scan 2016 - math.uu.se€¦ · schedule – scan 2016 as of september 12, 2016. sunday, september...

SCAN 201617th International Symposium on Scientific Computing,

Computer Arithmetic and Verified Numerics

Book of AbstractsDepartment of Mathematics

Uppsala University

Sweden

September 26 – 29, 2016

17th International Symposium onScientific Computing, Computer Arithmetic and

Verified Numerics

SCAN 2016

Book of Abstracts

Department of MathematicsUppsala University

Sweden

September 26–29, 2016

SCAN 2016

Scientific Committee

• G. Alefeld (Karlsruhe, Germany)• A. Bauer (Ljubljana, Slovenia)• G.F. Corliss (Milwaukee, USA)• T. Csendes (Szeged, Hungary)• R.B. Kearfott (Lafayette, USA)• V. Kreinovich (El Paso, USA)• W. Luther (Duisburg, Germany)• G. Mayer (Rostock, Germany)• S. Markov (Sofia, Bulgaria)• J.-M. Muller (Lyon, France)• M. Nakao (Fukuoka, Japan)• T. Ogita (Tokyo, Japan)• S. Oishi (Waseda, Japan)• K. Ozaki (Tokyo, Japan)• M. Plum (Karlsruhe, Germany)• A. Rauh (Rostock, Germany)• N. Revol (Lyon, France)• J. Rohn (Prague, Czech Republic)• S. Rump (Hamburg, Germany/Tokyo, Japan)• S. Shary (Novosibirsk, Russia)• W. Tucker (Uppsala, Sweden)• W. Walter (Dresden, Germany)• J. Wolff von Gudenberg (Wuerzburg, Germany)• N. Yamamoto (Tokyo, Japan)

Organzing Committee• Warwick Tucker (Chair)• Anna Belova

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SCAN 2016

Preface

SCAN 2016 will continue the long tradition of promoting the area ofrigorous computing, a field that has reached a great level of maturity.This conference will cover the research areas of

• reliable computer arithmetic• enclosure methods• self validating algorithms

and will bring many new ideas for future applications and research.The booklet you have before you contains the abstracts of the invitedand contributed talks.

The conference starts with awarding the R. E. Moore Prize for Ap-plications of Interval Analysis to Arnold Neumaier, Balazs Banhelyi,Tibor Krisztin, and Tibor Csendes for their paper Global attractivityof the zero solution for Wright’s equation. Each morning and after-noon begins with a plenary talk, and I am very proud that the fol-lowing distinguished experts have accepted to present their research:

• Maciej Capinski (Poland)• Martine Ceberio (USA)• Mioara Joldes (France)• Hiroshi Kokubu (Japan)• Jean-Philippe Lessard (Canada)• Weldon Lodwick (USA)• Kaori Nagatou (Germany)• Mark Stadtherr (USA)

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SCAN 2016

Organizing SCAN 2016 in Uppsala

Planning and organizing this conference has been a great honourand pleasure for me. With the help of the international scientificcommittee many practical issues have been greatly simplified – ahuge thanks to all of you!

Locally, I have enjoyed the fine assistance of Inga-Lena Assarsson,Anna Belova, and Susanne Gauffin – I thank you all for supportingme in all aspects. I also thank Akademikonferens for taking care ofmany of the practical matters. Special thanks to Anders Kallstrom

, for typesetting this book of Abstracts.With regards to financing, the Swedish Research Council, as well

as Uppsala University have been very generous. Without their con-tributions, it would have been very difficult to organise an event ofthis magnitude.

But most importantly it is all of you who have graciously agreedto participate, and to disseminate your knowledge to us that havemade the greatest contribution to SCAN2016. And for this I thankyou, and wish you all a very reliable SCAN2016!

Uppsala, September 14 Warwick Tucker

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Schedule – SCAN 2016As of September 12, 2016.

Sunday, September 25, 2016

18:00 – 20:00 Get-together and registration (Norrlands Nation)

Monday, September 26, 2016

08:00 – 09:00 Registration (Norrlands Nation)

09:00 – 09:30 Opening

09:30 – 10:30 R. E. Moore Prize Awarding Ceremony (Gamla salen; Chair:Baker Kearfott)

Tibor Csendes, University of Szeged

Interval based checking algorithm fit to the parallel architecture of GPUswith an application to circle covering problems

10:30 – 11:00 Coffee break

11:00 – 12:00 Plenary talk (Gamla salen)

Hiroshi Kokubu, Kyoto University

Computer-assisted methods for detecting global structure of dynamics

12:00 – 13:20 Lunch (Norrlands Nation)


Mioara Joldes, LAAS-CNRS

Validated numerics for robust space mission design or Beyond Gravity(2013)

14:20 – 15:10 Parallel Sessions

Session A1: Fuzzy computations (Inre las, Chair: Vladik Kreinovich)

14:20 – 14:45 Weldon A. Lodwick (University of Colorado Denver),Interval Methods in the Calculation of Solutions to Fuzzy Interval Lin-ear Systems

14:45 – 15:10 K.K. Semenov (Peter the Great St. Petersburg Poly-technic University), Interval computations in the metrology

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Session A2: ODEs (Gamla salen, Chair: Michael Plum)

14:20 – 14:45 Masahide Kashiwagi (Waseda University), A study onverified ODE solver from the standpoint of stiffness

14:45 – 15:10 Akitoshi Takayasu (University of Tsukuba), Verifiednumerical computations for blow-up solutions of ODEs



Session B1: Linear algebra (Inre las, Chair: Denis Gaidashev)

15:40 – 16:05 Roman Iakymchuk (KTH Royal Institute of Technol-ogy), Towards Fast, Accurate and Reproducible LU Factorization

16:05 – 16:30 Katsuhisa Ozaki (Shibaura Institute of Technology),Linear Systems with the Exact Solution for Numerical Tests

16:30 – 16:55 Yuka Yanagisawa (Waseda University), Verificationmethod for system of linear equations by QR factorization

16:55 – 17:20 Yuka Kobayashi (Tokyo Woman’s Christian University),An Accurate and Efficient Solution of Ill-conditioned Linear Systemsby Preconditioning Methods

17:20 – 17:45 Xuefeng Liu (Niigata University), A framework for high-precision verified eigenvalue bounds by using finite element methods

Session B2: Control (Gamla salen, Chair: Vladik Kreinovich)

15:40 – 16:05 Luc Jaulin (ENSTA Bretagne), Secure a zone withrobots

16:05 – 16:30 Simon Rohou (ENSTA Bretagne), Tube ProgrammingApplied to State Estimation

16:30 – 16:55 Andreas Rauh (University of Rostock), An Interval-Based Algorithm for Feature Extraction from Speech Signals

16:55 – 17:20 Andreas Rauh (University of Rostock), Interval-BasedIdentification of Friction and Hysteresis Models

17:20 – 17:45 Andreas Rauh (University of Rostock), Toward theOptimal Parameterization of Interval-Based Variable-Structure StateEstimation Procedures

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Tuesday, September 27, 2016


Mark A. Stadtherr, University of Notre Dame

Rigorous Method for Robust Optimization and Design of Nonlinear Dy-namic Systems



Session C1: Optimization (Inre las, Chair: Vladik Kreinovich)

10:30 – 10:55 Arnold Neumaier (University of Vienna), Generalizedintervals in global optimization

10:55 – 11:20 Ryo Kobayashi (Waseda University), A method of ver-ified computation for convex programming

11:20 – 11:45 Jurgen Garloff (University of Konstanz, HTWG Kon-stanz), Fast determination of the tensorial and simplicial Bernsteinenclosure

11:45 – 12:10 Ralph Baker Kearfott (University of Louisiana at Lafayette),Simplicial Branch and Bound in Interval Global Optimization

Session C2: Arithmetic (Gamla salen, Chair: Denis Gaidashev)

10:30 – 10:55 Nathalie Revol (ENS de Lyon), HPC and interval com-putations

10:55 – 11:20 Yusuke Morikura (Waseda University), Fast enclosurefor matrix multiplication on a GPU

11:20 – 11:45 Siegfried M. Rump (Hamburg University of Technol-ogy), The origin of interval arithmetic

11:45 – 12:10 Siegfried M. Rump (Hamburg University of Technol-ogy), Sharp error bounds for the Gamma function over the whole floating-point range



Kaori Nagatou, Karlsruhe Institute of Technology

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Orbital stability investigation for travelling waves in a nonlinearly sup-ported beam

14:30 – 17:50 Viking excursion (Gamla Uppsala)

Wednesday, September 28, 2016


Maciej Capinski, AGH University of Science and Technology

Geometric methods and computer assisted proofs for invariant mani-folds in dynamical systems



Session D1: Software (Inre las, Chair: Nathalie Revol)

10:30 – 10:55 Matthias Husken (University of Wuppertal), IeeeCC754++– an advanced tool to check IEEE 754-2008 conformity

10:55 – 11:20 Francois Fevotte (EDF R&D), VERROU: CESTACwithout recompilation

11:20 – 11:45 Romain Picot (Sorbonne Universites, EDF R&D), PROMISE:floating-point precision tuning with stochastic arithmetic

11:45 – 12:10 David P. Sanders (Universidad Nacional Autonoma deMexico (UNAM)), The Julia package ValidatedNumerics.jl and itsapplication to the rigorous characterization of open billiard models

Session D2: General (Gamla salen, Chair: Luc Jaulin)

10:30 – 10:55 Peter Franek (IST Austria), Zero Verification in Sys-tems of Equations: Interval-based Implementation of a Topological Test

10:55 – 11:20 David Romero i Sanchez (Universitat Autonoma deBarcelona), Numerical computation of invariant objects with wavelets

11:20 – 11:45 Denis Gaidashev (Uppsala University), Golden-meanuniversality for Siegel disks

11:45 – 12:10 Pedro Barragan (University of Texas at El Paso), Whysuperellipsoids: an explanation

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Martine Ceberio, University of Texas at El Paso

Using Interval Methods to handle Large Numerical Simulations


Session E1: PDEs (Inre las, Chair: Michael Plum)

14:30 – 14:55 Jonathan Wunderlich (Karlsruhe Institute of Technol-ogy), Computer-assisted existence proofs for one-dimensional Schrodinger-Poisson systems

14:55 – 15:20 Hussein Awala (Temple University), Validated Numer-ics Methods for the Mixed Boundary Value Problem for the System ofElastostatics

Session E2: General (Gamla salen, Chair: Denis Gaidashev)

14:30 – 14:55 Ivo List (University of Ljubljana), Efficient Dedekindreals in Haskell

14:55 – 15:20 Anastasia Volkova (Sorbonne Universites, UPMC),Computing the Worst-Case Peak Gain of Digital Filter in IntervalArithmetic



Session F1: ODEs (Inre las, Chair: Michael Plum)

15:50 – 16:15 Nobito Yamamoto (The University of Electro-Communications),Numerical verification of existence of homoclinic orbits in dynamicalsystems

16:15 – 16:40 Kaname Matsue (The Institute of Statistical Mathe-matics), Rigorous numerics of global trajectories for fast-slow systemswith an explicit range of multi-scale parameter

16:40 – 17:05 Alexandre Chapoutot (ENSTA ParisTech, UniversiteParis-Saclay), Runge-Kutta Theory and Constraint Programming

17:05 – 17:30 Irmina Walawska (Jagiellonian University), An implicitalgorithm for validated enclosures of the solutions to variational equa-tions for ODEs

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17:30 – 17:55 Florent Brehard (LAAS-CNRS), A New Efficient Al-gorithm for Computing Validated Chebyshev Approximations Solutionsof Linear Differential Equations

Session F2: PDEs (Gamla salen, Chair: Vladik Kreinovich)

15:50 – 16:15 Takuma Kimura (Saga University), Optimal order con-structive a priori error estimates for a full discrete approximation ofthe heat equation

16:15 – 16:40 Kouta Sekine (Waseda university), A norm estimationfor an inverse of linear operator using a minimal eigenvalue

16:40 – 17:05 Akitoshi Takayasu (University of Tsukuba), On ver-ification methods for parabolic partial differential equations using theevolution operator

17:05 – 17:30 Kazuaki Tanaka (Waseda University), On verified nu-merical computation for positive solutions to elliptic boundary valueproblems

17:30 – 17:55 Yoshitaka Watanabe (Kyushu University), Validatedconstructive error estimatations for bi-harmonic problems

19:00 – late Conference Banquet

Thursday, September 29, 2016


Weldon A. Lodwick, University of Colorado Denver

The Molecular Distance Geometry Problem Under Interval Uncertainty



Session G1: Multiprecision (Inre las, Chair: Denis Gaidashev)

10:30 – 10:55 Valentina Popescu (LIP, ENS-Lyon), Rigourous errorbounds for double-double operations

10:55 – 11:20 Nozomu Matsuda (The University of Electro-Communications),LILIB – Long Interval Library

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11:20 – 11:45 Hao Jiang (National University of Defense Technology),The Implementation of multi-precision package in multiple-componentformat in MATLAB

11:45 – 12:10 Clothilde Jeangoudoux (Sorbonne Universites, UPMC),A Decimal Multiple-Precision Interval Arithmetic Library

Session G2: General (Gamla salen, Chair: Vladik Kreinovich)

10:30 – 10:55 Evgenija D. Popova (Bulgarian Academy of Sciences),Enclosing the Solution Set to Interval Parametric Matrix EquationA(p)X = B(p)

10:55 – 11:20 Aymeric Grodet (Ehime University), Adaptive meshrefinement technique for the classical Plateau problem

11:20 – 11:45 Takuya Tsuchiya (Ehime University), Error Analysisof Lagrange Interpolation on Tetrahedrons

11:45 – 12:10 Ronald van Nooijen (Delft University of Technology),The properties of negation and zero in ringoids as defined by Kulisch



Jean-Philippe Lessard, Universite Laval

Rigorously verified computing for infinite dimensional nonlinear dy-namics: a functional analytic approach


Session H1: Control (Inre las, Chair: Andreas Rauh)

14:30 – 14:55 Shinya Miyajima (Iwate University), Fast validatedcomputation for solutions of algebraic Riccati equations arising in trans-port theory

14:55 – 15:20 Shinya Miyajima (Iwate University), Fast validatedcomputation for solutions of discrete-time algebraic Riccati equations

Session H2: (Gamla salen, Chair: Luc Jaulin)

14:30 – 14:55 Radoslav Paulen (Technische Universitat Dortmund),Model-based design of optimal experiments for guaranteed parameterestimation of nonlinear dynamic systems

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14:55 – 15:20 Harsh Purohit (Indian Institute of Technology Bom-bay), Automated tuning of robust fractional PID controller for intervalplants using Kharitonov’s theorem



Session I1: General (Inre las, Chair: Denis Gaidashev)

15:50 – 16:15 Milan Hladık (Charles University), When dependenciesdo not matter?

16:15 – 16:40 Antoine Plet (LIP, ENS Lyon) Sharp error bounds forcomplex floating-point inversion

16:40 – 17:05 Vladik Kreinovich (University of Texas at El Paso),Decision Making Under Interval Uncertainty as a Natural Example ofa Quandle

Session I2: Constraints (Gamla salen, Chair: Luc Jaulin)

15:50 – 16:15 Bart lomiej Kubica (Warsaw University of Life Sciences),A template-based C++ library for automatic differentiation and hullconsistency enforcing

16:15 – 16:40 Benoit Desrochers (ENSTA Bretagne), Relaxed inter-section of thick sets

16:40 – 17:05 Stefan Ratschan (Czech Academy of Sciences), SafetyVerification By Interval Based Quantified Constraint Solving

End of SCAN 2016

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SCAN 2016

Contents

Plenary speakersInterval based checking algorithm fit to the parallel archi-

tecture of GPUs with an application to circle coveringproblems, Balazs Banhelyi, Zsolt Bagoczki and TiborCsendes . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Geometric methods and computer assisted proofs for invari-ant manifolds in dynamical systems, Maciej Capinski . 23

Using Interval Methods to handle Large Numerical Simula-tions, Martine Ceberio . . . . . . . . . . . . . . . . . . 24

Validated numerics for robust space mission design or Be-yond Gravity (2013), Mioara Joldes . . . . . . . . . . . 26

Computer-assisted methods for detecting global structure ofdynamics, Hiroshi Kokubu . . . . . . . . . . . . . . . . 28

Rigorously verified computing for infinite dimensional non-linear dynamics: a functional analytic approach, Jean-Philippe Lessard . . . . . . . . . . . . . . . . . . . . . 30

The Molecular Distance Geometry Problem Under IntervalUncertainty, Weldon Lodwick, Tiago M. Costa, Hen-ricus Bouwmeester and Carlile Lavor . . . . . . . . . . 31

Orbital stability investigation for travelling waves in a non-linearly supported beam, Kaori Nagatou, P. Joseph McKenna(Storrs, USA) and Michael Plum (Karlsruhe, Germany) 33

Rigorous method for robust optimization and design of non-linear dynamic systems, Yao Zhao and Mark A. Stadtherr 35

Contributed talksValidated Numerics Methods for the Mixed Boundary Value

Problem for the System of Elastostatics, Hussein Awala 37Why superellipsoids: an explanation,

Pedro Barragan and Vladik Kreinovich . . . . . . . . . 38

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SCAN 2016

A New Efficient Algorithm for Computing Validated Cheby-shev Approximations Solutions of Linear DifferentialEquations, Florent Brehard, Nicolas Brisebarre andMioara Joldes . . . . . . . . . . . . . . . . . . . . . . . 40

Runge-Kutta Theory and Constraint Programming, JulienAlexandre dit Sandretto and Alexandre Chapoutot . . 42

Relaxed intersection of thick sets, Benoit Desrochers andLuc Jaulin . . . . . . . . . . . . . . . . . . . . . . . . . 44

