical with some bounds in [3]. Our algorithmautomatically defines suitable n0 and ↵, and obtain ↵ as fast as possible.Similar discussion can be applied into other methods in [2]. As a result,the computing time of the proposed algorithm is much smaller thanthat of the algorithm in [2], which will be shown in the presentation.
References:
[1] N. J. Higham, Accuracy and Stability of Numerical Algorithms,Second Edition, SIAM, Philadelphia, 2002.
[2] K. Ozaki, T. Ogita, S. Oishi, An Algorithm for AutomaticallySelecting a Suitable Verification Method for Linear Systems, Nu-merical Algorithms, 56 (2011), No. 3, pp. 363-382.
[3] A. M. Castaldo, R. C. Whaley, A. T. Chronopoulos, Re-ducing Floating Point Error in Dot Product using the SuperblockFamily of Algorithms, SIAM Journal on Scientific Computing, 31(2009), No. 2, pp. 1156-1174.
130
Bernstein branch-and-bound algorithmfor unconstrained global optimizationof multivariate polynomial MINLPs
Bhagyesh V. Patil1 and P. S. V. Nataraj2
1Laboratoire d’Informatique de Nantes Atlantique2, rue de la Houssiniere
BP 92208, Nantes44322, France
2Systems and Control Engineering GroupIndian Institute of Technology Bombay
Powai-400076, [email protected]
Keywords: Branch-and-bound, Bernstein polynomials, Global opti-mization, Mixed-integer nonlinear programming.
Optimization of mixed-integer nonlinear programming (MINLP)problems constitutes an active area of research. A standard strategyto solve MINLP problems is to use a branch-and-bound (BB) frame-work [1]. Specifically, a relaxed NLP is solved at each node of thebranch-and-bound tree. Di↵erent variants of the BB approach havebeen reported in the literature [2] and are widely adapted by severalstate-of-the-art MINLP solvers (cf. BARON , Bonmin, SBB). Albeitof the widespread enjoyed interest by the BB approach, the type ofNLP solver used has found to limit its performance in practice. Tosolve polynomial nonlinear programming (NLP) problems, an alterna-tive approach is provided by the Bernstein algorithms. The Bernsteinalgorithms are similar in philosophy to interval branch-and-bound pro-cedures. Several variants of the Bernstein algorithms to solve uncon-strained polynomial NLPs have been reported in the literature (see,for instance, work by Nataraj and co-workers). However, no work has
131
yet been reported in the literature for global optimization of uncon-strained polynomial MINLP problems using the Bernstein polynomialapproach.
In this paper, we propose a Bernstein algorithm for unconstrainedglobal optimization of multivariate polynomial MINLPs. The proposedalgorithm is similar to the classical Bernstein algorithm for the globaloptimization of unconstrained NLPs, but with several modificationslisted as follows. It uses tools, namely monotonicity and concavitytests, a modified subdivision procedure (to handle integer decision vari-ables in the given MINLPs), and the Bernstein box and Bernstein hullconsistency techniques to contract the search domain. The Bernsteinbox and Bernstein hull consistency techniques is applied to constraintsbased on the gradient and upper bound on the global minimum of theobjective polynomial to delete nonoptimal points from the given searchdomain of interest.
The performance of the proposed algorithm is numerically testedon a collection of 10 test problems (three to nine dimensional, andone to six integer variables) taken from [3]. These problems are con-structed as MINLPs, and the test results are compared with those ofthe classical Bernstein algorithm to solve polynomial NLPs 1. For thesetest problems, we first compare the performance of the proposed algo-rithm with and without accelerating devices (namely the cut-o↵, themonotonicity, and the concavity tests), combinations of the di↵erentaccelerating devices, and the combinations of the Bernstein box andthe Bernstein hull consistency techniques with the three acceleratingdevices. Based on our findings, the proposed algorithm is found tobe considerably more e�cient than the classical Bernstein algorithm,giving average reduction of 50 % in the number of boxes processed andthe computational time, depending on the tools used in the proposedalgorithm.
1It may be noted that the classical Bernstein algorithm is extended (based on simple rounding
heuristics borrowed from the MINLP literature) in this case to handle the integer variables of the
MINLPs.
132
References:
[1] C. A. Floudas, Nonlinear and mixed-integer optimization: Fun-damentals and applications, New York, U.S.A: Oxford UniversityPress, 1995.
[2] GAMS- The solver manuals, GAMS Development Corp., Wash-ington DC, 2003.
[3] J. Verschelde, PHC pack, the database of polynomial systems,Technical report, Mathematics Department, University of Illinois,Chicago, USA, 2001.
133
Improved Enclosure for ParametricSolution Sets with Linear Shape
Evgenija D. PopovaInst. of Mathematics and Informatics, Bulgarian Academy of Sciences
Acad. G. Bonchev str., block 8, 1113 Sofia, [email protected]
Keywords: linear systems, dependent data, solution set enclosure
Consider linear algebraic systems involving linear dependencies be-tween interval parameters. An interval method [1] for enclosing theparametric united solution set is known as the best one. Its e�cientapplication requires particular structure of the dependencies which isrepresentative for finite element models of truss structures. It is notknow which parametric systems (in general form) have the requiredrepresentation.
We generalize the method [1] for systems involving dependenciesbetween the matrix and the right-hand side vector. Some su�cientconditions for a parametric solution set to have linear boundary willbe presented. Via these conditions any parametric system that satisfiesthem is transformed into the form required by the method of Neumaierand Pownuk, and its generalization. Thus, an expanded scope of ap-plicability is achieved. Examples will demonstrate parametric solutionsets with linear boundary that appear in various application domains.The linear boundary of any parametric solution set (AE- in general)with respect to a given parameter, which is proven by the above su�-cient conditions, can be utilized for further improving an enclosure ofthe solution set.
References:
[1] A. Neumaier, A. Pownuk, Linear systems with large uncertain-ties, with applications to truss structures, Reliable Computing, 13(2007), pp. 149–172.
134
The architecture of the IEEE P1788 draftstandard for interval arithmetic
John PryceCardi↵ University, UK
Keywords: IEEE standard, Interval arithmetic, Interval exceptionhandling, Interval flavors
Interval arithmetic (IA) is the most used way of producing rigor-ously proven results in problems of continuous mathematics, usually inthe form of real intervals that (even in presence of rounding error) areguaranteed to enclose a value of interest, such as a solution of a di↵er-ential equation at some point. The basics of IA are generally agreed –e.g., to add two intervals x, y, find an interval containing all x+ y forx in x and y in y.
Many versions of IA theory exist, individually consistent but mutu-ally incompatible. They di↵er especially in how to handle operationsnot everywhere defined on their inputs, such as division by an intervalcontaining zero. In this situation a standard is called for, which notall will love but which is usable and practical in most IA applications.
The IEEE working group P1788 [1], begun in 2008, has produced adraft standard for interval arithmetic, currently undergoing the IEEEapproval process. The talk will concentrate on aspects of its architec-ture, especially:
• the levels structure, with a mathematical, a datum and an imple-mentation level;
• the decoration system, which notes when a library operation isapplied to input where it is discontinuous or undefined.
Time permitting, I may outline the P1788 flavor concept, by whichimplementations based on other versions of IA theory may be includedinto the standard in a consistent way.