VERROU: a CESTAC evaluation without recompilation, FrancoisFevotte and Bruno Lathuiliere . . . . . . . . . . . . . 46

Zero Verification in Systems of Equations: Interval-basedImplementation of a Topological Test, Peter Franek,Marek Krcal and Hubert Wagner . . . . . . . . . . . . 48

Golden-mean universality for Siegel disks, Denis Gaidashevand Michael Yampolsky . . . . . . . . . . . . . . . . . 50

Fast determination of the tensorial and simplicial Bernsteinenclosure, Jurgen Garloff and Jihad Titi . . . . . . . . 51

Adaptive mesh refinement technique for the classical Plateauproblem, Aymeric Grodet and Takuya Tsuchiya . . . . 53

When dependencies do not matter?, Milan Hladık . . . . . . 55IeeeCC754++ – an advanced tool to check IEEE 754-2008

conformity, Matthias Husken . . . . . . . . . . . . . . 57Towards Fast, Accurate and Reproducible LU Factorization,

Roman Iakymchuk, David Defour and Stef Graillat . . 59Secure a zone with robots, Luc Jaulin and Benoıt Zerr . . . 61A Decimal Multiple-Precision Interval Arithmetic Library,

Stef Graillat, Clothilde Jeangoudoux and ChristophLauter . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

The Implementation of multi-precision package in multiple-component format in MATLAB, Hao Jiang, PeibingDu, Kuan Li, Lin Peng . . . . . . . . . . . . . . . . . . 65

A study on verified ODE solver from the standpoint of stiff-ness, Masahide Kashiwagi . . . . . . . . . . . . . . . . 67

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SCAN 2016

Simplicial Branch and Bound in Interval Global Optimiza-tion, Ralph Baker Kearfott . . . . . . . . . . . . . . . 69

Optimal order constructive a priori error estimates for a fulldiscrete approximation of the heat equation, TakumaKimura, Teruya Minamoto and Mitsuhiro T. Nakao . 71

Decision Making Under Interval Uncertainty as a NaturalExample of a Quandle, Mahdokht Afravi and VladikKreinovich . . . . . . . . . . . . . . . . . . . . . . . . . 73

A template-based C++ library for automatic differentiationand hull consistency enforcing, Bart lomiej Kubica . . 75

Efficient Dedekind reals in Haskell, Ivo List . . . . . . . . . 77A framework for high-precision verified eigenvalue bounds

by using finite element methods, Xuefeng LIU . . . . . 79Interval Methods in the Calculation of Solutions to Fuzzy

Interval Linear Systems, Weldon A. Lodwick, LotfiTaher and Hana Veiseh . . . . . . . . . . . . . . . . . 81

LILIB – Long Interval Library, Nozomu Matsuda and No-bito Yamamoto . . . . . . . . . . . . . . . . . . . . . . 82

Rigorous numerics of global trajectories for fast-slow sys-tems with an explicit range of multi-scale parameter,Kaname Matsue . . . . . . . . . . . . . . . . . . . . . 84

Fast validated computation for solutions of algebraic Riccatiequations arising in transport theory, Shinya Miyajima 86

Fast validated computation for solutions of discrete-time al-gebraic Riccati equations, Shinya Miyajima . . . . . . 88

Fast enclosure for matrix multiplication on a GPU, YusukeMorikura, Yusuke Nozawa, Kouta Sekine, MasahideKashiwagi and Shi’nichi Oishi . . . . . . . . . . . . . . 90

Generalized intervals in global optimization, Arnold Neu-maier, Ferenc Domes, Tiago Montanher, Mihaly Markotand Hermann Schichl . . . . . . . . . . . . . . . . . . . 92

Linear Systems with the Exact Solution for Numerical Tests,Katsuhisa Ozaki and Takeshi Ogita . . . . . . . . . . . 93

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SCAN 2016

Model-based design of optimal experiments for guaranteedparameter estimation of nonlinear dynamic systems,Anwesh Reddy, Gottu Mukkula and Radoslav Paulen . 95

PROMISE: floating-point precision tuning with stochasticarithmetic, Stef Graillat, Fabienne Jezekiel, RomainPicot, Francois Fevotte and Bruno Lathuiliere . . . . . 97

Sharp error bounds for complex floating-point inversion, Claude-Pierre Jeannerod, Nicolas Louvet, Jean-Michel Mullerand Antoine Plet . . . . . . . . . . . . . . . . . . . . . 99

Rigourous error bounds for double-double operations, MioaraJoldes, Jean-Michel Muller and Valentina Popescu . . 100

Enclosing the Solution Set to Interval Parametric MatrixEquation A(p)X = B(p), Evgenija D. Popova . . . . . 102

Automated tuning of robust fractional PID controller for in-terval plants using Kharitonov’s theorem, Harsh Puro-hit, P.S.V. Nataraj . . . . . . . . . . . . . . . . . . . . 104

Safety Verification By Interval Based Quantified ConstraintSolving, Peter Franek, Jan Kur atko and Stefan Ratschan106

An Interval-Based Algorithm for Feature Extraction fromSpeech Signals, Andreas Rauh, Susann Tiede and Cor-nelia Klenke . . . . . . . . . . . . . . . . . . . . . . . . 108

Interval-Based Identification of Friction and Hysteresis Mod-els, Andreas Rauh and Harald Aschemann . . . . . . . 110

Toward the Optimal Parameterization of Interval-Based Variable-Structure State Estimation Procedures, Andreas Rauhand Harald Aschemann . . . . . . . . . . . . . . . . . 112

HPC and interval computations, Nathalie Revol . . . . . . . 114Tube Programming Applied to State Estimation, Simon Ro-

hou, Luc Jaulin, Lyudmila Mihaylova, Fabrice Le Bars,Sandor M. Veres . . . . . . . . . . . . . . . . . . . . . 116

Sharp error bounds for the Gamma function over the wholefloating-point range, Siegfried M. Rump . . . . . . . . 118

The origin of interval arithmetic, Siegfried M. Rump . . . . 119

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SCAN 2016

A method of verified computation for convex programming,Ryo Kobayashi, Takuma Kimura and Shin’ichi Oishi . 120

Numerical computation of invariant objects with wavelets,Lluıs Alseda i Soler and David Romero i Sanchez . . . 122

The Julia package ValidatedNumerics.jl and its applica-tion to the rigorous characterization of open billiardmodels, David P. Sanders, Luis Benet and NikolayKryukov . . . . . . . . . . . . . . . . . . . . . . . . . . 123

A norm estimation for an inverse of linear operator usinga minimal eigenvalue, Kouta Sekine, Kazuaki Tanakaand Shin’ichi Oishi . . . . . . . . . . . . . . . . . . . . 125

Interval computations in the metrology, Semenov K.K., SolopchenkoG.N. and Kreinovich V.Ya. . . . . . . . . . . . . . . . 127

Verified numerical computations for blow-up solutions ofODEs, Akitoshi Takayasu, Kaname Matsue, TakikoSasaki, Kazuaki Tanaka, Makoto Mizuguchi and Shin’ichiOishi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

On verification methods for parabolic partial differential equa-tions using the evolution operator, Akitoshi Takayasu,Makoto Mizuguchi, Takayuki Kubo and Shin’ichi Oishi 133

On verified numerical computation for positive solutions toelliptic boundary value problems, Kazuaki Tanaka, KoutaSekine, and Shin’ichi Oishi . . . . . . . . . . . . . . . . 135

Error Analysis of Lagrange Interpolation on Tetrahedrons,Kenta Kobayashi, Takuya Tsuchiya . . . . . . . . . . . 137

An implicit algorithm for validated enclosures of the solu-tions to variational equations for ODEs, Irmina Walawskaand Daniel Wilczak . . . . . . . . . . . . . . . . . . . . 139

The properties of negation and zero in ringoids as definedby Kulisch, Ronald van Nooijen and Alla Kolechkina . 141

Validated constructive error estimatations for bi-harmonicproblems, Yoshitaka Watanabe, Takehiko Kinoshita andMitsuhiro T. Nakao . . . . . . . . . . . . . . . . . . . 142

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SCAN 2016

Computing the Worst-Case Peak Gain of Digital Filter inInterval Arithmetic, Anastasia Volkova, Christoph Lauterand Thibault Hilaire . . . . . . . . . . . . . . . . . . . 143

Computer-assisted existence proofs for one-dimensional Schrodinger-Poisson systems, Jonathan Wunderlich . . . . . . . . . 145

Numerical verification of existence of homoclinic orbits indynamical systems, Nobito Yamamoto, Kaname Mat-sue and Shun Yamano . . . . . . . . . . . . . . . . . . 147

Verification method for system of linear equations by QRfactorization, Yuka Yanagisawa, Shin’ichi Oishi andFumi Noda . . . . . . . . . . . . . . . . . . . . . . . . 149

An Accurate and Efficient Solution of Ill-conditioned LinearSystems by Preconditioning Methods, Yuka Kobayashi,Takeshi Ogita and Katsuhisa Ozaki . . . . . . . . . . . 151

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Interval based checking algorithm fit tothe parallel architecture of GPUs with anapplication to circle covering problems

Balazs Banhelyi, Zsolt Bagoczki, and Tibor Csendes

University of SzegedP.O. Box 652, Szeged, Hungary

[email protected]

Keywords: circle covering, CUDA, parallel implementation

Videocards are not only for graphic display. With their high speedvideo memories, lots of math units and parallelism, they provide a verypowerful platform for general purpose computing tasks, in which wehave to deal with large datasets, which are highly parallelizable, havehigh computational complexity, etc. Our selected platform for testingis the CUDA (Compute Unified Device Architecture) [4], that grantsus direct reach to the virtual instruction set of the video card, and weare able to run our computations on dedicated computing kernels.

In this parallel environment we implemented a reliable method withproperly rounded interval arithmetic [2]. Our method is based on thebranch-and-bound algorithm, with the purpose to decide whether ornot any given property applies for an n-dimensional interval. This al-gorithm will give us the opportunity to use node level parallelization.This means, that our nodes are evaluated simultaneously on multiplethreads we start on the GPU. This is called low-level, or type 1 paral-lelization, since we don’t modify the searching trajectories, neither dowe modify the dimensions of the branch-and-bound tree.

For testing, we choose the circle covering problem [1, 3]. It is thedual of circle packing, where our goal is to find the densest packingof a given number of congruent circles with disjoint interiors in a unitsquare. In circle covering, we aim for finding the full covering of theunit square with congruent circles of minimal radii. Overlapping in-teriors are allowed. For achieving easily scalable test cases, we will

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decide the covering with precalculated circles, where the diameter ofthe circles is the diagonal of the unit square, divided by a given num-ber, n, and the number of the circles is the square of n. We scaled theproblem up to three dimensions, and ran tests with sphere coveringproblems, too, and we discuss the possibility of scaling the problem upto any dimensions easily. We report our parallelization results.

References:

[1] Balazs Banhelyi, Endre Palatinus, and Balazs L. Levai,Optimal circle covering problems and their applications, CentralEuropean J. Operations Research, 22(2014) pp. 815–832.

[2] Sylvain Collange, Marc Daumas, and David Defour, In-terval Arithmetic in CUDA, In: Wen-mei W. Hwu: GPU Com-puting Gems, Jade Edition, Morgan Kaufmann, 2011, pp. 99-107.

[3] Kari J. Nurmela and Patric R.J. Ostergard, Covering asquare with up to 30 equal circles, Research report, HUT-TCS-A62, Helsinki University of Technology, 2000 NP-hard classes oflinear algebraic systems with uncertainties,

[4] Jason Sanders and Edward Kandrot, CUDA by ExampleAn Introduction to General-Purpose GPU Programming, EdwardBrothers, Ann Arbor, 2010

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Geometric methods and computerassisted proofs for invariant manifolds in

dynamical systems

Maciej Capinski

AGH University of Science and Technologyal. Mickiewicza 30, 30-059 Krakow, Poland

[email protected]

Invariant manifolds can be used to determine global behaviour ofdynamical systems. They can often be proved using geometric or topo-logical arguments. In the talk we demonstrate how to apply such tech-niques for proofs of stable/unstable manifolds of fixed points and fornormally hyperbolic manifolds. We show how to combine these geo-metric methods with interval verified numerics, obtaining proofs forproblems inaccessible using standard techniques.

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Using Interval Methods to handle LargeNumerical Simulations

Martine Ceberio

Computer Science DepartmentThe University of Texas at El Paso

500 W. University, El Paso, Texas, [email protected]

Keywords: dynamic systems, interval computations, constraint solv-ing, proper-orthogonal decomposition (POD), reduced-order modeling(ROM).

Many natural phenomena can be modeled as ordinary or partialdifferential equations. A way to find solutions of such equations isto discretize them and solve the corresponding (possibly) nonlinearlarge systems of equations; see [1]. Major issues with solving suchnonlinear systems include the fact that their dimension can be verylarge and that uncertainty, often present, is tricky to handle. Model-Order Reduction (MOR) has been proposed as a way to overcome theissues associated with large dimensions, the most used approach fordoing so being Proper Orthogonal Decomposition (POD); see [2,3].The key idea of POD is to reduce a large number of interdependentvariables (snapshots) of the system to a much smaller number of uncor-related variables while retaining as much as possible of the variation inthe original variables. On the other hand, interval constraint-solvingtechniques (ICST) [4] allow to handle uncertainty and ensure reliableresults of systems of (possibly) nonlinear equations.

In this presentation, we show how intervals and constraint solvingtechniques (ICST) can be used to compute all the snapshots at once(IPOD). We take advantage of using interval techniques to also showhow IPOD can address not only dimension but also uncertainty. Asa result, this new process gives us two advantages over the traditionalPOD method: 1. handling uncertainty in some parameters or inputs;

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2. reducing the snapshots computational cost. We go over numeri-cal examples of the use of IPOD and we then take a glimpse at theimplications of this work: How does using interval constraint solvingtechniques and handling uncertainty affect the whole simulation pro-cess? How is the reduced-order model then solved?

Acknowledgment: Most of this work was supported by Stanford’sArmy High-Performance Computing Research Center (AHPCRC) fundedby the Army Research Lab (ARL), and by the National Science Foun-dation award 0953339.

References:

[1] J. Li and Y. Chen, Computational Partial Differential EquationsUsing MATLAB. CRC Press, Las Vegas (NV), 2008.

[2] W. H. Schilders and H. A. Vorst, Model Order Reduction:Theory, Research Aspects and Applications. Springer Science &Business Media, 2008.

[3] K. Carlberg, C. Bou-Mosleh, and C. Farhat, Efficientnon-linear model reduction via a least-squares Petrov–Galerkin pro-jection and compressive tensor approximations, Int. J. Numer.Meth. Engng., vol. 86, pp. 155–181, 2011.

[4] L. Granvilliers, and F. Benhamou, RealPaver: An IntervalSolver using Constraint Satisfaction Techniques. ACM Trans. onMathematical Software 32(1), 138–156, 2006.

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Validated numerics for robust spacemission design

or

Beyond Gravity (2013).

Mioara Joldes

LAAS-CNRS,7 Avenue du Colonel Roche, 31077 Toulouse, France

[email protected]

Keywords: validated numerics, symbolic-numeric method, space mis-sion, space rendez-vous, validated trajectories, collision probability

In this talk we present an overview of several symbolic-numericobjects, algorithms and software tools developed for applications inoptimal control and robust space mission design. In this domain, certi-fication of computations is at stake and we aim at providing computer-aided proofs of numerical values, with validated and reasonably tighterror bounds, without sacrificing efficiency.

One application is the computation of collision probabilities be-tween space objects in low Earth orbits. The large number of orbit-ing debris constitute a serious hazard for operational satellites. Thetrade-off between the collision risk and the inherent risks of performinga collision avoidance maneuver is a strong incentive to precisely esti-mate the collision probability between two orbiting objects, given theuncertainties on their locations. We present a new method for com-puting the probability of collision between two spherical space objectsinvolved in a short-term encounter under Gaussian-distributed uncer-tainty. In this model of conjunction, classical assumptions reduce theprobability of collision to the integral of a two-dimensional Gaussianprobability density function over a disk. The computational methodpresented here is based on an analytic expression for the integral, de-rived by use of Laplace transform and D-finite functions properties.Analytic bounds on the truncation error are also derived and are usedto obtain a very accurate algorithm.

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Another application is the computation of validated impulsive con-trol for the rendezvous problem of spacecrafts. The rendezvous (RdV)problem consists in designing a plan of maneuvers which takes oneactive spacecraft, originally moving on an initial orbit, to the finalreference orbit of a passive target spacecraft e.g., International SpaceStation. The impulsive nature comes from the way the thrusters ofmost satellites work. Since the ’60s many ideas were developed, andtoday, we are interested in successful RdV which minimizes fuel con-sumption, with increased autonomy (no human operator). This im-plies that validation of computations and solutions is at stake. Inthis talk we discuss the fixed-time minimum-fuel rendezvous betweenclose elliptic orbits, assuming a linear impulsive setting and a Kep-lerian relative motion. Firstly, the optimal velocity increments andimpulses locations are numerically obtained with a new iterative algo-rithm with proven convergence. This is based on discretizing a semi-infinite convex optimization problem. Secondly, the obtained numeri-cal solutions are validated by propagating the trajectories solutions ofthe linear differential equations of the dynamics. These are computedas truncated Chebyshev series together with rigorously computed errorbounds. Different realistic numerical examples illustrate these results.

This talk is based on joint works with D. Arzelier, F. Brehard, N.Brisebarre, J.-B. Lasserre, C. Louembet, A. Rondepierre, B. Salvy, R.Serra.

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Computer-assisted methods for detectingglobal structure of dynamics

Hiroshi Kokubu

Kyoto UniversityKyoto 606-8502, Japan

[email protected]

Keywords: dynamical system, Conley-Morse graph, ODE integration

Morse decomposition is a decomposition of the chain recurrent in-variant set of a dynamical system into finite collection of invariant sets,called Morse sets, that are related in a gradient-like manner. Morsegraph represents the gradient-like connection in terms of a finite di-rected graph whose nodes are Morse sets and edges exhibit (possibilityof) flow-defined connections. If each node is associated with the topo-logical information of the dynamics, more precisely the Conley index,of the corresponding Morse set, the graph is called the Conley-Morsegraph.

In [1] and [2] we have developed a computer-assisted method forobtaining the Conley-Morse graphs of a multi-parameter family of dis-sipative dynamical systems, in particular, in the form of iterated maps.This method can also be applied to families of ODEs, by using time-Tmaps of the flows generated by the ODEs. However, there are severalcomputational issues that need to be further investigated.

In this talk, we shall first review the method of Conley-Morsegraphs and show several examples of computations. We then discusssome issues of time integration of ODEs and show a benchmark com-parison of CAPD and COSY softwares, intended for the type of compu-tations for our purpose, namely multi-scale set-oriented computations([3]).

A modification of the method using time-series data from a dynam-ical system has been considered recently, which is mainly developed forapplication purposes, but is of theoretical interest as well.

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References:

[1] Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka,P. Pilarczyk, A database schema for the analysis of global dy-namics of multiparameter systems, SIAM Journal on Applied Dy-namical Systems, 8 (2009), pp. 757–789.

[2] J. Bush, M. Gameiro, S. Harker, H. Kokubu, K. Mis-chaikow, I. Obayashi, P. Pilarczyk, Combinatorial-topologicalframework for the analysis of global dynamics, Chaos 22 (2012),047508.

[3] T. Miyaji, P. Pilarczyk, M. Gameiro, H. Kokubu, K. Mis-chaikow, A study of rigorous ODE integrators for multi-scaleset-oriented computations, Applied Numerical Mathematics, 107(2016), pp. 34–47.

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Rigorously verified computing for infinitedimensional nonlinear dynamics:a functional analytic approach

Jean-Philippe Lessard

Universite LavalDepartement de mathematiques et de statistique

1045 av. de la Medecine, Quebec, QC, G1V 0A6, [email protected]

Studying and proving existence of solutions of nonlinear dynamicalsystems using standard analytic techniques is a challenging problem.In particular, this problem is even more challenging for partial dif-ferential equations, variational problems or functional delay equationswhich are naturally defined on infinite dimensional function spaces.The goal of this talk is to present rigorous numerical technique relyingon basic functional analytic tools to prove existence of steady states,time periodic solutions, traveling waves and connecting orbits for theabove mentioned dynamical systems. We will spend some time iden-tifying difficulties of the proposed approach as well as time to identifyfuture directions of research.

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The Molecular Distance GeometryProblem Under Interval Uncertainty

Weldon Lodwick∗, Tiago M. Costa∗, Henricus Bouwmeester∗∗,and Carlile Lavor∗∗∗

∗ University of Colorado Denver, Department of Mathematical andStatistical Sciences, USA

∗∗ Metropolitan State University of Denver∗∗∗ University of Campinas/[email protected]

More recent interval approaches are used to analyze the DistanceGeometry Problem (DGP) under interval uncertainty where we restrictourselves, in this presentation, to the molecular DGPs. The molecularDGP is the following:

Given the n atoms of a molecule and distances dij between atoms iand j in R3, find coordinates, xk, of the atoms k = 1, .., n, such that thedistances between xi and xj are equal to the given Euclidean distancesdij, that is,

‖xi − xj‖2 = dij.

The case which is of interest to this presentation focuses on anincomplete set of distances some of which are intervals due to mea-surement errors. This problem, for the case of incomplete real-valueddistances, is, in general, NP-Hard. When the problem is restricted tothe molecular distance geometry problem, for example, those arisingfrom protein molecules where the real-valued distances are obtainedfrom a nuclear magnetic resonance machine, the number of solutionsare finite though there are 2n possible solutions.

The inclusion of measurement errors from the nuclear resonancemachine is a more realistic approach to the molecular DGP. The mea-surement errors we assume to be intervals. Any approach that con-siders a distance as a real-number within the given real-valued interval

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error will produce only one of the possible conformations. We attemptto find all possible conformations using interval analytic techniques.

This presentation discusses the use of interval analytic methods infour facets of the molecular DGP:

1. The representation of the problem under interval uncertainty - werepresent all intervals as a one variable parameterization;

2. The reduction of the interval uncertainty - properties of distancesin R3 such as the triangle inequality and any molecule-specific extrainformation are applied to the given interval distance to obtain thetightest interval bounds on the distance data, though for this talk,we will assume that the interval distances, as given, are the tightestpossible;

3. The propagation of interval errors - given the tightest interval dis-tance bounds, the single-valued parameterization representation ispropagated through the 2n possible coordinate computations, thatis, we propagate the symbolic representation of the interval as asingle parameterization, not the interval itself;

4. The computation of a set of conformations - given the propagatedsymbolic values of the coordinates, the interval-values of the coor-dinates are computed as a one variable global optimization prob-lem.

This research was partially funded by CNPq grant 400754/2014-2.

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Orbital stability investigation fortravelling waves in a nonlinearly

supported beam

Kaori Nagatou∗, P. Joseph McKenna (Storrs, USA)and Michael Plum (Karlsruhe, Germany)

∗Institute for Analysis, Karlsruhe Institute of Technology∗Englerstrasse 2, 76131 Karlsruhe

[email protected]

We consider the fourth-order wave equation

ϕtt + ϕxxxx + f(ϕ) = 0, (x, t) ∈ R× R+,

with a nonlinearity f vanishing at 0. Solitary traveling waves ϕ =u(x− ct) satisfy the ODE

u′′′′ + c2u′′ + f(u) = 0 on R,

and for the case f(u) = eu − 1, the existence of at least 36 travellingwaves was proved in [1] by computer assisted means.

We investigate the orbital stability of these solutions via computa-tion of their Morse indicies and using results from [2] and [3]. In orderto achieve it we make use of both analytical and computer-assistedtechniques.

References:

[1] B. Breuer, J. Horak, P. J. McKenna, M. Plum, A computer-assisted existence and multiplicity proof for travelling waves ina nonlinearly supported beam, Journal of Differential Equations,224, pp. 60-97, 2006.

[2] J. Shatah and W. Strauss, Stability Theory of Solitary Wavesin the Presence of Symmetry, I, Journal of Functional Analysis,74, pp. 160-197, 1987.

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[3] J. Shatah and W. Strauss, Stability Theory of Solitary Wavesin the Presence of Symmetry, II, Journal of Functional Analysis,94, pp. 308-348, 1990.

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Rigorous method for robust optimizationand design of nonlinear dynamic systems

Yao Zhao and Mark A. Stadtherr

University of Notre DameDepartment of Chemical and Biomolecular Engineering

Notre Dame, Indiana, [email protected]

Keywords: robust design, robust optimization, nonlinear dynamicsystems, interval methods, Taylor models

In engineering product or process design it is common to have todeal with uncertainties, which may arise due to variability in problemparameters (e.g., physical properties of materials) or to the possibil-ity of external disturbances (e.g., change in temperature or flow rateof a feed stream). A design is considered robust if it will continueto meet specified constraints (e.g., on safety and/or performance) forall potential values of the uncertain quantities. Thus, the goal is toidentify a region in the design space for which this robustness propertyis achieved. If finding a minimum in some specified objective func-tion for performance is also desired, then robustness may be soughtby formulating the problem as a min-max optimization problem. Inthis case, the problem may be thought of as finding the best possibleperformance in the worst-case scenario with regard to the uncertain-ties. For systems with nonlinear dynamic constraints, these robustdesign and optimization problems may become particularly difficultto solve rigorously, and in many cases existing methods rely on prob-lem simplifications or modifications (e.g., linearization, discrete timeapproximation).

In this presentation we will review a framework for solving robustdesign and optimization problems using interval methods. This com-bines ideas from region transition modeling [1] and solution of staticmin-max optimization problems [2] with methods using Taylor models

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for rigorous solution of interval-valued ODE problems [3] and rigorousglobal optimization of nonlinear dynamic systems [4, 5]. A variety ofexample problems will be used.

References:

[1] H. Huang, C.S. Adjiman, N. Shah, Quantitative frameworkfor reliable safety analysis, AIChE J., 48 (2002), pp. 78-96.

[2] S. Zuhe, A. Neumaier, C. Eiermann, Solving min-max prob-lems by interval methods, BIT, 30 (1990), pp. 742–751.

[3] Y. Lin, M. A. Stadtherr, Validated solutions of initial valueproblems for parametric ODEs, Appl. Numer. Math., 57 (2007),pp. 1145–1162.

[4] Y. Lin, M. A. Stadtherr, Deterministic global optimization ofnonlinear dynamic systems, AIChE J., 53 (2007), pp. 866–875.

[5] Y. Zhao, M. A. Stadtherr, Rigorous global optimization fordynamic systems subject to inequality path constraints, Ind. Eng.Chem. Res., 50 (2011), pp. 12678–12693.

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Validated Numerics Methods for theMixed Boundary Value Problem for the

System of Elastostatics

Hussein Awala

Temple UniversityPhiladelphia, PA 19122, [email protected]

Elliptic boundary value problems with mixed Dirichlet and Neu-mann type boundary conditions arise naturally in connection withphysical phenomena such as conductivity, heat transfer, elastic de-formations, and electrostatics. In my talk I will discuss recent well-posedness results for the mixed boundary problem for the system ofelastostatics in infinite sectors in two dimensions. These results are ob-tained through a blend of Calderon-Zygmund theory methods, Mellintransform techniques, and validated numerics. This work is part of anongoing collaboration project with Irina Mitrea and Warwick Tucker.

References:

[1] I. Mitrea, W. Tucker, Some Counterexamples for the SpectralRadius Conjecture, Differential and Integral Equations, 16 (2003),No. 12, pp. 1409 -1439.

[2] I. Mitrea, W. Tucker, Interval Analysis Techniques for Bound-ary Value Problems of Elasticity in Two Dimensions, Journal ofDifferential Equations, 233 (2007), pp. 181 -198.

[3] H. Awala, I. Mitrea, K. Ott, On the Solvability of the ZarembaProblem in Infinite Sectors and the Invertibility of Associated Sin-gular Integral Operators, preprint 2016.

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Why superellipsoids: an explanation

Pedro Barragan and Vladik Kreinovich

University of Texas at El Paso500 W. University, El Paso, TX 79968, USA

[email protected]

Keywords: superellipsoid, uncertainty, probabilities

Need to describe uncertainty domains. The intent of the massproduction of a gadget is to produce gadgets with identical values(x1, . . . , xn) of the desired characteristics xi. In reality, of course, dif-ferent gadgets end up having slightly different values xi of these char-

acteristics: ∆xidef= xi − xi 6= 0. For each of these characteristics xi,

we usually have a tolerance bound ∆xi for which |∆xi| ≤ ∆i, so thatpossible values of ∆xi form an interval [−∆i,∆i]. Thus, possible val-ues of the deviation vector ∆x = (∆x1, . . . ,∆xn) are located in thebox [−∆1,∆1]× . . .× [−∆n,∆n]. In practice, not all vectors ∆x fromthis box are possible. It is therefore desirable to describe the set of allpossible deviation vectors ∆x. This set is known as the uncertaintydomain.

Shall not we also determine probabilities? At first glance, itseems that we should be interested not only in finding out which de-viation vectors ∆x are possible and which are not, but also in howfrequent different possible vectors are. In other words, we should beinterested not only in the uncertainty domain, but also on the probabil-ity distribution on this domain. In reality, however, it is not possibleto find these probabilities. Indeed, the manufacturing process mayslightly change (and often does change). After each such change, thetolerance intervals and the resulting uncertainty domain remain largelyunchanged, but the probabilities change (often drastically).

Empirical shapes of uncertainty domains. Empirical analysis hasshows that in many practical cases, the uncertainty domain can be well

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approximated by a super-ellipsoidn∑

i=1

( |∆xi|σi

)p

≤ C for some values

σi, p, and C, and the accuracy of this approximation is higher than forother approximation families with the same number of parameters.

What we do in this paper. In this paper, we provide a theoreticalexplanation for this empirical phenomenon.

Our idea. In reality, there is some probability distribution ρi(∆xi)for each of the random variables ∆xi. Since we have no reason to as-sume that positive values are more probable than negative values orvice versa, it makes sense to assume that they are equally probable,i.e., that each distribution ρi(∆xi) is symmetric: ρi(∆xi) = ρi(|∆xi|).Similarly, since we have no reasons to believe that different deviationsare statistically dependent, it makes sense to assume that the corre-sponding random variables are independent. In this case, the overall

probability density function (pdf) has the form ρ(∆x) =n∏

i=1

ρi(|∆xi|).Usually, we consider a deviation vector possible if its probability

exceed a certain threshold t. Thus, the desired set has the form Stdef=

∆x : ρ(∆x) ≥ t. Numerical values of the deviations ∆xi depend onthe choice of a measuring unit; if we replace the original unit by a unitwhich is λ times smaller, then for the exact same physical situation, weget the new numerical values ∆x′i = λ ·∆xi. Since the physics remainsthe same, it makes sense to require that the uncertainty domains do notchange under such a re-scaling. To be more precise, the pdf thresholdt may change, but the family of such sets should remain unchanged:S ′tt = Stt, where S ′t corresponds to the re-scaled pdf ρ′(∆x) =const · ρ(λ ·∆).

We prove that under this scale-invariance, the corresponding setsSt are exactly super-ellipsoids. Thus, we get the desired explanation.

References:

[1] I. Elishakoff, F. Elettro, Interval, ellipsoid, and super-ellipsoidcalculi for experimental and theoretical treatment of uncertainty:which one ought to be preferred?, International Journal of SolidStructures, 51 (2014), No. 7, pp. 1576–1586.

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A New Efficient Algorithm for ComputingValidated Chebyshev Approximations

Solutions of Linear Differential Equations

Florent Brehard∗, Nicolas Brisebarre∗∗ and Mioara Joldes∗∗LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse, France∗∗CNRS, LIP, ENS Lyon, 46 Allee d’Italie, 69364 Lyon, France

[email protected]

[email protected] [email protected]

Keywords: validated numerics, LODE, Chebyshev series, Newtonmethod, fixed-point operator, error bound

In this work we develop a validated numerics method for the solu-tion of linear ordinary differential equations (LODEs). It is well knownthat the Picard-Lindelof theorem ensures under mild assumptions theexistence and uniqueness of the solution to such an equation. How-ever, in general this solution lacks an easily computable closed form.A wide range of algorithms (i.e., Runge-Kutta, collocation, spectralmethods) exist for numerically computing approximations of the so-lutions. Most of these come with proofs of asymptotic convergence,but usually, provided error bounds are non-constructive. However, insome domains like critical systems and computer-aided mathematicalproofs, one needs validated effective error bounds. For that, one solu-tion is to use so-called a priori validated methods, which provide ateach iteration both an approximation and bounds on the error e.g.,validated Taylor approximations [3]. In contrast, a posteriori valida-tion methods take as argument an approximate solution (computed bysome numerical algorithm) and compute afterwards an error bound.Our contribution belongs to this second class and it relies on a fixedpoint argument of a contracting map [5].

The two main ingredients of our method are: (1) numerical approx-imation by Chebyshev truncated series of (regular enough) functions oncompact intervals. While appearing more natural to use, Taylor ap-proximation techniques may not be valid when the function is not suffi-ciently regular or has singularities in the complex disk surrounding the

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real interval considered. In contrast, Chebyshev approximations onlyrequire mild assumptions, typically a Lipschitz condition. Moreover,for a fixed degree, Chebyshev truncated series is guaranteed to be anear-best approximation. Also, standard operations on polynomialsare not much harder in Chebyshev basis than in the monomial one [2].

(2) Expressing the solution as the unique fixed point of a compactlinear integral operator, defined with multiplication by polynomials andintegration. In Chebyshev basis, it has the form of an ”almost-banded”operator: the corresponding (infinite) matrix has non-zero coefficientsonly on few upper/lower diagonals and initial rows. Based on this,in [4], a linear time (with respect to the truncation degree) numericalalgorithm has been proposed. Another linear complexity algorithm wasgiven in [1], observing that when applying the operator, the coefficientsof the expansions obey linear recurrence relations with polynomial co-efficients. For validating the solution, one possibility is to notice thatafter a finite number of iterations the operator becomes contracting [1].We use a different approach, namely a pseudo-Newton method [5]. Thekey idea is to multiply the operator by an approximate inverse in fi-nite dimension, so that the result is close to the identity map, and thedifference between them is hence a contracting operator. A technicalcontribution consists in correctly bounding the norm of this operator.

Our new approach is illustrated by validating solutions of LODEsappearing in robust guidance algorithms for space trajectories.

References[1] A. Benoit, M. Joldes, and M. Mezzarobba. Rigorous uniform approximation of d-finite functions

using chebyshev expansions. Mathematics of Computation, (To appear), 2016.

[2] N. Brisebarre and M. Joldes. Chebyshev interpolation polynomial-based tools for rigorouscomputing. In ISSAC ’10: Proceedings of the 2010 International Symposium on Symbolic andAlgebraic Computation, page 147–154, 2010.

[3] M. Neher, K. R. Jackson, and N. S. Nedialkov. On Taylor model based integration of ODEs.SIAM J. Numer. Anal., 45:236–262, 2007.