Invited talk
135
References:
[1] IEEE Interval Standard Working Group - P1788,http://grouper.ieee.org/groups/1788/.
136
Verified Parameter Identification forDynamic Systems with Non-Smooth
Right-Hand Sides
Andreas Rauh, Luise Senkel and Harald Aschemann
Chair of MechatronicsUniversity of Rostock
D-18059 Rostock, Germany{Andreas.Rauh,Luise.Senkel,Harald.Aschemann}@uni-rostock.de
Keywords: Non-smooth ordinary di↵erential equations, parameteridentification, mechanical systems, friction
Dynamic system models given by ordinary di↵erential equations(ODEs) with non-smooth right-hand sides are widely used in engi-neering. They can, for example, be employed to model transitionsbetween static and sliding friction in mechanical systems and to rep-resent variable degrees of freedom for dynamic applications in roboticswith contacts between at least two (rigid) bodies.
The verified simulation of such systems has to detect those pointsof time at which either one of the discrete model states (in a represen-tation of the ODEs by means of a state-transition diagram) becomesactive or at which one of the discrete states is deactivated [1,2]. Aslong as mechanical systems are taken into consideration that are de-scribed by position and velocity as corresponding state variables, it isguaranteed that the state trajectories (i.e., the solutions of the ODE)remain continuous at the before-mentioned switching points.
For practical applications, however, it is not only necessary to de-rive verified simulation techniques and to compute state variables thatcan be reached within a given time horizon under consideration of apredefined control law. Such a control law is usually given by theactuator signal (e.g. force) acting onto the (mechanical) system [3,4].In addition, a system identification is necessary to determine parame-ter values that comply with the non-smooth system model on the one
137
hand and the measured data on the other hand. In engineering appli-cations, these measurements are usually subject to uncertainty that isoften in the same order of magnitude as the measured data themselves.For that reason, it is in general not reliable to determine point valuesfor the system parameters. Instead, confidence intervals have to becomputed which satisfy both the constraints imposed by the dynamicsystem model (the ODEs with the variable-structure behavior) and themeasurements with the corresponding uncertainty.
In this contribution, an interval-based o✏ine system identificationroutine is presented and compared to a guaranteed stabilizing slidingmode state and parameter estimator. This estimator is proven to beasymptotically stable within a desired operating range, and may beemployed in real time within engineering applications. However, it hasthe drawback that it does not directly produce confidence intervalsthat are required for a guaranteed identification. Simulations and ex-perimental results are shown for a laboratory test rig representing theuncertain longitudinal dynamics of a vehicle.
References:
[1] E. Auer, S. Kiel, and A. Rauh, A Verified Method for SolvingPiecewise Smooth Initial Value Problems, Intl. J. of Applied Math-ematics and Computer Science, 23 (2013). No. 4, pp. 731–747.
[2] N.S. Nedialkov and M. v. Mohrenschildt, Rigorous Sim-ulation of Hybrid Dynamic Systems with Symbolic and IntervalMethods, Proc. of American Control Conference ACC, Anchor-age, USA, (2002). pp. 140–147.
[3] A. Rauh, M. Kletting, H. Aschemann, and E.P. Hofer,Interval Methods for Simulation of Dynamical Systems with State-Dependent Switching Characteristics, Proc. of the IEEE Intl. Conf.on Control Applications, Munich, Germany, (2006). pp. 355–360.
[4] A. Rauh, Ch. Siebert, H. Aschemann, Verified Simulationand Optimization of Dynamic Systems with Friction and Hystere-sis, Proc. of ENOC 2011, Rome, Italy, (2009).
138
Computation of Confidence Regions inReliable, Variable-Structure State and
Parameter EstimationAndreas Rauh, Luise Senkel and Harald Aschemann
Chair of MechatronicsUniversity of Rostock
D-18059 Rostock, Germany
{Andreas.Rauh,Luise.Senkel,Harald.Aschemann}@uni-rostock.de
Keywords: Ordinary di↵erential equations, sliding mode techniques,interval arithmetic, cooperativity of dynamic systems
Interval-based sliding mode controllers and estimators provide apossibility to stabilize the error dynamics despite not accurately knownparameters and bounded measurement uncertainty [2]. However, cur-rent implementations of both types of approaches are commonly char-acterized by the fact that they only provide point-valued estimateswithout any explicit computation of confidence intervals [4]. There-fore, this contribution aims at developing fundamental techniques foran extension towards the computation of guaranteed confidence inter-vals.
The corresponding procedure is based on the use of symbolic for-mula manipulation and interval arithmetic for the computation of thosesets of state variables (and estimated states, respectively) that can bereached in a finite time horizon. For that purpose, the nonlinear sys-tem is embedded into a (quasi-) linear state-space representation withpiecewise constant input signals for which the sets of reachable statescan be computed by using Muller’s theorem, or more generally, byexploiting the system property of cooperativity for linear parameter-varying finite-dimensional state equations [1,3]. The necessary prooffor cooperativity is performed by verified range computation proce-dures that rely on interval arithmetic software libraries. Basic build-ing blocks of these procedures are presented for the use of the Matlabtoolbox IntLab.
139
After a summary of the before-mentioned fundamental procedures,extensions are presented which show how these techniques can be em-ployed in a framework for designing sliding mode estimators. Theseestimators, extended by the use of interval arithmetic, determine thesets of state variables and parameters that are consistent with botha given dynamic system model and information about bounded mea-surement uncertainty.
Finally, necessary extensions are highlighted which allow for an ex-tension of the implementation in such a manner that the correspondingestimation schemes can make use of interval arithmetic in real time.Numerical results for estimation tasks related to the longitudinal dy-namics of a vehicle conclude this contribution.
References:
[1] M. Muller, Uber die Eindeutigkeit der Integrale eines Systemsgewohnlicher Di↵erenzialgleichungen und die Konvergenz einer Gat-tung von Verfahren zur Approximation dieser Integrale, Sitzungs-bericht Heidelberger Akademie der Wissenschaften, (1927). In Ger-man.
[2] L. Senkel, A. Rauh, H. Aschemann, Interval-Based SlidingMode Observer Design for Nonlinear Systems with Bounded Mea-surement and Parameter Uncertainty, In Proc. of IEEE Intl. Con-ference on Methods and Models in Automation and Robotics, Miedzyz-droje, Poland, 2013.
[3] H.L. Smith, Monotone Dynamical Systems; An Introduction tothe Theory of Competitive and Cooperative Systems, Mathemat-ical Surveys and Monographs. American Mathematical Society,Providence, USA, vol. 41 (1995).
[4] V. Utkin, Sliding Modes in Control and Optimization, (Springer-Verlag, Berlin, Heidelberg, (1992).
140
Exponential Enclosure Techniques forInitial Value Problems with Multiple
Conjugate Complex Eigenvalues
Andreas Rauh1, Ramona Westphal1, Harald Aschemann1 andEkaterina Auer2
1Chair of MechatronicsUniversity of Rostock
D-18059 Rostock, Germany
2Faculty of Engineering, INKOUniversity of Duisburg-EssenD-47048 Duisburg, Germany
{Andreas.Rauh,Ramona.Westphal,Harald.Aschemann}@uni-rostock.de,[email protected]
Keywords: Ordinary di↵erential equations, initial value problems,complex interval arithmetic, ValEncIA-IVP
ValEncIA-IVP is a verified solver providing guaranteed enclo-sures for the solution to initial value problems (IVPs) for sets of ordi-nary di↵erential equations (ODEs). In the basic version of this solver,the verified solution was computed as the sum of a non-verified ap-proximate solution (computed, for example, by Euler’s method) andadditive guaranteed error bounds determined using a simple iterationscheme [1,2].