[4] S. Olver and A. Townsend. A fast and well-conditioned spectral method. SIAM Review,55(3):462–489, 2013.

[5] N. Yamamoto. A numerical verification method for solutions of boundary value problemswith local uniqueness by banach’s fixed-point theorem. SIAM Journal on Numerical Analy-sis, 35(5):2004–2013, 1998.

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Runge-Kutta Theory and ConstraintProgramming

Julien Alexandre dit Sandretto and Alexandre Chapoutot

U2IS, ENSTA ParisTech, Universite Paris-Saclay,828 bd des Marechaux, 91762 Palaiseau cedex France

[email protected]

Keywords: Runge-Kutta, Butcher theory, Validated simulation

Introduction

There exist many Runge-Kutta methods (explicit or implicit), more orless adapted to a given class of problems. Some of them have interest-ing properties such as A-stability for stiff problems or symplecticity forproblems with energy conservation. Defining a new method, adaptedto a given class of problems, has become a challenge. Indeed, the num-ber of stages and the order don’t stop to increase. This race to the“best” method is interesting but forgot an important problem. Moreprecisely, the coefficients of a Runge-Kutta method are more and moredifficult to compute and the result is often given in floating-point num-bers, which may lead to violate their definition rules. We propose amethod using interval analysis tools to compute Runge-Kutta coeffi-cients by using a solver based on guaranteed constraint programming.Moreover, with a global optimization process and a well chosen costfunction, we propose a way to define some novel optimal Runge-Kuttamethods.

One step of a Runge-Kutta integration scheme, applied on an or-dinary differential equation y = f(t, y), is obtained with

yn+1 = yn + hs∑

i=1

biki, where ki = f

(t0 + cih, y0 + h

s∑

j=1

aijkj

). (1)

The coefficients ci, aij and bi, for i, j = 1, · · · , s, fully characterize theRunge-Kutta methods and they are usually synthesized in a Butcher

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tableau [1] of the form:c1 a11 . . . a1s...

.... . .

...cs as1 . . . ass

b1 . . . bs

Main idea

Our approach consists on the generation of constraints defined by theButcher theory in order to build a Runge-Kutta method. Then we solvethese constraints with a Branch&Prune algorithm. We also proposea Branch&Bound approach to define new optimal methods w.r.t. aneasy to obtain cost function.

Interval coefficients preserve properties

In a preliminary stage, we can verify that a Runge-Kutta method withinterval coefficients preserves the Butcher rule and then that a methodgiven for an order p has really a local truncature error in O(hp+1).We also propose three methods using interval tools to check the linearstability, algebraically stability and symplecticity properties.

Main results

First, our Branch&Prune based approach is used to find existing meth-ods and by the way re-discover the Runge-Kutta theory such as i)Gauss-Legendre is the only 2-stages 4-order method; ii) there is no2-stages 5-order method; etc. Our Branch&Bound method finds thesame results as Ralston [2]. Second, both of our methods is used todefine new validated Runge-Kutta methods.

References:

[1] Butcher, John C., Coefficients for the study of Runge-Kuttaintegration processes, Journal of the Australian Mathematical So-ciety, 5 (1963), No. 3, pp. 185–201.

[2] Ralston, Anthony, Runge-Kutta methods with minimum errorbounds, Mathematics of computation, (1962), pp. 431-437.

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Relaxed intersection of thick sets

Benoit Desrochers and Luc Jaulin

Lab-STICC, ENSTA Bretagne, Brest, France

[email protected]

Relaxed Intersection : The q-relaxed intersectionm sets X1, . . . ,Xm

of Rn, is the set of all x ∈ Rn which belong to all Xi’s except q at most.This notion is important in the context of robust bounded error esti-mation [1].

Thick set. Denote by (P(Rn),⊂), the powerset of Rn equippedwith the inclusion ⊂ as an order relation. A thick set JXKof Rn is aninterval of (P(Rn),⊂). If JXK is a thick set of Rn , there exist [2] twosubsets of Rn , called the subset bound and the supset bound such that

JXK = JX⊂,X⊃K = X ∈ P(Rn)|X⊂ ⊂ X ⊂ X⊃.

Thick test. Given a thick set JXK, the corresponding thick testJtK : Rn → 0, ?, 1 associates to any x: 0 if x /∈X⊃, 1 if x ∈ X⊂ and ?otherwise.

Algorithm. Using a paver and an arithmetic on thick tests weshow that we can easily obtain a guaranteed approximation of therelaxed intersection of m thick sets.

TestCase: Let us take the following system of interval linear equa-tions [3]:

[2, 4] [−2,−1][0, 2] [−3,−1]

[−6,−4] [−1, 0][−3,−2] [−3,−2][−1, 1] [−5,−4]

· x =

[−1, 1][−5, 0][−6, 0][2, 7]

[−9,−3]

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Each line corresponds to a thick set in R2. Using the relaxed in-tersection of thick tests, we obtain the characterization sets for q ∈1, 2, 3 as given by Figure 1. For q = 0, the solution set is empty.

Figure 1: Solution sets for q = 1 (Right), q= 2 (middle) and q = 3(Right). Red boxes are inside X⊂, while blue ones are outside X⊃ andorange ones belong to ∆X = X⊃ \ X⊂. Other boxes are too small tobe bisected and are undeterninated.

References:

[1] Q. Brefort, L Jaulin, M. Ceberio, V. Kreinovich, To-wards Fast and Reliable Localization of an Underwater Object:An Interval Approach, Journal of Uncertain Systems, (2015).

[2] B. Desrochers, L. Jaulin, Computing a guaranteed approxima-tion the zone explored by a robot, IEEE Transaction on AutomaticControl, (2016).

[3] V. Kreinovich and S. Shary, Interval Methods for Data Fittingunder Uncertainty: A Probabilistic Treatment, Reliable Comput-ing, (2016).

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VERROU: a CESTAC evaluation withoutrecompilation

Francois Fevotte and Bruno Lathuiliere

EDF R&D7 boulevard Gaspard Monge, F-91120, Palaiseau, FRANCEfrancois.fevotte, [email protected]

Keywords: V&V, Numerical Verification, CESTAC, MCA, Valgrind

As an industrial facility relying on numerical simulation to improvethe safety and efficiency of its electricity production units, EDF is com-mitted to ensure that all the numerical simulation codes it developsand uses are correctly validated and verified. Within this context, theaccuracy of floating-point operations has progressively become one ofthe important topics to study, especially since computing codes are ex-ploited on ever more powerful hardware to solve ever larger problems.The Verification and Validation (V&V) process should therefore in-clude the monitoring of inaccuracies introduced by floating-point arith-metic, as well as the verification that they are kept within acceptablelimits.

Numerous tools exist to diagnose floating-point problems, amongwhich cadna [2] is one of the most advanced. It allows to assessfloating-point inaccuracies, detect their origin in the source code, andfollow their propagation throughout the computation. To this end,cadna requires instrumentating the source code to replace standardfloating-point arithmetic by Discrete Stochastic Arithmetic (DSA),which is based on a synchronous cestac [4] method. cadna hasalready been successfully used on large industrial simulation codes [3],but instrumenting the source code of such tools can be hard, as theregenerally are numerous dependencies to third-party software librariesof which the development team only has limited understanding.

We present the verrou tool, a valgrind-based system which im-plements an asynchronous cestac method to monitor the accuracyof floating-point operations without needing to instrument the source

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code or even recompile it. This tool is therefore well-suited to be partof an industrial V&V process. It has been successfully tested both onsmall-scale, well understood numerical applications, and on large-scale,more complex industrial computing codes such as Athena [1] which isused to simulate the propagation of ultrasonic waves in steel welds.

References:

[1] B. Chassignole, R. E. Guerjouma, M.-A. Ploix, andT. Fouquet, Ultrasonic and structural characterization ofanisotropic austenitic stainless steel welds: Towards a higher re-liability in ultrasonic non-destructive testing. NDT & E Interna-tional 43, 4 (2010), 273 – 282.

[2] J.-L. Lamotte, J.-M. Chesneaux, and F. Jezequel,CADNA C: A version of CADNA for use with C or C++ programs.Computer Physics Communications 181, 11 (2010), 1925–1926.

[3] S. Montan, Sur la validation numerique des codes de calcul in-dustriels. PhD thesis, Universite Pierre et Marie Curie (Paris 6),France, 2013. In French.

[4] J. Vignes, A stochastic arithmetic for reliable scientific compu-tation. Mathematics and Computers in Simulation 35, 3 (1993),233–261.

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47

Zero Verification in Systems of Equations:Interval-based Implementation of a

Topological Test.

Peter Franek, Marek Krcal and Hubert Wagner

IST AustriaAm Campus 1, 3400 Klosterneuburg, Austria

[email protected]

Keywords: Verification, Nonlinear equations, Computational Topol-ogy

Assume that B is a box in Rm and f : B → Rn is continuous andour knowledge of f is limited. The function may come from imprecisemeasurements or previous numerical computations. One way to dealwith the uncertainty is to represent f via an interval function. Ourgoal is to verify the existence of zero of f , if we only have access to anoracle I(f) that for a box B′ ⊆ B outputs a box containing f(B′).

If the dimensions of the domain and codomain are equal, then theproblem is reduced to the computation of the topological degree and hasbeen described in [3]. The degree test is provably stronger then oldertests based on Brouwer, Miranda’s or Borsuk’s theorem. For under-determined systems, however, the situation is much more complex.

A closely related problem asks whether f has a zero, if f is approx-imated via a given piecewise linear function g on a simplicial complexand we know that ‖f − g‖ < 1. In [1] we showed that while this isundecidable in full generality, there exists a sequence of tests of in-creasing complexity, whereas a positive results of any of these testscertifies the zero of f . These tests can easily be adjusted to even ap-proximate the robustness of zero, that is, the maximal r so that eachh with ‖h− f‖ < r has a zero. The first and most accessible of thesetests is called the primary obstruction.

In our current work, we implement the primary obstruction test inthe framework of interval arithmetic. First experiments with random

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vector fields indicate that the primary obstruction is typically suffi-cient to detect zeros. In other words, higher obstructions (that we candetect in a slightly different simplicial setting) never occurred in theexperiments [2].

References:

[1] Peter Franek, Marek Krcal, Robust Satisfiability of Systemsof Equations, J. of the ACM, 2015

[2] Peter Franek, Marek Krcal, Hubert Wagner, Robust-ness of Zero Sets: Implementation, Preprinthttp://www.cs.cas.cz/~franek/rob-sat/experimental.pdf

[3] Peter Franek, Stefan Ratschan, Piotr Zglizcynski, Quasi-Decidability of a Fragment of the First-order Theory of Real Num-bers, J. of Automated Reasoning, 2015

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49

Golden-mean universality for Siegel disks

Denis Gaidashev and Michael Yampolsky

University of UppsalaBox 480, Uppsala, [email protected]

Keywords: Siegel disks, renormalization, rigidity

We will describe a computer-assisted proof of one of the centralopen questions in one-dimensional renormalization theory – universal-ity of the golden-mean Siegel disks. We further show that for everyfunction in the stable manifold of the golden-mean renormalizationfixed point the boundary of the Siegel disk is a quasicircle which coin-cides with the closure of the critical orbit, and that the dynamics onthe boundary of the Siegel disk is rigid.

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Fast determination of the tensorial andsimplicial Bernstein enclosure

Jurgen Garloff 1,2, Jihad Titi 1

1Department of Mathematics and Statistics, University of Konstanz2Institute for Applied Research, University of Applied Sciences/

HTWG Konstanz

Konstanz, Germany

[email protected] [email protected]

Keywords: Bernstein coefficients, tensorial Bernstein form, simplicialBernstein form, range enclosure, subdivision

The underlying problems of our talk are unconstrained and con-strained global polynomial optimization problems over boxes and sim-plices. One approach for their solution is based on the expansion of a(multivariate) polynomial into Bernstein polynomials of the objectivefunction and the constraints polynomials. This approach has the ad-vantage that it does not require function evaluations which might becostly if the degree of the polynomials involved is high.

The coefficients of this expansion are called the Bernstein coeffi-cients. These coefficients can be rearranged in a multi-dimensionalarray, the so-called Bernstein patch. From these coefficients we getbounds for the range of the objective function and the constraints overa box or a simplex, see [1, 2]. We can improve the enclosure for therange of the polynomial under consideration by elevating the degree ofits Bernstein expansion or by subdivision of the region. Subdivision ismore efficient than degree elevation. The complexity of the traditionalapproach for the computation of these coefficients is exponential in thenumber of the variables.

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In [1] Garloff proposed a method for computing the Bernstein coef-ficients of a bivariate polynomial over the unit box and the unit trian-gle by using a forward difference operator. We propose some efficientmatrical methods that involve only matrix operations such as multipli-cation, transposition, and reshaping as Ray and Nataraj’s method, see[3]. Our methods are superior over Garloff’s and Ray and Nataraj’smethod for the computation of the Bernstein coefficients over the unitand a general box and the standard simplex. We also propose a ma-tricial method for the computation of the Bernstein coefficients oversubboxes and subsimplices when the original box and simplex are sub-divided, respectively.

References:

[1] J. Garloff, Convergent bounds for the range of multivariate poly-nomials, Interval Mathematics 1985, K. Nickel, Ed., Lecture Notesin Computer Science, 212(1986), Springer, Berlin, Heidelberg, NewYork, pp. 37–56.

[2] J. Titi and J. Garloff, Matrix methods for the tensorial Bern-stein form and for the evaluation of multivariate polynomials, sub-mitted.

[3] R. Leroy, Convergence under subdivision and complexity of poly-nomial minimization in the simplicial Bernstein basis, ReliableComputing, 17(2012), pp. 11–21.

[2] S. Ray and P.S.V. Nataraj, A matrix method for efficient com-putation of Bernstein coefficients, Reliable Computing, 17(2012),No. 1, pp. 40–71.

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Adaptive mesh refinement technique forthe classical Plateau problem

Aymeric Grodet and Takuya Tsuchiya

Graduate School of Science and Engineering, Ehime UniversityMatsuyama, Ehime, [email protected]

Keywords: mesh refinement, error estimation, minimal surface

Let D =

(u, v) ∈ R2∣∣u2 + v2 < 1

be the unit disk with boundary

∂D = S1 the unit circle. Let Γ ⊂ Rd be an arbitrary Jordan curve. Wewould like to compute a minimal surface ϕ : D → Rd with ϕ(∂D) = Γ.For such a minimal surface, it exists the following variational principle:H1(D;Rd) being the ordinary Sobolev space, let

XΓ =ψ ∈ C(D;Rd) ∩H1(D;Rd)

∣∣ψ(∂D) = Γ, ψ|∂D : monotone

where monotone means that if ∂D is traversed in a given direction, Γis also traversed in the corresponding direction, although we allow arcsof ∂D to map onto single points of Γ. To such a map ϕ, we denote itsDirichlet integral by

D(ϕ) =

∫

D

|∇ϕ|2 dx =

∫ ∫

D

d∑

k=1

[(∂ϕk

∂u

)2

+

(∂ϕk

∂v

)2]

dudv.

Then, ϕ ∈ XΓ is a minimal surface if and only if it is a stationarypoint of D(ϕ). The problem of finding such points is called the clas-sical Plateau problem [1]. The mapping ϕ is actually obtained byminimizing its Dirichlet integral.

We need a normalization condition to determine a minimal surface(locally) uniquely. Several exist. For example, one can specify theimages of three points on ∂D to define the same orientation on ∂Dand Γ. In order to find a minimum, we have to move the interior

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points (inside the domain defined by the curve Γ) but also boundarypoints (on Γ), except for the ones we set as fixed points.

If one apply a finite element approximation of minimal surfaces onthe unit disk, with piecewise linear functions on each triangle of onetriangulation, some inaccuracies can occur, in particular at singular-ities of Γ. We propose to overcome this difficulty by introducing aposteriori error estimation and an adaptive mesh refinement. This al-lows us to refine the mesh only in the local area corresponding to theproblematic one on the domain defined by Γ. In this talk, we explainhow to perform the finite element approximation and present severalnumerical examples to highlight what problems can occur and how tosolve them.

References

[1] U. Dierkes, S. Hildebrandt, F. Sauvigny, Minimal Sur-faces, Springer, Berlin, 2010.

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When dependencies do not matter?

Milan Hladık

Charles University, Faculty of Mathematics and Physics, Departmentof Applied Mathematics,

Malostranske nam. 25, 118 00, Prague, Czech Republic,[email protected]

Keywords: dependencies, interval linear equations, interval linear in-equalities

Dependencies are one of the most intractable obstacles in inter-val computation because they cause overestimation when evaluatingexpressions by interval arithmetic. Nevertheless, not always depen-dencies must cause overestimation. Hansen [1] presented an endpointanalysis to check if the enclosure computed by a natural interval ex-tension is optimal.

There are also classes of interval problems, where dependencies donot matter. As shown by [3, 4], the (united) solution set of the intervalsystem

Ax = b

is the same as the solution set of interval inequalities (ignoring depen-dencies)

Ax ≤ b, Ax ≥ b.

This property was then generalized [2] to the so called AE solutionsand under some assumptions to AE solvability.

As another example, a system Ax = b is strongly solvable if andonly if the system

Ax1 −Ax2 = b, x1, x2 ≥ 0

is strongly solvable.We will show more examples of problems, where dependencies can

be ignored and the problem will not change. Moreover, we will show

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possible transformations of interval linear systems from equations toinequalities and vice versa, related to weak and strong solvability. Suchtransformations can be very useful in deriving new properties or simpli-fying the complicated proofs characterizing various kinds of solvability.

We also discuss possible extensions to parametric systems.