The disadvantage of this iteration scheme, however, is that thewidths of the resulting state enclosures might get larger even for asymp-totically stable ODEs [2,3]. This phenomenon is caused by the so-calledwrapping e↵ect which arises if non-axis-parallel state enclosures aredescribed by axis-aligned interval boxes in a state-space of dimensionn > 1. To avoid the resulting overestimation, it is useful to trans-form the ODEs into a suitable canonical form. For the case of linearODEs with real eigenvalues of multiplicity one, the canonical form isgiven by the Jordan normal form. It results in a decoupling of the
141
vector-valued set of state equations. A solutions of this transformedIVP can then be determined by an exponential enclosure techniquewhich guarantees that asymptotically stable solutions are representedby contracting interval bounds if a suitable time discretization stepsize is chosen. For real eigenvalues, this property holds as long as thevalue zero is not included in any vector component of the solution in-terval. This advantageous property can be preserved for linear ODEswith conjugate complex eigenvalues if a transformation into the com-plex Jordan normal form is employed. Then, a complex-valued intervaliteration scheme is used to determine state enclosures [4].
This contribution extends the solution procedure, described in [4]for eigenvalues of multiplicity one, to more general situations withseveral multiple real and complex eigenvalues. Simulation results fortechnical system models from control engineering, containing boundeduncertainty in initial values and parameters, conclude this contribu-tion.
References:
[1] E. Auer, A. Rauh, E.P. Hofer, and W. Luther, ValidatedModeling of Mechanical Systems with SmartMOBILE: Improve-ment of Performance by ValEncIA-IVP, Lecture Notes in Com-puter Science 5045, (2008), Springer, pp. 1–27.
[2] A. Rauh and E. Auer, Verified Simulation of ODEs and DAEs inValEncIA-IVP, Reliable Computing, 15 (2011). No. 4, pp. 370–381.
[3] A. Rauh, M. Brill, C. Gunther, A Novel Interval Arith-metic Approach for Solving Di↵erential-Algebraic Equations withValEncIA-IVP, International Journal of Applied Mathematicsand Computer Science, 19 (2009). No. 3, pp. 381–397.
[4] A. Rauh, R. Westphal, E. Auer, and H. Aschemann, Ex-ponential Enclosure Techniques for the Computation of Guaran-teed State Enclosures in ValEncIA-IVP, Reliable Computing, 19(2013). No. 1, pp. 66–90.
142
Numerical Validation of Sliding ModeApproaches with Uncertainty
Luise Senkel, Andreas Rauh and Harald AschemannChair of Mechatronics University of Rostock
18059 Rostock, Germany{Luise.Senkel,Andreas.Rauh,Harald.Aschemann}@uni-rostock.de
Keywords: Sliding Mode Techniques, Stochastic and bounded uncer-tainty, Interval Arithmetic
Many technical systems are a↵ected by bounded and stochasticdisturbances, which are usually summarized as uncertainty in general.Bounded uncertainty comprises, for example, lack of knowledge aboutspecific parameters as well as manufacturing tolerances. In contrast,stochastic disturbances have to be taken into consideration in controland estimation tasks if only inaccurate sensor measurements are avail-able and if random e↵ects influencing the stability of the system are tobe modeled, as for example friction. To cope with these phenomena, asliding mode approach is derived in this presentation that takes thesetypes of uncertainty into account and stabilizes the error dynamicseven if system parameters are not exactly known and measurementsare a↵ected by noise processes.
The sliding mode approach consists of two parts: a quasi-linear anda variable structure part. This provides the possibility to take not onlylinear but also nonlinear systems into account because the first partstabilizes the linear dynamics, and the second one counteracts nonlin-ear influences on the system. In contrast to some known sliding modeapproaches, restrictive matching conditions are avoided. Additionally,intervals for uncertain parameters as well as control, estimation andmeasurement errors are used for the calculation of the switching am-plitudes based on the Ito di↵erential operator in combination with asuitable candidate for a Lyapunov function.
To show the applicability, consider a dynamic system a↵ected bystochastic disturbance inputs dw that act on the system dynamics as
143
a standard Brownian motion. Then, the system can be described bythe stochastic di↵erential equation
dx = f (x(t),p,u(x(t))) dt+ g (x(t),p) dw . (1)
Applying the Ito di↵erential operator L(V (x,p)), cf. [3], to the Lya-punov function candidate V (x) = 1
2
· xT · P · x with P = PT � 0, itstime derivative becomes
L(V (x,p)) =@V
@t+
✓@V
@x
◆T
·f(x,p)+1
2·trace
⇢gT (x,p) · @
2V
@x2
· g(x,p)�
(2)with the vector of interval parameters p 2 [p ; p] where p
i< pi < pi
holds for each parameter pi. In (2), x1 is the equilibrium state withx = x� x1. If the condition L(V (x,p)) < 0 holds in the interior of adomain V (x) < c, c > 0, containing x1, the equilibrium can be provenasymptotically stable in a stochastic sense.
This procedure can be exploited for both control as well as state andparameter estimation tasks. Numerical results conclude this presenta-tion and show the practical applicability of the sliding mode approachconsidering bounded and stochastic disturbances.
References:
[1] L. Senkel, A. Rauh, H. Aschemann: Interval-Based Sliding ModeObserver Design for Nonlinear Systems with Bounded Measure-ment and Parameter Uncertainty , In Proc. of IEEE Intl. Con-ference on Methods and Models in Automation and Robotics,Miedzyzdroje, Poland, 2013.
[2] V. Utkin, Sliding Modes in Control and Optimization, (Springer-Verlag, Berlin, Heidelberg, 1992).
[3] H. Kushner: Stochastic Stability and Control (Academic Press,New York, 1967).
144
Reserve as recognizing functional forAE-solutions to interval system of linear
equationsIrene A. Sharaya and Sergey P. Shary
Institute of Computational Technologies SD RAS6, Lavrentiev ave., 630090 Novosibirsk, Russia
{sharaya,shary}@ict.nsc.ru
Keywords: interval linear systems, AE-solutions, reserve
It is shown in [1] that, for interval systems (A8 +A
9)x = b
8 + b
9
with A
8,A9 2 IRm⇥n, b8, b9 2 IRm such that A8ij ·A9
ij = 0, b8i · b9j = 0for every i, j, any AE-solution set
⌅ = {x 2 Rn | 8A0 2 A
8, 8b0 2 b
8, 9A00 2 A
9, 9b00 2 b
9
(A0 + A00)x = b0 + b00}
can be characterized in Kaucher interval arithmetic by the inclusion
Cx ✓ d, where C = A
8 + dual(A9), d = dual(b8) + b
9. (1)
Definition of reserve z. We call by reserve of the inclusion (1)the maximal real number z such that Cx + e [�z, z] ✓ d for them-vector e = (1 1 . . . 1)>.
Formulas for z. From the above definition, we get
z = mini
min�Ci:x� di, �Ci:x+ di
=
= mini
min�Ci:x
+ �Ci:x� � di, �Ci:x
+ +Ci:x� + di
=
= mini
min
(nX
j=1
C�sgn(xj)
ij xj � di, �nX
j=1
Csgn(xj)
ij xj + di
)=
= mini
⇣raddi � radCi: |x|�
��middi �midCi: x��⌘=
= mini
⇣rad(A9
i:)|x|+ rad(b9i )���A
8i:x� b
8i +mid(A9
i:)x�mid(b9i )��⌘
145
using the notation x+, x� 2 Rn+
, x+ = max{0, x}, x� = max{0,�x},
C�sgn(xj)
ij =
(Cij, if xj � 0,
Cij, otherwise,C
sgn(xj)
ij =
(Cij, if xj � 0,
Cij, otherwise.
Geometrical properties of the functional z(x). The functionalz(x) is defined on the entire Rn, continuous and piecewise-linear. It isconcave in each orthant of Rn.
For C = A
8 (in particular, for the tolerable solution set and theset of strong solutions), the functional z(x) is concave on the whole ofRn and bounded from above by the real number mini rad(b
9i ).
Recognizing properties of the functional z(x). Judging on thevalue of z(x) at a point x, we can decide on whether the point x belongsto the solution set ⌅, its topological interior int⌅ or the boundary @⌅:
z(x)�0 , x2⌅, z(x)>0 ) x2 int⌅, z(x)=0 ( x2@⌅. (2)
Examining the value of maxx2Rn z(x), we can recognize whether thesets ⌅ and int⌅ are empty or not:
maxx2Rn
z(x) � 0 , ⌅ 6= ?, maxx2Rn
z(x) > 0 ) int⌅ 6= ?. (3)
We have derived necessary and su�cient conditions on C, A8, A9,b
8, b9 for changing “)” and “(” in (2) and (3) to “,”.The set Argmaxx2Rn z(x) turns out to be quite useful too:1) if maxx2Rn z(x) � 0, then Argmaxx2Rn z(x) consists of the ‘best’
points of ⌅ at which the inclusion (1) holds with maximum reserve;2) if maxx2Rn z(x) > 0, then Argmaxx2Rn z(x) ✓ int⌅;3) if maxx2Rn z(x) < 0, then the set Argmaxx2Rn z(x) consists of
the points where the inclusion (1) is violated in the minimum amount.The latter enables one to use such points as ‘type ⌅ pseudosolutions’.
References:
[1] S.P. Shary, A new technique in systems analysis under intervaluncertainty and ambiguity, Reliable Computing, 8 (2002), No. 5,pp. 321–418. — Electronic version is downloadable fromhttp://interval.ict.nsc.ru/shary/Papers/ANewTech.pdf
146
Maximum Consistency Methodfor Data Fitting
under Interval UncertaintySergey P. Shary
Institute of Computational Technologiesand Novosibirsk State University
Novosibirsk, [email protected]
Keywords: interval uncertainty, data fitting, interval linear systems,solution set, recognizing functional, maximum consistency method
For the linear regression model b = a1
x1
+ a2
x2
+ . . . + anxn, weconsider the problem of data fitting under interval uncertainty. Let aninterval m ⇥ n-matrix A = (aij) and an interval m-vector b = (bi)represent, respectively, the input data and output responses of themodel, such that a
1
2 ai1, a2 2 ai2, . . . , an 2 ain, b 2 bi in the i-thexperiment, i = 1, 2, . . . ,m. It is necessary to find the coe�cients thatbest fit the above linear relation for the data given.
A family of values of the parameters is called consistent with theinterval data (ai1,ai2, . . . ,ain), bi, i = 1, 2, . . . ,m, if, for every indexi, there exist such point representatives ai1 2 ai1, ai2 2 ai2, . . . , ain 2ain, bi 2 bi that ai1x1 + ai2x2 + . . . + ainxn = bi. The set of allthe parameter values consistent with the data given form a parameteruncertainty set. As an estimate of the parameters, it makes sense totake a point from the parameter uncertainty set providing that it isnonempty. Otherwise, if the parameter uncertainty set is empty, thenthe estimate should be a point where maximal “consistency” (in aprescribed sense) with the data is achieved.
The parameter uncertainty set, as defined above, is nothing but thesolution set ⌅(A, b) to the interval system of linear equations Ax = b
introduced in interval analysis:
⌅(A, b) = { x 2 Rn | Ax = b for some A from A and b from b }.
147
For the data fitting problem under interval uncertainty, we propose,as the consistency measure, the values of the recognizing functional ofthe solution set ⌅(A, b) which is defined as
Uss(x,A, b) = min1im
(rad bi +
nX
j=1
(rad aij) |xj|
������ mid bi �
nX
j=1
(mid aij) xj
�����
),
where “mid” and “rad” mean the midpoint and radius of an interval.The functional Uss “recognizes” the points of ⌅(A, b) by the sign ofits values: x 2 ⌅(A, b) if and only if Uss (x,A, b) � 0. Additionally,Uss has reasonably good properties as a function of x and A, b.
As an estimate of the parameters in the data fitting problem, wetake the value of x = (x
1
, x2
, . . . , xn) that provides maximum of therecognizing functional Uss (Maximum Consistency Method). Then,• if the parameter uncertainty set is nonempty, we get its point,
• if the parameter uncertainty set is empty, we get a point that stillhas maximum possible consistency (determined by values of thefunctional Uss) with the data given.
In our work, we discuss properties of the recognizing functional Uss,interpretation and features of the estimates obtained by the MaximumConsistency Method. Also treated is correlation with the other ap-proaches to data fitting under interval uncertainty.
References:
[1] S.P. Shary, Solvability of interval linear equations and data anal-ysis under uncertainty, Automation and Remote Control, 73 (2012),No. 2, pp. 310–322.
[2] S.P. Shary and I.A. Sharaya, Recognizing solvability of inter-val equations and its application to data analysis, ComputationalTechnologies, 18 (2013), No. 3, pp. 80–109. (in Russian)
148
An Implementation of CompleteArithmetic
Stefan Siegel
University of Wurzburg97074 Wurzburg, Germany
Keywords: Exact dot product, Correctly rounded sum, Long accu-mulator
To enlarge the acceptance of interval arithmetic the IEEE intervalstandard working group P1788 [1] has been founded in 2008.
For this upcoming standard, complete arithmetic as described in [2]should be provided by implementing libraries to provide (bit) losslessarithmetics. The so called complete format C(F), with F describingthe number format, uses a long accumulator to ensure precise results.
As opposed to the more commonly used fixed (double) precisioninterval arithmetic the 2nd arithmetical feature dynamic precision in-terval arithmetic is able to provide
• conversion of several number formats to and from the accumulator,e.g. from floating-point format to complete format, vice versa orfrom one complete format to another,
• addition and subtraction, e.g. add or subtract two complete orfloating-point formats, of which at least one is complete,
• multiply-add, to compute a = x ⇤ y + z with x, y 2 F and a, z 2C(F) and
• a dot product to calculate an exact result of two vectors a, b 2 Fwith length n. The exact results of
Pnk=1
ak ⇤ bk is available fromthe accumulator.