References

[1] E.R. Hansen, Sharpness in interval computations, Reliable Com-puting, 3 (1997), No. 1, pp. 17–29.

[2] M. Hladık, Transformations of interval linear systems of equationsand inequalities, submitted.

[3] W. Li, A note on dependency between interval linear systems,Optimization Letters, 9 (2015), No. 4, pp. 795–797.

[4] J. Rohn, Miscellaneous results on linear interval sys-tems, Freiburger Intervall-Berichte 85/9 (1985), Albert-Ludwigs-Universitat, Freiburg.

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IeeeCC754++ – an advanced tool to checkIEEE 754-2008 conformity

Matthias Husken

University of WuppertalGaußstr. 20, 42119 Wuppertal, Germany

[email protected]

Keywords: IeeeCC754++, IEEE 754-2008, compliance checker

Running numerical algorithms on a given computing platform in-volves a lot more components than visible at first glance, amongstwhich are the programming language of choice, the compiler, and thehardware on which the program is executed. Ideally, it should not benecessary for a user to fully understand all parts involved and the al-gorithms should give reproducible, platform-independent results. Theuser should thus rely on the means provided by the chosen program-ming language and trust that all other components work together fol-lowing the philosophy of the IEEE 754-2008 standard, or more pre-cisely, that all components adhere to this standard as far as needed tosupply a conforming platform as a whole.

With this in mind, IEEE 754-2008 explicitly specifies that it “pro-vides a discipline for performing floating-point computation that yieldsresults independent of whether the processing is done in hardware, soft-ware, or a combination of the two.”[IEEE2008, p. iv], and it states that“conformance to this standard is a property of a specific implementa-tion of a specific programming environment, rather than of a languagespecification.”[IEEE2008, p. 2]

In this talk, we introduce IeeeCC754++, a flexible testing tool thattogether with an extensive evaluation framework enables the user toevaluate the IEEE 754-2008 conformity of the computing platformthat is used to run the target numerical application. Furthermore,IeeeCC754++ comes with a large selection of ports that enable testinga wide variety of architectures, specific hardware floating-point units,and software libraries such as MPFR.

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IeeeCC754++ is based on IeeeCC754 [VCV01a, VCV01b] and closelyfollows its testing philosophy: checking for standard conformity by ex-ecuting a large number of test vectors whose correct results are hardto achieve, e. g. because they are hard to round. Furthermore, testvectors that check for special representations like NaNs or infinitiesas well as test vectors to verify standard-conforming use of exceptionflags are included. With this systematic approach of checking partic-ularly difficult cases, it is possible to identify flaws in a floating-pointimplementation which might go undetected otherwise.

We present the following extensions that were implemented in ournew tool IeeeCC754++:

• Support for the IEEE 754-2008 standard..

• A default testing mode to check the current user environment,taking the chosen compiler and compiler options into account.

• Ports for a number of hardware architectures and their floating-point units.

• An evaluation framework including advanced analysis capabilities.

We will conclude with test results from selected architectures that em-phasize the need for testing tools like IeeeCC754++.

References:

[IEEE08] IEEE Std 754-2008, Standard for Floating-point Arithmetic,IEEE, 2008.

[VCV01a] B. Verdonk, A. Cuyt, and D. Verschaeren. A precision-and range-independent tool for testing floating-point arithmetic I:basic operations, square root and remainder. ACM Transactionson Mathematical Software, 27:92–118, 2001.

[VCV01b] B. Verdonk, A. Cuyt, and D. Verschaeren. A precision-and range-independent tool for testing floating-point arithmeticII: conversions. ACM Transactions on Mathematical Software,27:119–140, 2001.

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Towards Fast, Accurate and ReproducibleLU Factorization

Roman Iakymchuk, David Defour and Stef Graillat

CST/PDC, CSC, KTH Royal Institute of TechnologyLindstedtsvagen 5, 11428 Stockholm, Sweden

[email protected]

DALI–LIRMM, Universite de Perpignan52 avenue Paul Alduy, F-66860 Perpignan, France

[email protected]

Sorbonne Universites, UPMC Univ Paris 06, UMR 7606, LIP64 place Jussieu, F-75005 Paris, France

[email protected]

Keywords: Triangular solver, matrix-vector product, LU factoriza-tion, reproducibility, accuracy, long accumulator, error-free transfor-mation

The process of finding the solution of a linear system of equationsis often the core of many scientific applications. Usually, this pro-cess relies upon the LU factorization, which is also the most compute-intensive part of it. Although current implementations of the LU fac-torization may reach 70% of the peak performance, their accuracyand, even more, reproducibility cannot be guaranteed, mainly, due tothe non-associativity of floating-point operations and dynamic threadscheduling.

In this work, we address the problem of reproducibility of the LUfactorization due to cancelations and rounding errors, resulting fromfloating-point arithmetic. Instead of developing a completely inde-pendent version of the LU factorization, we benefit from the hier-archical structure of linear algebra libraries and start from develop-ing/enhancing reproducible algorithmic variants for the kernel oper-ations – like the ones included in the BLAS library – that serve asbuilding blocks in the LU factorization. In addition, we aim at ensur-ing the accuracy of these underlying BLAS routines.

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We consider an unblocked algorithmic variant of the LU factoriza-tion that can be expressed in terms of BLAS level 1 and 2 routines:triangular solver (TRSV), dot product (DOT) and matrix-vector prod-uct (GEMV). We prevent cancellations and rounding errors duringthe accumulation stage thanks to a long accumulator and error-freetransformations [1]. Consequently, we tackle the problem of accuracyin the reproducible triangular solver (ExTRSV) [2] through iterativerefinement. Moreover, we provide an accurate and reproducible al-gorithmic variant as well as an implementation of the matrix-vectorproduct. Thus, following the bottom-up approach, we construct thereproducible algorithmic variant of the LU factorization. Up to ourknowledge, this is the first implementation of a reproducible LU fac-torization. Finally, we present initial evidence that both accurate andreproducible higher-level linear algebra routines can be constructedfollowing our hierarchical bottom-up approach.

References:

[1] S. Collange, D. Defour, S. Graillat, R. Iakymchuk, Nu-merical Reproducibility for the Parallel Reduction on Multi- andMany-Core Architectures, Parallel Computing, 49 (2015), pp. 83–97.

[2] R. Iakymchuk, S. Collange, D. Defour, S. Graillat, Re-producible Triangular Solvers for High-Performance Computing, Inthe Proceedings of the 12th International Conference on Informa-tion Technology: New Generations (ITNG 2015), Special track on:Wavelets and Validated Numerics, April 13-15, 2015, Las Vegas,Nevada, USA, pp. 353-358.

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Secure a zone with robots

Luc Jaulin and Benoıt Zerr

Lab-STICC, ENSTA Bretagne, Brest, France

[email protected]

Problem. We consider n underwater robots R1, . . . ,Rn at po-sitions a1, . . . , an and moving in a 2D world [1]. Each robot has avisibility zone. If an intruder is inside the visibility zone of one robot,it is detected. The robots have to collaborate to guarantee that theyis no moving intruder inside a subzone of a compact subset O of R2,representing the 2D ocean.

Complementary approach. We assume that there exists a vir-tual intruder moving inside O satisfying the differential inclusion

x(t) ∈ F(x(t)),

where x(t) is the state vector. Moreover, we assume that each robotRi has a visibility zone of the form g−1ai

([0, d]) where d is the scope. Ourcontribution is to show that characterizing the secure zone translatesinto a set-membership set estimation problem [2] where x(t) is shownto be inside the set X(t) returned by our set-membership observer.Then we conclude that x(t) cannot be inside the complementary ofX(t). This result can be formalized by the following theorem.

Theorem. The virtual intruder has a state vector x(t) inside theset

X(t) = O ∩ dt · F(X(t− dt)) ∩⋂

i

g−1ai(t)([d(t),∞]),

where X(0) = O. As a consequence, the secure zone is

S(t) = projworld(X(t)).

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Figure 1: (left) Paving of the Biscay Bay, (right) Green: Secured zone

Proof (sketch). Two cases should be considered. If no actualintruder exists then S(t) cannot is secured. If the virtual intruder is areal one, its state x(t) is inside X(t) and its position (which is a partof the state) is inside projworld(X(t)). In both situations, the intrudercan not be inside S(t).

Method. Each robot follows a reference point. All reference pointsform a flat ellipsoid which plays the role of a barrier. The strategy isillustrated by Figure 1 for 10 robots. The set O corresponds to theblue area (left). On the right, S(t) is painted green. The observer hasbeen implemented using interval analysis.

References:

[1] L. Jaulin, Mobile robotics, ISTE editions, London, 2016.

[2] M. Kieffer, L. Jaulin, E. Walter, D. Meizel, NonlinearIdentification Based on Unreliable Priors and Data, with Applica-tion to Robot Localization, Robustness in Identification and Con-trol, 3 (1999), LNCIS 245, pp. 190–203.

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A Decimal Multiple-Precision IntervalArithmetic Library

Stef Graillat, Clothilde Jeangoudoux, and Christoph Lauter

Sorbonne Universites, UPMC Univ Paris 06, CNRS, LIP6 UMR 76064 place Jussieu, 75005 Paris, France

[email protected], [email protected],

[email protected]

Keywords: interval arithmetic, multi-precision, IEEE 754-2008 deci-mal format.

We describe the mechanisms and implementation of a library thatdefine a decimal multiple-precision interval arithmetic. The aim ofsuch a library is to provide guaranteed and accurate results in dec-imal. This library contains correctly rounded elementary operationsand some transcendental functions such as sin, cos or exp.

The application fields of such a library are wide from financial cal-culation to aeronautic engineering. In those fields, complex algorithmsare generally designed in high level with decimal numbers. Thus thereis a need for correctly rounded decimal arithmetic multiple-precisionlibrary. The IEEE 754-2008 [2] revision introduced the definition ofdecimal floating-point format, with the purpose of providing a basisfor a robust and correctly rounded decimal arithmetic.

Apart from being IEEE 754-2008 compliant, our library holds sev-eral useful properties, such as correct rounding for decimal arbitraryprecision. The correctness of some functions will be proven. Further-more the interval arithmetic will be compliant with the new IEEE1788-2015 Standard for Interval Arithmetic [3].

Studies of implementation of decimal arithmetic libraries have shownthat one of the simplest and efficient way to implement decimal func-tions is to use their binary counterparts [1]. This method makes itpossible to rely on the efficiency of the binary arithmetic and focus

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more effort only on some critical points. The implementation of thislibrary is based on GMP and MPFR libraries.

References

[1] J. Harrison, Decimal Transcendentals via Binary, Proceedingsof the 19th IEEE Sympoisum on Computer Arithmetic, 2009, pp.187-194.

[2] IEEE Std 754-2008, IEEE Standard for Floating-Point Arithmetic,2008.

[3] IEEE Std 1788-2015, IEEE Standard for Interval Arithmetic, 2015.

[4] GNU MPFR Library, multiple-precision floating-point computa-tions with correct rounding based on GMP.

[5] MPFI Library, A multiple precision interval arithmetic library basedon MPFR.

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The Implementation of multi-precisionpackage in multiple-component format in

MATLAB

Hao Jiang, Peibing Du, Kuan Li, Lin Peng,

College of Computer, National University of Defense Technology410072 Changsha, China

haojiang,[email protected]

Keywords: multi-precision, multiple-component, error-free transfor-mation, overloading.

High precision computations are required in many application ar-eas, such as algorithm’s verification. There are serval methods andresponding libraries to obtain high precision, including arbitrary pre-cision library MPFUN, ARPREC, GMP, and MPFR. These librariesstore numbers in a multiple-digit format similar to the symbolic com-putation package Mathematica and the symbolic toolbox and VPA inMatlab. Another well-known and interesting package is QD library[1], which uses the multiple-component format. The QD library de-fines a double-double number and a quad-double number, which arerepresented as an unevaluated sum of two and four double numbers,respectively. Trip-double number and its responding arithmetic areproposed by Lauter [2]. The packages described above are written in Cor Fortran codes. Meanwhile, Matlab provides a familiar and efficientproblem solving environment for many engineers. Some multi-precisionmatlab toolboxes based on the above arbitrary precision libraries havebeen proposed such as MPL[3] and mp[4]. Advanpix[5] is a commercialpackage having good performance, but the algorithms involved are in-visible. The literature [6] implements real double-double number andthe corresponding operations in Matlab.

In this paper, we implemented a multi-precision package in Mat-lab based on multiple component format. In the package, double-double(DD), triple-double(TD), and quad-double(QD) numbers are

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Table 1: Basic operations and functions implemented

z = x+ y z = exp(x) z = DD(x) ==,∼=z = x− y z = sqrt(x) z = TD(x) <,>,<=, >=z = x ∗ y z = sqr(x) z = QD(x) z = real(x)z = x/y z = ln(x) z = DDrand(x) z = imag(x)z = xn z = log10(x) z = TDrand(x) z = floor(x)z = xy z = inv(x) z = QDrand(x) z = fix(x)z = abs(x) z = conj(x) z = round(x) z = ceil(x)

defined as a Matlab class including real and complex format. The ba-sic arithmetic operations, relational operation, some Matlab functionsand so on are overloaded based on this class shown in Table 1. Thispackage allows Matlab obtain the more accurate numerical results andrun much faster than VPA, MPL and mp with some required accuracy.

References:

[1] Y. Hida, X.S. Li, D. H. Bailey, Algorithms for quad-doubleprecision floating point arithmetic, ARITH, San Diego, CA, 2001.IEEE Computer Society, pp. 155–162.

[2] C.Q. Lauter, Basic building blocks for a triple-double interme-diate format, Technical report LIP6,UCMP, Paris,France, 2005..

[3] W. Schreppers, F. Backeljauw, and A. Cuyt, MPL: AMultiprecision Matlab-Like Environment, LNCS, vol:3514, 295-303, 2005.

[4] www.sorceforge.net/projects/mptoolbox/. Multiple precision tool-box for MATLAB.

[5] www.advanpix.com. Advanpix multiprecision computing toolbox

[6] D. Tsarapkina and D. J. Jeffrey, Exploring Rounding Errorsin Matlab using Extended Precision, Procedia Computer Science,vol:29, 1423-1432, 2014.

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A study on verified ODE solver from thestandpoint of stiffness

Masahide Kashiwagi

Waseda UniversityTokyo, Japan

[email protected]

Solving initial value problems of ordinary differential equations isvery important and several studies for them have been proposed. Wehave proposed an algorithm for verified ODE solver for IVP usingpower series arithmeric and affine arithmetic [1]-[4].

Power series arithmetic defines operations between order-p polyno-mials like

x(t) = x0 + x1t+ x2t2 + · · · xp−1tp−1 +Xpt

p

where the last coefficient Xp is interval. For IVP:

x′(t) = f(x(t), t), x(t0) = x0 ,

we use (t0-shifted) Picard type fixed point form:

x(t) = x0 +

∫ t

0

f(x(t), t+ t0) ≡ P (x(t))

and if P (x∗(t)) ⊂ x∗(t) for some interval polynomial, then x∗(t) en-closes true solution.

For avoiding so-called wrapping effect, we always use Jacobian ofthe operator φti,ti+1

, where φti,ti+1: x(ti) 7→ x(ti+1) . We use mean value

form and affine arithmetic to connect solutions over multi steps. Forcalculating φ′, we use variational equation w.r.t. initial value:

y′(t) = fx(x∗(t), t)y(t), y(t0) = I .

We have tested two algorithms (Algo1, Algo2) for calculating y(t).Details will be shown at presentation.

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As a numerical example, we take up van der Pol equation:

x′′ − µ(1− x2)x′ + x = 0 .

It is known that the equation is non-stiff for small µ and become stifffor large µ. We show calculation results for µ = 1 and µ = 100 underthe following environment: Intel Xeon CPU E5-2687W (3.10GHz),Ubuntu 14.04, gcc 4.8, kv-0.4.36, initial value (x, x′) = (1, 1), domain= [0, 200], order of polynomial is 24 and local error is machine epsilon.

µ = 1 µ = 100

0.01

0.1

1

10

0 50 100 150 200

x

t

x

0.01

0.1

1

0 50 100 150 200

ste

psi

ze

t

algorithm1algorithm2

awa

-2.5-2

-1.5-1

-0.5 0

0.5 1

1.5 2

2.5

0 50 100 150 200x

t

x

0.0001

0.001

0.01

0.1

1

0 50 100 150 200

ste

psi

ze

t

algorithm1algorithm2

awa

algo1 2.276sec 53.820secalgo2 1.798sec 13.478secAWA 3.815sec 176.78sec

We can see that algo2 can take large step size and calculation becomefaster consequently in stiff case.

References:

[1] M, Kashiwagi, S. Oishi, Numerical Validation for Ordinary Dif-ferential Equations — Iterative Method by Power Series Arith-metic —, NOLTA’94, pp.243–246 (1994.10.7).

[2] M, Kashiwagi, kv – C++ Numerical Verification Libraries, http://verifiedby.me/kv/ .

[3] M, Kashiwagi, Verified ODE(IVP) Solver, http://verifiedby.me/kv/demo/ode.cgi .

[4] M, Kashiwagi, An algorithm to reduce the number of dummyvariables in affine arithmetic, SCAN2012 (2012.9.28).

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Simplicial Branch and Bound in IntervalGlobal Optimization

Ralph Baker Kearfott

University of Louisiana at LafayetteDepartment of Mathematics, U.L. Box 4-1010, Lafayette, LA

[email protected]

Keywords: rigorous global optimization, branch and bound, rangeenclosure

The predominant shape chosen for search regions in branch andbound (B&B) algorithms for global optimization of continuous func-tions is a box x = x ∈ Rn | xi ≤ xi ≤ xi, 1 ≤ i ≤ n, while the pre-dominant method of branching has been bisection of one of the coordi-nate directions. This scheme has been popular because of its simplicity,because a sub-region provides clear error bounds on individual param-eters, and because bounds on the coordinates occur naturally in manyproblems.