149
In this talk we will give a short overview of the complete arith-metic as suggested by the upcoming P1788 standard and present ourstandard implementation which provide this claimed function. Fur-thermore we will talk about our testing environment.
References:
[1] IEEE Interval Standard Working Group - P1788, April 2014, http://grouper.ieee.org/groups/1788/.
[2] Kulisch, Ulrich and Snyder, Van The exact dot product asbasic tool for long interval arithmetic, Position paper, P1788 Work-ing Group, version 11, July 2009.
150
Non-arithmetic approach to dealing withuncertainty in fuzzy arithmetic
Igor SokolovMoscow State University124482 Moscow, [email protected]
Keywords: Fuzzy Arithmetic, Interval Arithmetic, Fuzzy Sets
Interval arithmetic is one of the most common ways to deal withfuzzy numbers. In the interval arithmetic each fuzzy number is repre-sented as a set of intervals for each ↵-level, where ↵-levels are chosenbased on required precision. Such representation of fuzzy numbers al-lows usage of all interval arithmetic tools, including basic arithmeticoperations.
But such representation of fuzzy numbers doesn’t form either a fieldor even a group with any basic arithmetic operations. It lacks bothtransitivity and inverse element.
Some researchers try to construct unnatural and counter-intuitiveoperations to deal with these problems. For example - we can findoperations like ”inverse-addition” and ”inverse-multiplication”.
Such operations make it possible to use inverse element for ad-dition and multiplication respectively, but have very strict limits ofusage. But in this paper it is proposed that seeking for inverse ele-ment is unnecessary, because it ruins the intuition of fuzzy arithmetic.
The following intuition is used as basic idea of this paper. SupposeA and B are fuzzy numbers, so that A = B. We declare that:
1. A - B = 0, if both A and B reflect the same measurement of thesame object or event;
2. A - B 6= 0, if A and B do not reflect the same measurement of thesame object or event.
151
So before using interval arithmetic to deal with problem on fuzzynumbers, it is proposed to start with analysis of variables in terms oftheir relations. Such analysis allows to dramatically reduce uncertaintycaused by lack of transitivity and inverse element.
Let’s consider a brief example. Numbers are provided only for ↵0
for simplification, but it is valid for any ↵-level:
X = [100, 150], profit of a company.Y = 0.3*X = [30, 45], a 30% tax this company has to pay.Z = X - Y, profit left after the taxes.
While calculating Z we face the issue - there are two di↵erent waysto calculate it:
1. Z = X - Y = [100, 150] - [30, 45] = [55, 120]2. Z = X - Y = X - 0.3*X = 0.7*X = 0.7*[100, 150] = [70, 105]
Thinking about this example will definitely lead us to the conclu-sion, that 1st way is wrong and 2nd way is right.
This paper suggests, that by applying proposed analysis of relation-ship between variables we can reduce uncertainty caused by choosingimproper way to solve such conflicts. Several common rules and morecomplex examples are provided.
References:
[1] M. Hanss, Applied Fuzzy Arithmetic, 2004.
[2] R. Boukezzoula, S. Galichet, L. Foulloy, Inverse arith-metic operators for fuzzy intervals, LISTIC, Universite de SavoieDomaine Universitaire, 2007
152
True orbit simulation of dynamicalsystems and its computational complexity
Christoph Spandl
Computer Science Department, Universitat der Bundeswehr Munchen85577 Neubiberg, Germany
Keywords: Dynamical systems, Lyapunov exponents, computationalcomplexity
Due to respectable power of computers nowadays, molecular dy-namics simulation is meanwhile state-of-the-art practice in academicand industrial research. The complexity of some classes of systems tosimulate, e.g. proteins, lead to the development of simulation softwarepackages. This situation in turn raises the question of validation [1].One aspect of validation concerns the implementation of the numer-ical integration scheme. A widespread algorithm in use is the Verletmethod, a discretized version of Newton’s second law. Despite thefact that the Verlet method only approximates the true solution ofthe ODE, is has some pleasant properties inherited from the originalequations of motion. Discretization is one source of error, but the trueproblem in simulating orbits in molecular dynamics is another [2]. Asexamined in the one dimensional case in [3], chaotic behavior leads,when iterating the (discrete) equations of motion, asymptotically toa constant loss of significant bits per iteration step in the state spacevariable. Thus, using standard IEEE-754 floating-point arithmetic foriteration, as typically is done in molecular dynamics, rounding errorsoverwhelm the dynamics even after iteration times lying of orders be-low the times required for obtaining a reasonable simulation.
In this contribution, the results obtained in [3] are generalized tothe multidimensional case. The starting point is a discrete dynami-cal system (M, f), where M ✓ Rn is a box and f : M ! M a C2-
153
di↵eomorphism. The time evolution is governed by the iteration equa-tion
x(k+1) = f(x(k)), x(0) 2 M.
To handle numerical errors, the iteration is reformulated on boxes in-stead of points and f is replaced by an appropriate centered form in thesense of [4]. The candidate chosen for a centered form is obtained byusing techniques coming from ergodic theory [5]. Specifically, methodsused for proving the existence of Lyapunov exponents for dynamicalsystems are applied. As the main result, an algorithm for computingtrue orbits (x(k))
0kN of lengthN of the dynamical system with prede-fined accuracy is obtained, supposed that a computable expression forf exists. Moreover, the asymptotic loss of precision in each iterationstep is shown to be given by the Lyapunov exponents. These resultsmay form the starting point for developing more accurate integrationschemes.
References:
[1] W.F. van Gunsteren, A.E. Mark, Validation of moleculardynamics simulation, Journal of Chemical Physics, 108 (1998),pp. 6109–6116.
[2] R.D. Skeel, What makes molecular dynamics work? SIAM Jour-nal on Scientific Computing, 31 (2009), pp. 1363–1378.
[3] Ch. Spandl, Computational complexity of iterated maps on theinterval, Mathematics and Computers in Simulation, 82 (2012),pp. 1459–1477.
[4] R. Krawczyk, Centered forms and interval operators, Computing,34 (1985), pp. 243–259.
[5] L. Barreira, Y.B. Pesin, Introduction to Smooth Ergodic The-ory, AMS, Providence, Rhode Island, 2013.
154
Numerical verification for periodicstationary solutions to the Allen-Cahn
equation
Kazuaki Tanaka1 and Shin’ichi Oishi2,3
1Graduate School of Faculty of Science and Engineering, WasedaUniversity.
2Faculty of Science and Engineering,Waseda University.3CREST, JST.
1,2Building 63, Roeom 419, Okubo 3-4-1, Shinjuku, Tokyo 169-8555,Japan.
Keywords: Allen-Cahn equation, Numerical verification, Periodic so-lutions
The main purpose of this talk is to propose and discuss the resultsof numerical verification for some periodic stationary solutions to theAllen-Cahn equation, which form attractive patterns like a kaleido-scope.
To derive stationary solutions to the Allen-Cahn equation, we tryto solve the following equation:
( �"2�u = u� u3 in ⌦,@u
@n= 0 on @⌦,
(1)
where " is a given positive number and ⌦ is a square domain (0, 1)2.Here, we set V = H1(⌦) and denote the dual space of V by V ⇤. Defin-ing operator F : V ! V ⇤ by
hF (u) , vi := (ru,rv)L2 � "�2
�u� u3, v
�L2 , 8v 2 V,
the equation (1) can be transformed into the equation
F(u) = 0 in V ⇤.