However, other region shapes also have advantages and have beenconsidered. One alternative is an n-simplex S = 〈P0, P1, . . . , Pn〉, theconvex hull of n + 1 points Pi ∈ Rn. A relatively early B&B methoddealing with continuous nonlinear systems was proposed by Stenger[4], a method subsequently enhanced with more elegant formulas byStynes [5, 6] and the present author [1, 2], although those techniquesused a heuristic to decide when to branch.

Due to certain advantages, simplexes as domains in B&B algo-rithms have received renewed scrutiny in more recent work. As sur-veyed in [3], Zilinskas et al develop simplicial B&B to take advantage ofcertain naturally occurring symmetries is the objective and constraints.

Here, in the spirit of Paulavcius and Zilinskas, we investigate sim-plex-based B&B algorithms in contexts in which the feasible region isbest described by a simplex. However, we develop representations that

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allow interval evaluations (and hence mathematically rigorous boundson ranges) with relatively small overestimation. We present appro-priate ways to represent the simplexes, we derive associated relativelysharp mean-value type interval extensions, and we propose and discussbranching processes related to these representations.

References

[1] Ralph Baker Kearfott. Computing the degree of maps and a general-ized method of bisection. Ph.D. thesis, Department of Mathematics,University of Utah, Salt Lake City, UT, USA 84112, 1977.

[2] Ralph Baker Kearfott. An efficient degree-computation methodfor a generalized method of bisection. Numerische Mathematik,32(2):109–127, June 1979.

[3] Remigijus Paulavcius and Julius Zilinskas. Simplicial Global Opti-mization. Springer Verlag, 2014.

[4] Frank Stenger. Computing the topological degree of a mapping inRn. Numerische Mathematik, 25(1):23–38, March 1975.

[5] Martin Stynes. A simplification of Stenger’s topological degree for-mula. Numerische Mathematik, 33(2):147–155, June 1979.

[6] Martin Stynes. On the construction of sufficient refinementsfor computation of topological degree. Numerische Mathematik,37(3):453–462, July 1981.

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Optimal order constructive a priori errorestimates for a full discrete

approximation of the heat equation

Takuma Kimura∗, Teruya Minamoto† and Mitsuhiro T.Nakao‡

∗† Saga University, Saga 840-8502, Japan.‡ Kyushu University, Japan.∗ [email protected]

Keywords: parabolic problem, Galerkin methods, constructive a pri-ori error estimates

We consider the constructive a priori error estimates for an approx-imate solution of the following equations with homogeneous initial andboundary conditions:

∂u∂t − ν4u = f(x, t) in Ω× J,u(x, t) = 0 on ∂Ω× J,u(x, 0) = 0 in Ω.

(1)

where Ω ⊂ Rd(d ∈ 1, 2, 3) is bounded polygonal or polyhedral do-mains, J := (0, T ) ⊂ R (for a fixed T < ∞) is an open interval, ν isa positive constant and f ∈ L2(J ;L2(Ω)). Let P k

hu be a full discreteapproximation for (1) studied in [1], [2], where h and k are mesh sizefor Ω and J , respectively. Our method is based on the finite elementGalerkin method with an interpolation in time that uses the fundamen-tal solution for semidiscretization in space. In [1], the authors studiedthe optimal order L2(J ;H1

0(Ω)) and L2(J ;L2(Ω)) error estimates forP khu with the assumption that h = k2.

In this talk, assuming that f is sufficiently smooth, we show theoptimal order error estimates can be obtained even if h 6= k2. Inthe below, let CΩ(h) and CJ(k) denote constants determined by theapproximation property with order of O(h) and O(k), respectively.

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Then we prove the following error estimates.H1

0-error estimates:

||u− P khu||L2H1

0

≤ C1(h, k)(

2ν ||f ||L2L2 + 1√

2ν

√||f(·, 0)||2L2(Ω) + ||f ||2L2L2 + || ∂∂tf ||2L2L2

)

where C1(h, k) := maxCΩ(h), CJ(k) .L2-error estimates:

‖u− P khu‖L2L2 ≤ C0(h, k)

(‖f‖L2L2 + ‖ ∂∂tf‖L2L2 + ‖∇P0f(·, 0)‖L2(Ω)

)

where C0(h, k) := max

8νCΩ(h)2,

√νCJ(k)2, CJ(k)2

and P0 is the L2-

projection.V 1-error estimates:

‖ ∂∂t(u− P khu)‖L2L2 ≤ C3(h, k)C(||f ||L2L2, || ∂∂tf ||L2L2, ||∇f(·, 0)||L2(Ω))

where C3(h, k) := maxCΩ(h), CJ(k)(≡ C1(h, k)),

C(||f ||L2L2, || ∂∂tf ||L2L2, ||∇f(·, 0)||L2(Ω))

:=√

2ν

(CΩ(h)2 + ν)(2|| ∂∂tf ||2L2L2 + ν||∇f(·, 0)||2L2(Ω)) + 2||f ||2L2L2

12

+ν||∇f(·, 0)||2L2(Ω) + || ∂∂tf ||2L2L2

12

In the talk, some numerical examples which confirm the expected rateof convergence will be presented.

References:

[1] M. T. Nakao, T. Kimura, T. Kinoshita, Constructive a priorierror estimates for a full discrete approximation of the heat equa-tion, SIAM Journal on Numerial Analysis, 51 (2013), pp. 1525–1541.

[2] T. Kinoshita, T. Kimura, M.T. Nakao, On the a posterioriestimates for inverse operators of linear parabolic equations withapplications to the numerical enclosure of solutions for nonlinearproblem, Numerische Mathematik, 126 (2014), pp. 679–701.

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Decision Making Under IntervalUncertainty as a Natural Example of a

Quandle

Mahdokht Afravi and Vladik Kreinovich

University of Texas at El PasoEl Paso, Texas 79968, USA

[email protected]

Keywords: decision making, interval uncertainty, quandle

Need for decision making. In many real-life problems, we need toselect an alternative a from the list of possible alternatives – e.g., wewant to select a design and/or location of a plant, a financial invest-ment, etc. In many such situations, we have a well-defined objectivefunction u(a) that describes our preferences. If we know the exactvalue of u(a) for each alternative a, then we select the alternative withthe largest value of u(a).

Decision making under interval uncertainty. In practice, we usu-ally only know the consequences of each decision with some uncertainty.Often, the only information that we have about the corresponding val-ues of u(a) is that it is somewhere between the known bounds u(a) andu(a), i.e., that u(a) ∈ [u(a), u(a)]. In such situations, to make a definitedecision, we need to assign, to each such interval, a single numericalvalue u0 describing the quality of the corresponding alternative. Wewill denote this value u0 by u(a) B u(a).

Natural properties of the corresponding operation B. Whatare the natural properties of the operation aB b?

First, if we know the exact value of u(a), i.e., if the correspondinginterval has the form [x, x] for some x, then the corresponding equiva-lent value is simply equal to x: xB x = x.

Another reasonable property is monotonicity: if x < x′, thenx B y > x′ B y. (For continuous functions, monotonicity is equivalentto invertibility of x→ xB y.)

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To get the third property, let us consider a slightly more complexsituation, when we know the lower bound z of the corresponding in-terval, but we do not know its upper bound: we only know that thisupper bound is between y and x. We can analyze this situations intwo different ways.

First, we can say that since all we know about the upper bound isthat it is between y and x, this upper bound is therefore equivalent tothe value yB x. Now, after we have thus reduced the uncertain upperbound to a single number, the original information becomes simply aninterval with an exact lower bound z and an exact upper bound xB y.We can now apply the operation B to estimate the equivalent value ofthis interval as (xB y) B z.

There is also an alternative approach. For each possible value vbetween y and x, we have an interval [z, v] with equivalent value vB z.Due to the natural monotonicity, this equivalent value is the smallestwhen v is the smallest, i.e., when v = y, and it is the largest when v isthe largest, i.e., when v = x. Thus, possible equivalent values form aninterval [yB z, xB z]. The equivalent value of this interval is therefore(xB z) B (y B z).

It is reasonable to require that these two approaches lead to thesame value, i.e., that (xB y) B z = (xB z) B (y B z).

Hurwicz optimism-pessimism criterion satisfies these proper-ties. One can easily check that the widely used Hurwicz criterionxB y = α · x+ (1− α) · y, with α > 0, satisfies the above properties.

This is a quandle. Interestingly, the above three natural propertiesare well known in knot theory: sets with operations satisfying theseproperties are knows as quandles [1]. Since decision making underinterval uncertainty naturally leads to a quandle, maybe we will beable to apply results from quandle theory to make better decisions?

References:

[1] M. Elhamdadi, S. Nelson, Quandles: An Introduction to theAlgebra of Knots, American Mathematical Society, Providence,Rhode Island, 2015.

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A template-based C++ library forautomatic differentiation and hull

consistency enforcing

Bart lomiej Kubica

Department of Applied Informatics,Warsaw University of Life SciencesNowowiejska 159, Warsaw, Polandbartlomiej [email protected]

Keywords: automatic differentiation, hull consistency, C++ templatemeta-programming

Automatic differentiation (AD) is often used in interval algorithmsfor optimization, nonlinear systems of equations or solving other deci-sion problems (see, e.g., [4]). The existing software for this purpose,like the one from the Toolbox of C-XSC library [1], has several limita-tions and drawbacks. As for this specific package:

• there are distinct classes for computing first or second derivativesand for univariate or multivariate functions;

• there are global variables to distinguish the order of computedderivative – these variables have to be checked at runtime, sev-eral times during the computation; they also affect multithreadedimplementations [3];

• computing derivatives of higher order would require to implementa separate (but analogous) class;

• although, the developers of C-XSC have provided several usefulclasses for sparse matrices and vectors, their AD code makes nouse of it.

What is more, if we need to use some hull consistency based narrowing(HC3, HC4), the representation of functions using AD classes is oflittle use to us.

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Many of these drawbacks can be overcome by using C++ templatemeta-programming [2]. This allows automatic generation of similarcode for distinct data types (single intervals or their vectors/matrices,sparse or dense types, etc.).

The author is going to present a novel C++ template library, hav-ing, in particular, the following features:

• a single template class used for generating all derivatives and gen-erating the procedure to create syntactic tree of the function – allof them at compile time;

• generating (at compile time) separate functions to compute variousderivatives;

• possibility of using both dense and sparse formats for gradientsand Hesse matrices;

• potential possibility to computing higher derivatives;

• integration with procedures enforcing hull consistency.

References:

[1] C-XSC library, http://www.xsc.de.

[2] A. Alexandrescu, Modern C++ Design: Generic Programmingand Design Patterns Applied, Addison-Wesley, 2001.

[3] B.J. Kubica, A. Wozniak, A multi-threaded interval algorithmfor the Pareto-front computation in a multi-core environment, LNCS,6126, published online.

[4] B.J. Kubica, A class of problems that can be solved using intervalalgorithms, Computing, 94 (2012), No. 2, pp. 271–280.

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Efficient Dedekind reals in Haskell

Ivo List

Faculty of Mathematics and PhysicsUniversity of Ljubljana

Ljubljana, [email protected]

Keywords: Dedekind reals, ASD, Haskell

Exact reals may be described using Dedekind construction as cuts.In such construction the cut is described by two predicates that de-scribe lower and upper bounds. Predicates may be composed of arith-metics, functions, inequalities and quantifiers.

Haskell with its flexible type system allows us to descibe concretepredicates without implementating them. On top of those descriptionswe can implement different evaluation strategies.

Concrete Haskell implementation will be presented with exampledescriptions of exact reals and evaluating them using different strate-gies.

The basic evaluation strategy employs bisection and splitting ofintervals. Although it is inefficient it serves as a proof of concept.

More efficient strategy uses Kaucher arithmetics with back-to-frontintervals and employs extended interval Newton method. Improvingthose strategies is currently in progress.

On the other hand Haskell has some pitfalls, most pronouncedin lack of support for efficient low-level floating-point and interval li-braries. Quite some time has been invested to overcome these.

References:

[1] A. Bauer, P. Taylor, The Dedekind reals in abstract Stoneduality, Mathematical Structures in Computer Science; MSCS, 19(2009), No. 4, pp. 757–838.

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[2] P. Taylor, A lambda calculus for real analysis, Journal of Logicand Analysis, 2 (2010), No. 5, pp. 1–115.

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A framework for high-precision verifiedeigenvalue bounds by using finite element

methods

Xuefeng LIU

Niigata University8050 Ikarashi 2-no-cho, Nishi-ku, Niigata City, Niigata 950-2181

[email protected]

Keywords: differential operators, eigenvalue bounds, finite elementmethod, Lehmann–Goerisch’s theorem

For the purpose of high-precision eigenvalue bounds for differentialoperators, a general framework based on finite element methods isproposed. The high-precision eigenvalue bounds are obtained throughthe following two steps.

First, rough but exact eigenvalue bounds can be obtained by us-ing lower order finite element methods along with a fundamentaltheorem in [1]. For example, for the Laplace operator definedon a bounded domain, the eigenvalues can be easily estimated byapplying the Crouzeix–Raviart finite element. For biharmonic op-erators, the Fujino–Morley FEM can be adopted to give expliciteigenvalue bounds.

Second, to obtain high-precision eigenvalue bounds, Lehmann–Goerisch’s theorem along with high-order finite element methodsis adopted, where the rough eigenvalue bounds obtained in thefirst step is important [3, 2]. See Table 1 for a sample computa-tion result for eigenvalues of Laplacian with homogeneous Dirichletboundary condition over square-minus-square domain, where thereexist singularities of eigenfunction around the reentrant corners.

By further adopting the interval arithmetic, the explicit eigenvaluebounds from numerical computations can be mathematically correct.

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As a computer-assisted proof, the verified eigenvalue bounds have beenused to investigate the solution existence of semi-linear elliptic differ-ential equations; see, e.g., [4].

Table 1: Bounds for the leading 5 eigenvalues of Laplacian over square-minus-square domain [2]

λi Eigenvalue bounds

1 9.1602164372 9.1700889613 9.1700889614 9.1805680525 10.0898433714

(0, 0)

(7, 1)

(8, 8)

(1, 7)

References

[1] Liu, X., A framework of verified eigenvalue bounds for self-adjointdifferential operators, Applied Mathematics and Computation,267, pp.341-355, 2015

[2] Liu, X., Okayama, T. and Oishi, S., High-precision eigenvaluebound for the Laplacian with singularities. Computer Mathemat-ics, pp.311-323, Springer, 2014

[3] Liu, X. and Oishi, S., Guaranteed high-precision estimation forP0 interpolation constants on triangular finite elements, JapanJournal of Industrial and Applied Mathematics, 30, pp.635-652,2013

[4] Takayasu, A., Liu, X. and Oishi, Verified computations to semi-linear elliptic boundary value problems, Nonlinear Theory and ItsApplications, IEICE, 4(1), pp.34-61. doi:10.1588/nolta.4.34, 2013

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Interval Methods in the Calculation ofSolutions to Fuzzy Interval Linear

Systems

Weldon A. Lodwick1, Lotfi Taher2 and Hana Veiseh2

1. University of Colorado Denver, Department of Mathematical andStatistical Sciences

2. Islamic Azad University, Hamedan Branch, Hamedan, Iran,Department of Applied [email protected]

We present methods to obtain proper fuzzy interval vector solutionsets (fuzzy intervals with nested α−levels) arising from systems of fuzzyinterval linear equations. Since computing solutions, at times, do notproduce proper fuzzy interval solutions, special care must be takenwhen obtaining solutions that are indeed fuzzy interval solutions.

The theory of how to obtain proper fuzzy interval solutions to fuzzylinear systems is discussed. In particular, we present the conditionsnecessary for the solutions to fuzzy interval linear systems (both equal-ity and inequality) to exist as proper fuzzy interval vectors. We usemore recent results that unify fuzzy interval linear systems as well assome older, early 1960tees, results from Oettli as it pertains to thecomputation of solutions to interval linear systems and the α−levelinterval systems in fuzzy interval linear systems.

References:

[1] R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction toInterval Analysis, SIAM, Philadelphia, 2009.

[2] A.V. Lakeyev, V. Kreinovich, NP-hard classes of linear al-gebraic systems with uncertainties, Reliable Computing, 3 (1997),No. 1, pp. 51–81.

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LILIB – Long Interval Library

Nozomu Matsuda and Nobito Yamamoto

The University of Electro-Communications1-5-1 Chofu-ga-oka, Chofu, 182-8585, Tokyo, Japan

[email protected]

Keywords: verified numerics, multiple precision arithmetic, center-radius form

We have been developing a new computer program library on multi-ple precision arithmetic with verified computation named LILIB, LongInterval Library. It adopts interval arithmetic for verified computation.Whereas most of existing library for interval arithmetic, e.g. MPFI [1],represent an interval by its infimum and supremum, LILIB uses center-radius form in which an interval is represented by its center and radius.Center-radius form has advantages against inf-sup form in memory ca-pacity and speed on computation if the digits of the numbers is longenough.

MPFI represents an interval number by floating point numbers ofMPFR [2] as infimum and supremum and exploits the advantage of cor-rect rounding of MPFR for verified computation. Since the processesof computation for infimum and supremum are almost same, the com-putational time of interval arithmetic in MPFI takes twice of floatingpoint number arithmetic of MPFR as well as the memory capacity.