155
We derived the approximate solutions to this equation with spectralmethod and verified these solutions using Newton-Kantorovich’s the-orem (the verification method with this theorem summarized in [1]).One of the most important things for verification is how to estimatethe norm of inverse of linearized operator kF 0 [u]�1 k, where u 2 V isan approximate solution and F 0 [u] is the Frechet derivative of F atu. We estimated the operator norm using the theorem in [2] basedon Liu-Oishi’s theorem [3], which is an e↵ective theorem to evaluateeigenvalues of the Laplace operator on arbitrary polygonal domains.
The Allen-Cahn equation has various solutions, which constituteinteresting patterns, especially when " is small. Since a small " makessolutions to (1) singular, a more accurate basis becomes necessary toobtain an appropriate approximate solution for small ". Of course,numerical verification also becomes di�cult when " is small.
This type of solutions often have periodicity and therefore theycan be composed by solutions on some small domain. In this talk,we would like to show the verification results using periodicity anddiscuss its e↵ectiveness. A consideration about behavior of solutionsto (1) with respect to " also will be performed.
References:
[1] A. Takayasu and S. Oishi, A Method of Computer AssistedProof for Nonlinear Two-point Boundary Value Problems UsingHigher Order Finite Elements, NOLTA IEICE, 2 (2011), No. 1,pp. 74–89.
[2] K. Tanaka, A. Takayasu, X. Liu, and S. Oishi, Verified normestimation for the inverse of linear elliptic operators using eigen-value evaluation, submitted in 2012. (http://oishi.info.waseda.ac.jp/˜takayasu/preprints/LinearizedInverse.pdf)
[3] X. Liu and S. Oishi, Verified eigenvalue evaluation for Laplacianover polygonal domain of arbitrary shape, SIAM J. Numer. Anal,51 (2013), No. 3, pp. 1634–1654.
156
Choice of metrics in interval arithmetic
Philippe Theveny
Ecole Normale Superieure de Lyon – Universite de LyonLIP (UMR 5668 CNRS - ENS de Lyon - INRIA - Universite Claude
Bernard), Universite de LyonENS de Lyon, 46 allee d’Italie 69007 Lyon, France
Keywords: Interval analysis, error analysis, matrix multiplication
The correctness of algorithms computing with intervals depends onthe respect of the inclusion principle. So, for a given problem, di↵erentalgorithms may give di↵erent solutions, as long as each output containsthe mathematically exact result. This raises the problem of comparingthe computed approximations. When the exact solution is a real point,several measures of the distance to the exact result have been proposed:for instance, Kulpa and Markov define relative extent [KM03], Rumpdefines relative precision [Rum99]. When the exact solution itself is aninterval, the ratio of radii is often used.
In this work, we discuss the possible choices for such metrics. Weintroduce the notion of relative accuracy for quantifying the amountof information that an interval conveys with respect to an unknownexact value it encloses. This measure is similar, yet not equivalent, tothe relative precision, the relative extent, or the relative approximationerror proposed by Kreinovich [Kre13].
We then advocate the use of the Hausdor↵ metric for measuringabsolute error as well as relative error between two intervals. We showhow ratios of radii simplify the analysis with the Hausdor↵ distancein the case of nested intervals. We also point out that this simplerapproach may overlook some important phenomena and we illustratethis shortcoming on the example of interval matrix product.
157
References:
[Rum99] Siegfried M. Rump. Fast and parallel interval arithmetic.BIT Numerical Mathematics, 39:534–554, 1999.
[KM03] Zenon Kulpa and Svetoslav Markov. On the inclusion prop-erties of interval multiplication: A diagrammatic study. BIT Nu-merical Mathematics, 43:791–810, 2003.
[Kre13] Vladik Kreinovich. How to define relative approximationerror of an interval estimate: A proposal. Applied MathematicalSciences, 7(5):211–216, 2013.
158
Numerical Verification for EllipticBoundary Value Problem with
Nonconforming P1 Finite Elements
Tomoki Uda
Department of Mathematics, Kyoto UniversityKitashirakawa Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan
Keywords: Nakao’s method, numerical verification, elliptic boundaryvalue problem, nonconforming P1 finite element
In 1988, M. T. Nakao [1] developed a method to verify the ex-istence of solutions to an elliptic boundary value problem. Nakao’smethod, which is based on a finite element method (FEM), implic-itly assumed the finite element space to be conforming. We generalizeNakao’s method for the nonconforming P1 FEM.
Let us consider the following boundary value problem:
⇢�4 u = f(u) in ⌦,
u = 0 on @⌦,(1)
where 4 denotes the Laplace operator, ⌦ is a bounded convex polygonand f : H2(⌦) ! L2(⌦) is a continuous map. If we use the conformingP1 FEM for the problem (1), a finite element basis function i belongsto H1
0
(⌦), that is, i 2 H1(⌦) and i|@⌦ = 0. Thus, for any v 2H2(⌦) \H1
0
(⌦), we get the following formula by integration by parts:
Z
⌦
r i ·rv dx dy = �Z
⌦
i4 v dx dy. (2)
In original (or classical) Nakao’s method, this formula (2) plays an im-portant role. On the other hand, if we use the nonconforming P1 FEM,a finite element basis function 'i does not vanish on the boundary of
159
Ki = supp'i. Therefore, by integration by parts, we get the followingformula:
Z
Ki
r'i ·rv dx dy =
Z
@Ki
'i@nv ds�Z
Ki
'i4 v dx dy, (3)
where n denotes the outward unit normal vector on the boundary@Ki. In order to deduce one of su�cient conditions for verification,we apply v = 4�1 'j to the formula (3). Hence, it is di�cult tocalculate the boundary integration accurately by numerical means. Wehere use an upper and lower estimate of width O(h) for the boundaryintegration instead, where h denotes the mesh size. That is to say,we get constructive inequalities �C(Ki)h |
R@Ki
'i@n(4�1 'j)ds| C(Ki)h. For this purpose, we apply a similar analysis to the estimateof error in nonconforming P1 FEM [2].
Finally, we show some numerical results of our method. Practically,a naive implementation of our method tends to fail verification or tomake the candidate set too large even if the verification is successful.Those are because the interval vector derived from the boundary inte-grations is enlarged by the wrapping e↵ect. We also propose a deviceto avoid this problem, which improves the numerical results.
References:
[1] Mitsuhiro T. Nakao, A Numerical Approach to the Proof ofExistence of Solutions for Elliptic Problems, Japan Journal of Ap-plied Mathematics, 5 (1988), Issue. 2, pp. 313–332.
[2] Philippe G. Ciarlet, The Finite Element Method for EllipticProblems, SIAM, Philadelphia, 2002.
160
Two applications of interval analysis toparameter estimation in hydrology.