An interval number of LILIB is represented by a center value oflong digits number and a radius of short digits. This structure comesfrom the multiple precision interval numbers in INTLAB and obviouslyit reduces the memory capacity of an interval number. Moreover it issupposed to reduce the computational time since the amount of com-putation of long center and short radius seems to be less than twice oflong digits computation. On the other hand the processes of intervalarithmetic for center-radius form are more complicated than inf-supform. This makes it unclear for us whether less computational timewould be attained or not. In order to get an answer to the question, we

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implemented center-radius form and floating point system in a previousversion of LILIB [3] [4] , in which we had developed our original meth-ods for multiple precision arithmetic without using MPFR. Despite thefact that the previous LILIB is actually so slow in computational time,we have confirmed that the center-radius form is faster than twice ofsingle floating point arithmetic in our system as long as the digits arelong enough, which implies center-radius form is faster than inf-supform.

We have improved LILIB to be faster in computation. Our newversion of LILIB is based on MPFR which is used in calculation ofmultiple precision numbers for both of center and radius. When thedigits are long enough, the new version will be faster than MPFI. Inour talk, we will explain the details of specification and implementationof the new version and show some numerical examples. Comparisonwith MPFI will be also mentioned if possible.

References:

[1] N.Revol and F.Rouillier, MPFI,http://perso.ens-lyon.fr/nathalie.revol/software.html.

[2] Laurent Fousse, Guillaume Hanrot, Vincent Lefevre,Patrick Pelissier and Paul Zimmermann, MPFR:A Multiple-Precision Binary Floating-Point Library With Correct Rounding,ACM Transactions on Mathematical Software, Vol. 33, No. 2, 2007,Article 13.

[3] Nozomu Matsuda and Nobito Yamamoto, On the basic oper-ations of Interval Multiple-Precision Arithmetic with center-radiusform, Nonlinear Theory and Its Applications, IEICE 2(1), 2011,pp. 54–67.

[4] Nozomu Matsuda, Development of interval multiple precisionarithmetic library with center-radius form, doctor’s thesis for grad-uate school of The University of Electro-Communications, in Japanese,2016.

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Rigorous numerics of global trajectoriesfor fast-slow systems with an explicit

range of multi-scale parameter

Kaname Matsue1

1The Institute of Statistical Mathematics,10-3, Midori-Cho, Tachikawa, Tokyo, 190-8562, Japan

[email protected]

Keywords: geometric singular perturbation theory, rigorous numer-ics, covering relations, invariant foliations.

My talk is concerned with validations of global trajectories for thefast-slow system

x′ = f(x, y, ε),

y′ = εg(x, y, ε)(1)

with computer assistance, where ′ = d/dt is the time derivative andf, g are Cr-functions with r ≥ 1. The factor ε is a nonnegative realnumber and is considered as the multi-scale parameter. Unless other-wise noted, the slow variable y is assumed to be one-dimensional. Ifocus on validations of

(i) the singular limit trajectory H0 for (1) with ε = 0;

(ii) global trajectories Hε near H0 for all ε ∈ (0, ε0]

at the same time, where ε0 > 0 is an explicitly given number. “Grobaltrajectories” here mean periodic, homoclinic and heteroclinic orbits.

The strategy is based on topological notions such as covering rela-tions ([5]), isolating blocks and cones ([1]) as well as geometric singularperturbation theory (e.g. [2]). Invariant foliations of stable and unsta-ble manifolds are also applied to validating homoclinic and heteroclinictrajectories. Using these notions, normally hyperbolic slow manifolds

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can be validated in the standard way with computer assistance. Addi-tionally, I propose a new technique called the covering-exchange, whichis the singular perturbation version of covering relations and is viewedas a topological counterpart of the Exchange Lemma (e.g. [2]). Thecovering-exchange consists of three parts: slow shadowing, drop andjump. These concepts topologically describe true trajectories whichshadow the reference connecting orbits (with ε = 0) and normally hy-perbolic slow manifolds of (1). The covering-exchange automaticallysolves the matching problem between fast scale and slow scale dy-namics. Moreover, the assistance of rigorous numerics in reasonableprocesses enables us to validate (i) and (ii) simultaneously.

Details of this talk is shown in [3]. If the time permits, I also talkseveral extensions of the current work (e.g. [4]).

References:

[1] C. Conley, Isolated invariant sets and the Morse index, volume 38of CBMS Regional Conference Series in Mathematics, AmericanMathematical Society, Providence, R.I., 1978.

[2] C.K.R.T. Jones, Geometric singular perturbation theory, In Dy-namical systems (Montecatini Terme, 1994), volume 1609 of Lec-ture Notes in Math., 44–118. Springer, Berlin, 1995.

[3] K. Matsue, Rigorous numerics for fast-slow systems with one-dimensional slow variable: topological shadowing approach, arXiv:1507.01462, to appear in Topological Methods in Nonlinear Analy-sis.

[4] K. Matsue, Rigorous verification of smooth neighborhoods aroundslow manifolds via (un)stable normal bundles, in preparation.

[5] P. Zgliczynski, Covering relations, cone conditions and the sta-ble manifold theorem, J. Differential Equations, 246(5):1774–1819,2009.

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Fast validated computation for solutionsof algebraic Riccati equations arising in

transport theory

Shinya Miyajima

Iwate University4-3-5 Ueda, Morioka, Iwate, Japan

[email protected]

Keywords: algebraic Riccati equation, verified computation, M -matrix

Consider the following nonsymmetric algebraic Riccati equation(NARE) arising in transport theory:

XCX −XE − AX +B = 0, (1)

where A,B,C,E ∈ Rn×n are given by

A = ∆− eqT , B = eeT , C = qqT , E = D − qeT .

Here e = (1, . . . , 1)T , q = (q1, . . . , qn)T with qi = ci/(2ωi), ∆ =diag(δ1, . . . , δn) with δi = 1/(cωi(1 + α)), D = diag(d1, . . . , dn) withdi = 1/(cωi(1 − α)), 0 < c ≤ 1, 0 ≤ α < 1, 0 < ωn < · · · < ω1 < 1,∑n

i=1 ci = 1 and ci > 0, i = 1, . . . , n. For descriptions on how thisequation arises in transport theory, see [1] and references cited therein.

In this talk, we consider enclosing the solution of (1). To the au-thor’s best knowledge, an algorithm for enclosing a solution of theNARE has not been written down in literatures. For continuous-timealgebraic Riccati equation, well-established algorithms for enclosingthe solution are proposed in [2–7]. In [4,5], the algorithms for discrete-time algebraic Riccati equation are also proposed. These algorithmsrequire O(n3) or larger operations.

The purpose of this talk is to propose an algorithm for enclosing thesolution of (1). This algorithm enables us to enclose the solution withonly O(n2) operations under a reasonable assumption by exploiting the

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special structure of (1). We finally report numerical results to showthe effectiveness and efficiency of this algorithm.

References:

[1] J. Juang, W-W. Lin, Nonsymmetric algebraic Riccati equationsand Hamiltonian-like matrices, SIAM Journal on Matrix Analysisand Applications, 20 (1998), pp. 228–243.

[2] T. Haqiri, F. Poloni, Methods for verified solutions to continuous-time algebraic Riccati equations, arXiv:1509.02015 [math.NA].

[3] B. Hashemi, Verified computation of symmetric solutions to continuous-time algebraic Riccati matrix equations, Proceedings of SCAN con-ference, Novosibirsk, 2012, pp. 54–56. http://conf.nsc.ru/files/conferences/scan2012/139586/Hashemi-scan2012.pdf

[4] W. Luther, W. Otten, Verified calculation of the solution ofalgebraic Riccati equation, in: T. Csendes (ed.), Developmentsin Reliable Computing, Kluwer Academic Publishers, Dordrecht,1999, pp. 105–118.

[5] W. Luther, W. Otten, H. Traczinski, Verified calculationof the solution of continuous- and discrete time algebraic Riccatiequation, Schriftenreihe des Fachbereichs Mathematik der Gerhard-Mercator-Universitat Duisburg, (1998), SM-DU-422.

[6] S. Miyajima, Fast verified computation for solutions of continuous-time algebraic Riccati equations, Japan Journal of Industrial andApplied Mathematics, 32 (2015), pp. 529–544.

[7] K. Yano, M. Koga, Verified numerical computation in LQ con-trol problem, Transactions of SICE, 45 (2009), pp. 261–267 (inJapanese).

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Fast validated computation for solutionsof discrete-time algebraic Riccati

equations

Shinya Miyajima

Iwate University4-3-5 Ueda, Morioka, Iwate, Japan

[email protected]

Keywords: discrete-time algebraic Riccati equation, verified compu-tation, stabilizing solution

Consider the following discrete-time algebraic Riccati equation (DARE):

AHXA−X − AHXB(R +BHXB)−1BHXA+Q = 0,

where A,Q ∈ Cn×n, B ∈ Cn×m and R ∈ Cm×m are given, Q and R areHermitian, AH denotes the conjugate transpose of A, and X ∈ Cn×n isto be solved. The DARE appears in many important problems in sci-ence and technology, e.g., discrete-time LQ-optimal control problemsand an equation in ladder networks [1]. The solution X of interest is anHermitian matrix, and the Hermitian solution X is called stabilizing ifall the eigenvalues of A−B(R+BHXB)−1BHXA are inside the unitdisk. The stabilizing solution is required in the practical applications.

In this talk, we consider enclosing the solution of the DARE, specif-ically, computing interval matrices containing the solution. The pio-neering work is the algorithms proposed in [2,3]. In these algorithms,the special structure of the DARE is skillfully exploited. These algo-rithm require O(n6) operations.

The purpose of this talk is to propose an algorithm for enclosingthe solution of the DARE. This algorithm utilizes numerical spectraldecomposition of A − B(R + BHXB)−1BHXA, where X denotes anumerical solution, requires only O(n3) operations, and is applicablewhen A−B(R+BHXB)−1BHXA is diagonalizable. The assumption

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of the applicability is made just to accelerate the enclosure. The al-gorithm moreover verifies that the solution contained in the intervalmatrix is unique and the stabilizing one. We finally report numericalresults to show the effectiveness and efficiency of this algorithm.

References:

[1] D.A. Bini, B. Iannazzo, B. Meini, Numerical Solution of Al-gebraic Riccati Equations, SIAM, Philadelphia, 2012.

[2] W. Luther, W. Otten, Verified calculation of the solution ofalgebraic Riccati equation, in: T. Csendes (ed.), Developmentsin Reliable Computing, Kluwer Academic Publishers, Dordrecht,1999, pp. 105–118.

[3] W. Luther, W. Otten, H. Traczinski, Verified calculationof the solution of continuous- and discrete time algebraic Riccatiequation, Schriftenreihe des Fachbereichs Mathematik der Gerhard-Mercator-Universitat Duisburg, (1998), SM-DU-422.

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Fast enclosure for matrix multiplicationon a GPU

Yusuke Morikura1), Yusuke Nozawa1), Kouta Sekine1),Masahide Kashiwagi1) and Shi’nichi Oishi1,2)

1) Department of Applied Mathematics, Waseda University,2) JST/CREST

1) 419 room Building 63, 3–4–1 Ookubo, Shinjuku-ku, Tokyo, [email protected]

Keywords: matrix multiplication, interval arithmetic, GPU

In this talk, we are concerned with a fast inclusion of matrix mul-tiplication on a GPU. Let F be a set of floating-point number definedby IEEE 754 standard [1]. Let A ∈ Fm×n and B ∈ Fn×p be given. Wedefine C, C ∈ Fm×p by

C ≤ A ·B ≤ C. (1)

The inclusion of (1) is one of the significant methods in verified numer-ical computation of linear problems (e.g. linear systems and eigenvalueproblems [2]).

On CPU, since settled rounding modes hold until we change itagain, we can compute C and C such as the following by BLAS rou-tines1 [2]:

#include <fenv.h> //Include fenv headerfesetround(FE DOWNWARD); //Rounding upwardC = dgemm routine; //Calculate Cfesetround(FE UPWARD); //Rounding downwardC = dgemm routine; //Calculate C

1We assume that dgemm routine does not use a fast matrix multiplication algorithm such asStrassen’s method

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Here, the function fesetround is defined by fenv.h in C99, andchanges the selected rounding mode. To compute matrix multipli-cation two times, we can obtain the inclusion of matrix multiplication.

In this talk, we consider the other simple approach based on MAGMABLAS routine on a GPU [3]. MAGMA BLAS is open source routinesfor numerical linear problems on GPU. MAGMA BLAS is written byCUDA [4] created by NVIDIA. Since recent NVIDIA’s GPUs supportIEEE 754 standard, we can perform computation with rounding modecontrol using intrinsic functions. However, rounding modes of GPUcan not change such as the function fesetround on CPU. Therefore, wedirectly revise the dgemm function in MAGMA BLAS in order to addthe inclusion part of a matrix multiplication. In this method, sinceit requires no extra memory transfer from global memory to cachememory, we achieved less than twice of computing time, though wecalculate two times of matrix mulplication. In addition, we show theoptimization of the function dgemm in MAGMA BLAS for the case ofthe inclusion on a GPU. Finally, we illustrate details and the resultsof computation in the talk.

References:

[1] ANSI/IEEE 754-1985, Standard for Binary Floating-Point Arith-metic, 1985.

[2] S. Oishi and S.M. Rump, Fast verification of solutions of matrixequations, Numer. Math., 90(4):755–773, 2002.

[3] MAGMA., http://icl.cs.utk.edu/magma/icl.cs.utk.edu/magma/,(2016, 03, 30)

[4] CUDA C Programming Guide, NVIDIA, http://docs.nvidia.

com/cuda/cuda-c-programming-guide/, (2016, 03, 30)

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Generalized intervals in globaloptimization

Arnold Neumaier, Ferenc Domes, Tiago Montanher, MihalyMarkot and Hermann Schichl

University of Vienna Faculty of Mathematics,Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria

[email protected]

Keywords: global optimization, union arithmetic, projective trans-formation

Union arithmetic and projective arithmetic are two generalizationsof interval arithmetic that were found useful for speeding up branch andbound algorithms for solving nonlinear systems, constraint satisfactionproblems, and global optimization problems.

Union arithmetic represents sets of reals as unions of one or moredisjoint intervals, and is thus able to take advantage of division byintervals containing zero in interval Gauss-Seidel methods.

Projective arithmetic is a way to handle unbounded regions inconstraint satisfaction problems and global optimization problems byusing projective transformations that map unbounded regions intobounded ones [1].

References:

[1] H. Schichl, A. Neumaier, M.C. Markot, F. Domes, Onsolving mixed-integer constraint satisfaction problems with un-bounded variables, pp. 216–233 in: Integration of AI and ORTechniques in constraint programming for combinatorial optimiza-tion problems, Springer, Berlin 2013.

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Linear Systems with the Exact Solutionfor Numerical Tests

Katsuhisa Ozaki and Takeshi Ogita

Shibaura Institute of Technology307 Fukasaku, Minumaku, Saitama 337-8570, Japan

[email protected]

Keywords: Numerical Linear Algebra, Linear System, NumericalComputations

This talk is concerned with a linear system Ax = b for numericalcomputations, where A is a coefficient matrix and b is a right-hand sidevector. Let F be a set of binary floating-point numbers as defined bythe IEEE 754 standard [1]. Let fl(·) denote that each operation in theparenthesis is evaluated by floating-point arithmetic. All elements inthe matrix and the vector are represented by the floating-point num-bers, i.e., A ∈ Fn×n and b ∈ Fn. Accuracy of a numerical solution xof Ax = b is sometimes monitored by the residual Ax− b. However, abasic numerical analysis says that the residual is not enough to showthe accuracy of numerical results. If we know the exact solution x suchthat Ax = b in advance, then it is very useful to examine robustnessof algorithms for solving linear systems, tendency of convergence foriterative methods, and overestimation of an error bound obtained byverified numerical computations. One may think that a right-handside vector b is generated by Ax from given A ∈ Fn×n and x ∈ Fn.However, if rounding errors occur in fl(Ax), then Ax = b may not besatisfied.

Miyajima, Ogita and Oishi proposed a method [2] which producesA′ ∈ Fm×m, x′, b′ ∈ Fm from given A ∈ Fn×n and x ∈ Fn such thatA′x′ = b′. As advantage of their method, we can set arbitrary vector xand condition number of A. On the other hand, the size of the matrixincreases, i.e., m ≥ n in many cases. Moreover, the structure of A′ isdifferent from that of A. For example, A′ becomes unsymmetric, evenif A is symmetric.

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Assume that A is non-singular, and a condition number of A isless than u−1, where u is a roundoff unit, e.g., u = 2−53 for IEEE 754binary64. For a brief introduction, we set x = (1, 1, . . . , 1)T in thisabstract. Our algorithm produces A′ ∈ Fn×n and b ∈ Fn such thatA′x = b with A′ ≈ A. We adopt ideas of error-free transformation ofmatrix multiplication [3] to this problem. The vector σ ∈ Fn is definedas

σi = 2β · 2vi, β = dlog2 ne, vi = dlog2 maxj|aij|e.

Let qij be a set of indices. The matrix A′ is obtained as follows.

a′ij = fl((fij + aij)− fij), δij = fl(aij − a′ij), fij = maxk∈qij

σk,

where the set qij depends on the structure of the matrix, e.g., qij =i, j for a symmetric matrix. Then,

A = A′ + ∆, b := fl(A′x) = A′x.

The structure of A′ is the same to that of A.In the presentation, we introduce a method which produces A′ and

b, assuming that each element in given x is a power of two. In addition,we demonstrate how to make ∆ as small as possible and how to preservepositive definiteness.

References:

[1] ANSI/IEEE, IEEE Standard for Binary Floating Point Arith-metic, IEEE, New York, 2008.