Ronald van Nooijen1 and Alla Kolechkina2
1Delft University of TechnologyStevinweg 1, 2628 CN, Delft, Netherlands
[email protected], Den Hoorn Z.H., Netherlands
Keywords: Hydrology, parameter estimation, Gamma distribution,interval analysis
This paper concerns two applications of interval analysis to param-eter estimation in hydrology: an e↵ort to develop an interval analysisbased optimization code for parameter estimation related to ground-water tracer experiments and a code for parameter estimation for prob-ability distributions in a hydrological context by maximizing the like-lihood [1]. At the time the work started in 2008 there was no intervalfunction library that contained a specialized interval extension for thepdf of the Gamma distribution with the distribution parameters asvariables or even an easily available version for the digamma function.
References:
[1] R. van Nooijen, T. Gubareva, A. Kolechkina, and B. Gartsman.Interval analysis and the search for local maxima of the log like-lihood for the Pearson III distribution. In Geophysical ResearchAbstracts, volume 10, pages EGU2008–A–05006, 2008.
161
Combining Interval Methods withEvolutionary Algorithms for Global
OptimizationCharlie Vanaret, Jean-Baptiste Gotteland, Nicolas Durand
and Jean-Marc AlliotInstitut de Recherche en Informatique de Toulouse
31000 Toulouse, [email protected]
Keywords: global optimization, interval analysis, evolutionary algo-rithms
Reliable global optimization is dedicated to solving problems tooptimality in the presence of rounding errors. The most successfulapproaches for achieving a numerical proof of optimality in global op-timization are mainly exhaustive interval-based algorithms [1] that in-terleave pruning and branching steps. The Interval Branch & Prune(IBP) framework has been widely studied [2] and has benefitted fromthe development of refutation methods and filtering algorithms stem-ming from the Interval Analysis and Interval Constraint Programmingcommunities.
In a minimization problem, refutation consists in discarding subdo-mains of the search-space where a lower bound of the objective func-tion is worse than the best known solution. It is therefore crucial: i)to compute a sharp lower bound of the objective function on a givensubdomain; ii) to find a good approximation (an upper bound) of theglobal minimum. Many techniques aim at narrowing the pessimisticenclosures of interval arithmetic (centered forms, convex relaxation,local monotonicity, etc.) and will not be discussed in further detail.
State-of-the-art solvers are generally integrative methods, that isthey embed local optimization algorithms (BFGS, LP, interior points)to compute an upper bound of the global minimum over each subspace.In this presentation, we propose a cooperative approach in which in-terval methods collaborate with Evolutionary Algorithms (EA) [3] on
162
a global scale. EA are stochastic algorithms in which a population ofindividuals (candidate solutions) iteratively evolves in the search-spaceto reach satisfactory solutions. They make no assumption on the ob-jective function and are equipped with nature-inspired operators thathelp individuals escape from local minima. EA are thus particularlysuitable for highly multimodal nonconvex problems. In our approach[4], the EA and the IBP algorithm run in parallel and exchange boundsand solutions through shared memory: the EA updates the best knownupper bound of the global minimum to enhance the pruning, while theIBP updates the population of the EA when a better solution is found.We show that this cooperation scheme prevents premature convergencetoward local minima and accelerates the convergence of the intervalmethod. Our hybrid algorithm also exploits a geometric heuristic toselect the next subdomain to be processed, that compares well withstandard heuristics (best first, largest first).
We provide new optimality results for a benchmark of di�cult mul-timodal problems (Michalewicz, Egg Holder, Rana, Keane functions).We also certify the global minimum of the (open) Lennard-Jones clus-ter problem for 5 atoms. Finally we present an aeronautical applicationto solve conflicts between aircraft.
References:
[1] Moore, R. E. (1966). Interval Analysis. Prentice-Hall.
[2] Hansen, E. and Walster, G. W.(2003). Global optimization usinginterval analysis: revised and expanded. Dekker.
[3] Goldberg, D. E. (1989). Genetic algorithms in search, optimiza-tion, and machine learning. Addison-Wesley Reading Menlo Park.
[4] Vanaret, C., Gotteland, J.-B., Durand, N., and Alliot, J.-M. (2013).Preventing premature convergence and proving the optimality inevolutionary algorithms. In International Conference on ArtificialEvolution (EA-2013), pages 84–94.
163
Dynamic Load Balancing for RigorousGlobal Optimization of Dynamic Systems
Yao Zhao*, Gang Xu** and Mark Stadtherr*
*University of Notre DameNotre Dame, Indiana, USA
**Schneider Electric/SimSciLake Forest, California, USA
Keywords: Dynamic Load Balancing, Global Optimization, IntervalMethods, Dynamic Systems, Parallel Computing
Interval methods [e.g., 1, 2] that provide rigorous, verified enclo-sures of all solutions to parametric ODEs serve as powerful boundingtools for use in branch-and-bound methods for the deterministic globaloptimization of dynamic systems [e.g., 3, 4]. In practice, however, thenumber of decision variables that can be handled by this approach isoften severely limited by considerations of computation time, especiallyin real-time or near real-time applications.
Since the early 2000s, parallel computing (multiprocessing), gener-ally in the form of multicore processors, has replaced frequency scalingto become the dominant cause of processor performance gains. Today,parallel computing hardware is ubiquitous, but the extent to which itis well exploited depends significantly on the application. There aremany opportunities to exploit fine-grained parallelism in applicationsof interval methods—for example, in basic interval arithmetic [5]. Wewill focus here on the coarse-grained parallelism that arises naturally inthe interval branch-and-bound procedure for global optimization. It iswell known that this provides the opportunity for superlinear speedups,and so has been a topic of continuing interest [e.g., 6, 7]. A key is-sue is the ability to dynamically balance workload, thus avoiding idlecomputational resources. We describe here a dynamic load balancing(DLB) framework, and implement two versions of it, one using MPI
164
(Message Passing Interface) and one using POSIX multi-threads, forsolving global dynamic optimization problems on a multicore, multi-processor server. We will use computational results to compare the twoimplementations and to evaluate several DLB design factors, includingthe communication scheme and virtual network used. Through thisframework it is possible to significantly reduce problem solution times(wall-clock) and increase the number of decision variables that can beconsidered within reasonable computation times.
References:
[1] Y. Lin, M.A. Stadtherr, Validated solutions of initial valueproblems for parametric ODEs, Appl. Numer. Math., 57 (2007),pp. 1145–1162.
[2] A.M. Sahlodin, B. Chachuat, Discretize-then-relax approachfor convex/concave relaxations of the solutions of parametric ODEs,Appl. Num. Math., 61 (2011) pp. 803–820.
[3] Y. Zhao, M.A. Stadtherr, Rigorous global optimization fordynamic systems subject to inequality path constraints, Ind. Eng.Chem. Res., 50 (2011), pp. 12678–12693.
[4] B. Houska, B. Chachuat B, Branch-and-lift algorithm for de-terministic global optimization in nonlinear optimal control, J. Op-timiz. Theory App., in press (2014).
[5] S.M. Rump, Fast and parallel interval arithmetic, BIT, 39 (1999),pp. 534–554.
[6] J.F. Sanjuan-Estrada, L.G. Casado, I. Garcıa, Adaptiveparallel interval branch and bound algorithms based on their per-formance for multicore architectures, J. Supercomput., 58 (2011),pp. 376–384.
[7] J.L. Berenguel, L.G. Casado, I. Garcıa, E.M.T. Hendrix,On estimating workload in interval branch-and-bound global opti-mization algorithms, J. Glob. Optim., 56 (2013), pp. 821–844.
165
Author index
Adm, Mohammad, 53Alliot, Jean-Marc, 162Alt, Rene, 26Anguelov, Roumen, 28Aschemann, Harald, 137, 139, 141,
143Auer, Ekaterina, 30, 141
Banhelyi, Balazs, 31, 33Balzer, Lars, 35Bauer, Andrej, 37Behnke, Henning, 38Boldo, Sylvie, 39Brajard, Julien, 47
Chabert, Gilles, 72Chohra, Chemseddine, 40Collange, Sylvain, 42Csendes, Tibor, 31
Defour, David, 42Demmel, James, 123Dobronets, Boris S., 44Dongarra, Jack, 46Du, Yunfei, 76Duracz, J., 87Durand, Nicolas, 162
Eberhart, Pacome, 47Elskhawy, Abdelrahman, 49
Fahmy, Hossam A. H., 103Farjudian, A., 87Fortin, Pierre, 47
Garlo↵, Jurgen, 53, 56Golodov, Valentin, 58Gotteland, Jean-Baptiste, 162Graillat, Stef, 42, 60Gustafson, John, 62
Hamadneh, Tareq, 56Hartman, David, 64Hiwaki, Tomohirio, 66Hladık, Milan, 64, 68, 70Horacek, Jaroslav, 70Horvath, Zoltan, 105
Iakymchuk, Roman, 42Ismail, Kareem, 49
Jezequel, Fabienne, 47Jaulin, Luc, 72Jeannerod, Claude-Pierre, 75Jiang, Hao, 76
Kambourova, Margarita, 26Kearfott, Ralph Baker, 78Kinoshita, Takehiko, 81Kobayashi, Kenta, 83, 85Kolechkina, Alla, 161Konecny, M., 87Kreinovich, Vladik, 96, 98Krisztin, Tibor, 31Kubica, Bart lomiej Jacek, 89Kubo, Takayuki, 117, 119Kumkov, S. I., 90Kupriianova, Olga, 92
166
Levai, Balazs Laszlo, 33Langlois, Philippe, 40Lauter, Christoph, 60, 92Le Doze, Vincent, 72Le Menec, Stephane, 72Liu, Xuefeng, 94Longpre, Luc, 96Lorkowski, Joe, 98Luther, Wolfram, 100
Maher, Amin, 103Markot, Mihaly Csaba, 105Markov, Svetoslav, 26, 28Mascarenhas, Walter, 121Mayer, Gunter, 107Minamihata, Atsushi, 109Miyajima, Shinya, 111, 114Mizuguchi, Makoto, 117, 119Montanher, Tiago, 121
Nakao, Mitsuhiro T., 81Nataraj, P. S. V., 131Neumaier, Arnold, 31Nguyen, Hong Diep, 123Ninin, Jordan, 72, 125
Ogita, Takeshi, 109, 127, 129Oishi, Shin’ichi, 60, 94, 109, 117,
119, 129, 155Ozaki, Katsuhisa, 129
Parello, David, 40Patil, Bhagyesh V., 131Peng, Lin, 76Popova, Evgenija D., 134Popova, Olga A., 44
Pryce, John, 135
Radchenkova, Nadja, 26Rauh, Andreas, 137, 139, 141, 143Revol, Nathalie, 125Rump, Siegfried M., 75, 109
Saad, Mohamed, 72Sekine, Kouta, 109Senkel, Luise, 137, 139, 143Sharaya, Irene A., 145Shary, Sergey P., 145, 147Siegel, Stefan, 149Sokolov, Igor, 151Spandl, Christoph, 153Stadtherr, Mark, 164Stancu, Alexandru, 72
Taha, W., 87Takayasu, Akitoshi, 117, 119Tanaka, Kazuaki, 155Tang, Ping Tak Peter, 60Theveny, Philippe, 157Tsuchiya, Takuya, 85
Uda, Tomoki, 159
van Nooijen, Ronald, 161Vanaret, Charlie, 162Vassilev, Spasen, 26
Wang, Feng, 76Watanabe, Yoshitaka, 81Westphal, Ramona, 141
Xu, Gang, 164
Yamamoto, Nobito, 66
167
Yamanka, Naoya, 60
Zhao, Yao, 164Zohdy, Maha, 49
168
169
List of participants
Country Participants Country Participants
Germany 18 United Kingdom 2Japan 17 Austria 1France 16 Brazil 1United States 7 China 1Russia 6 India 1Czech Republic 3 Netherlands 1Bulgaria 2 Poland 1Egypt 2 Slovenia 1Hungary 2 Sweden 1
Last Name First Name Country
Alefeld Gotz GermanyAuer Ekaterina GermanyBalzer Lars GermanyBanhelyi Balazs HungaryBauer Andrej SloveniaBehnke Henning GermanyBohlender Gerd GermanyBoldo Sylvie FranceCeberio Martine United StatesChohra Chemseddine FranceCsendes Tibor HungaryDallmann Alexander GermanyDobronets Boris RussiaDongarra Jack United StatesEberhart Pacome FranceElskhawy Abdelrahman EgyptGarlo↵ Jurgen GermanyGolodov Valentin Russia
170
Last Name First Name Country
Graillat Stef FranceGustafson John United StatesHartman David Czech RepublicHiwaki Tomohiro JapanHladık Milan Czech RepublicHoracek Jaroslav Czech RepublicIakymchuk Roman FranceJaulin Luc FranceJeannerod Claude-Pierre FranceJezequel Fabienne FranceJiang Hao ChinaKearfott Ralph Baker United StatesKimura Takuma JapanKinoshita Takehiko JapanKobayashi Kenta JapanKonecny Michal United KingdomKreinovich Vladik United StatesKubica Bartlomiej PolandKulisch Ulrich GermanyKumkov Sergey I. RussiaKupriianova Olga FranceLanglois Philippe FranceLiu Xuefeng JapanLouvet Nicolas FranceLuther Wolfram GermanyMaher Amin EgyptMarkot Mihaly Csaba AustriaMarkov Svetoslav BulgariaMayer Gunter GermanyMelquiond Guillaume FranceMezzarobba Marc FranceMinamihata Atsushi JapanMiyajima Shinya Japan
171
Last Name First Name Country
Mizuguchi Makoto JapanMontanher Tiago BrazilNakao Mitsuhiro JapanNehmeier Marco GermanyNguyen Hong Diep United StatesOgita Takeshi JapanOtsokov Shamil RussiaOzaki Katsuhisa JapanPaluri Nataraj IndiaPopova Evgenija BulgariaPryce John United KingdomRauh Andreas GermanyRevol Nathalie FranceSchulze Friederike GermanySenkel Luise GermanyShary Sergey RussiaSiegel Stefan GermanySokolov Igor RussiaSpandl Christoph GermanyStadtherr Mark United StatesTakayasu Akitoshi JapanTanaka Kazuaki JapanTheveny Philippe FranceTsuchiya Takuya JapanTucker Warwick SwedenUda Tomoki Japanvan Nooijen Ronald NetherlandsVanaret Charlie FranceWatanabe Yoshitaka JapanWol↵ von Gudenberg Jurgen GermanyYamamoto Nobito JapanZiegler Martin Germany
172