[2] S. Miyajima, T. Ogita, S. Oishi, A method of generating linearsystems with an arbitrarily ill-conditioned matrix and an arbitrarysolution, Proc. 2005 International Symposium on Nonlinear The-ory and its Applications (NOLTA 2005), Bruges, Belgium, (2005),pp. 741–744.

[3] K. Ozaki, T. Ogita, S. Oishi, S. M. Rump, Error-Free Trans-formation of Matrix Multiplication by Using Fast Routines of Ma-trix Multiplication and its Applications, Numerical Algorithms,59:1 (2012), pp. 95–118.

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Model-based design of optimalexperiments for guaranteed parameter

estimation of nonlinear dynamic systems

Anwesh Reddy Gottu Mukkula and Radoslav Paulen

Process Dynamics and Operations Group, Technische UniversitatDortmund

Emil-Figge-Strasse 70, Dortmund 44227, GermanyAnwesh-Reddy.Gottu-Mukkula,

[email protected]

Keywords: Optimal experiment design, Parameter estimation, Boundednoise

Mathematical modeling has become an integral part of modernprocess design methodologies as well as in control system design andoperations optimization. A typical model development procedure isdivided into two main phases, namely specification of the model struc-ture and estimation of the unknown/uncertain model parameters. Thelatter phase, often referred to as model fitting, normally proceeds bydetermining parameter values for which the model predictions closelymatch (for example in least-squares sense) the available process mea-surements.

Tailored approaches exist nowadays to strike against certain prob-lems encountered in classical parameter estimation. A guaranteed pa-rameter estimation of non-linear dynamic systems has been formu-lated [1] in a context of parameter estimation techniques that accountfor bounded measurement error. The problem consists of finding—orapproximating as closely as possible—the set of all possible param-eter values such that the predicted values of certain outputs matchtheir corresponding measurements within prescribed error bounds. Aset-inversion algorithm is applied, whereby the parameter set is succes-sively partitioned into smaller boxes and exclusion tests are performedto eliminate some of these boxes, until a given threshold on the ap-proximation level is met.

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Optimal experiment design has been extensively studied in liter-ature as an approach that provides best available conditions for col-lection of information-rich data of a dynamic system [2]. Naturallythis problem leads to dynamic optimization problem. The classicaloptimal experiment design is, however, not consistent with guaranteedparameter estimation as the results of the (classical) least-squares andguaranteed parameter estimation are not the same while the expec-tations of the realization of measurement noise differ in least-squares(unbounded normal distribution) and guaranteed (bounded arbitrarydistribution) estimation.

The work presented in this contribution provides a methodologyfor performing optimal experiment design in the context of guaranteedparameter estimation. The set of guaranteed parameter estimates isfirst over–approximated by a box using non-linear optimization. Theproblem of design of experiments, which determines a sequence of con-trol inputs for the optimal experiment design in the context of guar-anteed parameter estimation, is then formulated as a dynamic opti-mization problem over the non-linear optimization problem that over-approximates the set of guaranteed parameter estimates. The arisingbi-level program is regularized which makes the resulting non-linearoptimization with complementarity constraints well-conditioned.

Acknowledgments: The research leading to these results has re-ceived funding from the European Commission under grant agreementnumber 291458 (ERC Advanced Investigator Grant MOBOCON).

References:

[1] G. Franceschini, S. Macchietto, Model-based design of ex-periments for parameter precision: State of the art, Chemical En-gineering Science, 63 (2008), pp. 4846–4872.

[2] M. Kieffer, E. Walter, Guaranteed estimation of the param-eters of nonlinear continuous-time models: contributions of inter-val analysis, International Journal of Adaptive Control and SignalProcessing, 25 (2011), No. 3, pp. 191–207.

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PROMISE: floating-point precisiontuning with stochastic arithmetic

Stef Graillat1, Fabienne Jezequel1,2, Romain Picot1,3,Francois Fevotte3 and Bruno Lathuiliere3

1 Sorbonne Universites, UPMC Univ Paris 06, CNRS, UMR 7606,Laboratoire d’Informatique de Paris 6 (LIP6), Paris, France

2 Universite Pantheon-Assas, 12 place du Pantheon, 75231 ParisCEDEX 05, France

3 EDF R&D 7 boulevard Gaspard Monge, F-91120, Palaiseau, FranceStef.Graillat,Fabienne.Jezequel,[email protected]

Francois.Fevotte,[email protected]

Keywords: auto-tuning, Discrete Stochastic Arithmetic, floating-pointarithmetic, mixed precision, numerical validation, round-off errors

Nowadays, most floating-point computations in numerical simula-tions are performed in IEEE 754 binary64 precision (double preci-sion). This means that a relative accuracy of about 10−16 is providedfor every arithmetic operation. Indeed, in practice, programmers tendto use the highest precision available in hardware which is the doubleprecision on current processors. This approach can be costly in termsof computing time, memory transfer and energy consumption [1]. Abetter strategy would be to use no more precision than needed to getthe desired accuracy on the computed result. The challenge of usingmixed precision is to find some parts of codes (and so variables) thatmay be executed with lower precision. Unfortunately the amount ofpossible configurations is exponential in the number of variables.

To overcome this difficulty, we propose an algorithm and a toolcalled PROMISE (PRecision OptiMISEd) based on the delta debug-ging search algorithm [2] that provide a mixed precision configurationwith a worst-case complexity quadratic in the number of variables.From an initial C or C++ program and a required accuracy on thecomputed result, PROMISE automatically modifies the precision of

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variables. To estimate the numerical quality of results, PROMISEuses Discrete Stochastic Arithmetic (DSA) [3] which controls round-off errors in simulation programs. Unlike Precimonious [4], PROMISEdoes not focus on performance constraints, it can rather be seen as atool helping developers to reduce the cost of double precision variablesand improve the memory usage of their code. The PROMISE tool hasbeen successfully tested on programs implementing several numericalalgorithms including linear system solving and also on an industrialcode that solves the neutron transport equations [5].

References:

[1] M. Baboulin, A. Buttari, J. Dongarra, J. Kurzak, J.Langou, J. Langou,P. Luszczek, and S. Tomov, Accelerat-ing scientific computations with mixed precision algorithms, Com-puter Physics Communications, vol. 180, No. 12, pp. 2526–2533,2009.

[2] A. Zeller and R. Hildebrandt, Simplifying and isolating fail-ure-inducing input, IEEE Trans. Softw. Eng., vol. 28, No. 2,pp. 183–100, Feb. 2002.

[3] J. Vignes, Discrete Stochastic Arithmetic for validating resultsof numerical software, Numerical Algorithms, vol. 37, no. 1–4,pp. 377– 390, Dec. 2004.

[4] C. Rubio-Gonzalez, C. Nguyen, H. D. Nguyen, J. Dem-mel, W. Kahan, K. Sen, D. H. Bailey, C. Iancu, and D.Hough, Precimonious: Tuning assistant for floating-point preci-sion, Proceedings of the International Conference on High Perfor-mance Computing, Networking, Storage and Analysis, ser. SC’13.New York, NY, USA: ACM, 2013, pp. 27:1– 27:12.

[5] F. Fevotte and B. Lathuiliere, MICADO: Parallel implemen-tation of a 2D–1D iterative algorithm for the 3D neutron trans-port problem in prismatic geometries, Proceedings of Mathematics,Computational Methods & Reactor Physics, May 2013.

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Sharp error bounds for complexfloating-point inversion

Claude-Pierre Jeannerod1, Nicolas Louvet2,Jean-Michel Muller3, and Antoine Plet4

1 INRIA, 2 UCBL, 3 CNRS, 4 ENS LyonLaboratoire LIP, Universite de Lyon

ENS Lyon, 46, allee d’Italie, 69364 Lyon cedex 07, France

[email protected]

Keywords: floating-point arithmetic, numerical error, complex inver-sion, rounding error analysis.

We study the accuracy of the classic algorithm for inverting a com-plex number given by its real and imaginary parts as floating-pointnumbers. Our analyses are done in binary floating-point arithmeticwith an unbounded exponent range in precision p, and we assume thatthe basic arithmetic operations (+, −, ×, /) are rounded to nearest, sothat the roundoff unit is u = 2−p. We prove the componentwise relativeerror bound 3u for the complex inversion algorithm (assuming p ≥ 4),and we show that this bound is asymptotically optimal (as p → ∞)when p is even, and reasonably sharp when using one of the basic IEEE754 binary formats with an odd precision (p = 53, 113). This compo-nentwise bound obviously leads to the same bound 3u for the normwiserelative error. However we prove that the significantly smaller bound2.707131u holds (assuming p ≥ 24) for the normwise relative error, andwe illustrate the sharpness of this bound using numerical examples forthe basic IEEE 754 binary formats (p = 24, 53, 113). This work hasbeen published in [1].

References:

[1] C.-P. Jeannerod, N. Louvet, J.-M. Muller, A. Plet, Sharperror bounds for complex floating-point inversion, Numerical Al-gorithms, 2016.

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Rigourous error bounds for double-doubleoperations

Mioara Joldes∗, Jean-Michel Muller∗∗ andValentina Popescu∗∗

∗LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse, France∗∗LIP, ENS Lyon, 46 Allee d’Italie, 69364 Lyon, France

[email protected]

Keywords: floating-point arithmetic, high precision arithmetic, multiple-precision arithmetic, double-double

Most of today’s numerical computations are done using floating-point (FP) formats for representing real numbers. The two most widelyimplemented formats defined by the IEEE 754-2008 standard [1] forFP arithmetic are single-precision (binary32 ) and double-precision (bi-nary64 ). For some critical problems, such precisions do not suffice.Since higher precisions, such as quad -precision (binary128 ) is not hard-ware implemented on widely distributed processors, the only solutionis to use software emulation for extended precision.

There are mainly two ways of representing numbers in higher pre-cision. The first one, the multiple-digit method, uses integers for rep-resenting numbers as a sequence of possibly high-radix digits coupledwith an exponent. The second one, the multiple-term method, repre-sents real numbers as unevaluated sums of several FP numbers, allow-ing for computations to be carried out naturally using the underlyingFP hardware level. The later one is called a floating-point expansionwhen made up with an arbitrary number of terms and it makes use ofwell known error-free-transforms algorithms such as 2Sum, Fast2Sumand Fast2Mult (see [2]).

In this work we focus on the “double-double” (DD) precision, i.e.,the number x is represented as the sum of two double-precision FPnumbers, xh+x`, the high and the low part, that satisfy |x`| ≤ 1

2ulp|xh|.As a matter of fact, we should better talk about “double-word” preci-sion, since the algorithms and proofs we discuss can be used with any

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precision-p underlying FP format (p = 53 for double-precision). Moreprecisely we talk about two algorithms that were first presented by Ka-han et. al. in [3]. The first one (Alg. 1) computes the sum of two DDnumbers and returns also a DD number. In the initial paper they claimto have a relative error bound of 2 ·2−2p, but no rigorous proof is given.During our tests we observed that this bound is too optimistic, so wepresent here a slightly larger bound, 2−2p ·(5+9 ·2−p+7 ·2−2p+6 ·2−3p),for which we can provide a rigorous proof.

Alg. 2 computes the product of a DD number with a standard dou-ble-precision one and returns also a DD number. In this case we proveda tighter error bound that the one given in [3]. The initial error boundwas 4 · 2−2p and we provide a proof for 2−2p|xy|

(3 + 4 · 2−p + 2 · 2−2p

).

Algorithm 1 (zh, z`) = (xh, x`) +(yh, y`).

1: (sh, s`)← 2Sum(xh, yh)2: (th, t`)← 2Sum(x`, y`)3: c← RN(s` + th)4: (vh, v`)← Fast2Sum(sh, c)5: w ← RN(t` + v`)6: (zh, z`)← Fast2Sum(vh, w)7: return (zh, z`)

Algorithm 2 (zh, z`) = (xh, x`)∗y.

1: (ch, c`1)← Fast2Mult(xh, y)2: c`2 ← RN(x` · y)3: (th, t`1)← Fast2Sum(ch, c`2)4: t`2 ← RN(t`1 + c`1)5: (zh, z`)← Fast2Sum(th, t`2)6: return (zh, z`)

References:[1] IEEE Computer Society, IEEE Standard for Floating-Point Arithmetic, IEEE Standard

754-2008, Aug. 2008.

[2] J.-M. Muller, N. Brisebarre, F. de Dinechin, C.-P. Jeannerod, V. Lefevre, G.Melquiond, N. Revol, D. Stehle, and S. Torres, Handbook of Floating-Point Arith-metic, Birkhauser Boston, 2010.

[3] X.S.Li, J.W.Demmel, D.H.Bailey, G.Henry, Y.Hida, J.Iskandar, W.Kahan, A.Kapur,M.C.Martin, T.Tung, D.J.Yoo, Design, Implementation and Testing of Extended andMixed Precision BLAS, ACM TOMS, vol. 28, no. 2, 2002.

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Enclosing the Solution Set to IntervalParametric Matrix Equation A(p)X = B(p)

Evgenija D. Popova

Institute of Mathematics and InformaticsBulgarian Academy of Sciences

Acad. G. Bonchev str., block 8, 1113 Sofia, [email protected]

Keywords: linear matrix equation, dependent data, enclosure meth-ods

Consider the parametric matrix equation

A(p)X = B(p), p ∈ [p],

where A(p) and B(p) are known m × m and m × n matrices whoseelements are linear functions of uncertain parameters p = (p1, . . . , pk)varying within given intervals, and X = X(p) is the unknown matrix.

We prove that the united solution set to the matrix equation andthis solution set to the same parametric system considered as a sys-tem with multiple right hand sides have the same exact interval hullalthough, as shown in [1], they are different as sets. This implies thatall known methods for enclosing AE-solution sets to parametric linearsystems can be generalized to enclose the corresponding AE-solutionsets to the parametric matrix equation without penalties on the com-putational cost. We present a generalization of the method from [2,3],which does not require strong regularity of the parametric matrix, andsome improvements of the parametric Krawczyk iteration proposed in[1]. Comparison between some of the enclosure methods with respectto their scope of applicability and the quality of the solution enclosurewill demonstrate some properties of the methods that have not beenmentioned before.

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References:

[1] M. Dehghani-Madiseh, M. Dehghan, Parametric AE-solutionsets to the parametric linear systems with multiple right-hand sidesand parametric matrix equation A(p)X = B(p), Numerical Algo-rithms, (2016), doi: 10.1007/s11075-015-0094-3.

[2] A. Neumaier, A. Pownuk, Linear systems with large uncertain-ties, with applications to truss structures, Reliable Computing, 13(2007), pp. 149–172.

[3] E.D. Popova, Improved enclosure for some parametric solutionsets with linear shape, Comput. Math. Appl., 68, (2014), pp. 994–1005.

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Automated tuning of robust fractionalPID controller for interval plants using

Kharitonov’s theorem

Harsh Purohit, P.S.V. Nataraj

Systems and Control Engineering GroupIndian Institute of Technology Bombay

Powai-400076, [email protected],

[email protected]

Keywords: Fractional PID controller, Interval plants, Kharitonov’stheorem

In this article, we propose a new automated method for synthesizingrobust fractional PID controller for interval plant using Kharitonov’stheorem [1]. Only four Kharitonov polynomials are required to beHurwitz and we can determine the stability of the interval system un-der the given parameter variation. A gain and phase margin testercompensator is incorporated to guarantee the concerned system withcertain robust safety margins. This kind of controller synthesis pro-cedures have not closed form solutions and usually they are iterativeand/or based on graphical interpretations. Specifically, a new graphicalmethod for synthesizing robust PID controllers to achieve pre-specifiedsafety margin is proposed for uncertain time delay systems [2].How-ever, with easy availability of computation power, we can automatethis design procedure.

We have chosen fractional order controller because of its promis-ing recent developments and application to engineering problems[3].Considering the fractional order PID controller, we can obtain char-acteristic equation of the close loop system. And after selecting thedesign frequencies and gain-phase margin, we can pose this controllersynthesis problem as a constrained satisfaction problem (CSP) withall vertex Kharitonov polynomials where unknowns for this CSP are

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controller parameters (Kp, Ki, Kd, λ, µ) with pre-specified initial searchinterval. In this work, we have solved this problem using constraintsolver Realpaver [4]. Different consistency techniques like box, hull and3B has been implemented in Realpaver. Finally, illustrative examplewith computer simulations is demonstrated to confirm the validity ofthe proposed methodology.

References:

[1] V.L. Kharitonov, Asymptotic stability of an equilibrium postionof a family systems of linear differential equations, Differential’nyeUraveniya , (2008), No. 14, pp. 2086-2088.

[2] Yuan-Jay Wang, Graphical computation of gain and phase mar-gin specifications-oriented robust PID controllers for uncertain sys-tems with time-varying delay, Journal of Process Control, (2011),No. 21, pp. 475–488.

[3] C. Monje, B. Vinagre, V. Feliu and Y. Chen, Tuning andautotuning of fractional order controllers for industry applications,Control Engineering Practice, (2008), No. 16(7), pp. 798–812.

[4] L. Granvilliers and F. Benhamou, Algorithm 852: Real-paver: an Interval Solver using Constraint Satisfaction Techniques,ACM TOMS , (2006), No. 32(1), pp. 138-156.

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Safety Verification ByInterval Based

Quantified Constraint Solving

Peter Franek, Jan Kuratko, and Stefan Ratschan

Institute of Computer Science, Czech Academy of SciencesPod Vodarenskou vezi 2, 182 07 Praha, Czech Republic

[email protected]

Keywords: ordinary differential equations, safety verification, con-straints, constraint solving

We consider the following problem (“safety verification”):

Given:

• an ordinary differential equation

• a set of states I (the initial states)

• a set of states U (the unsafe states)

Verify: It is not possible for a solution of the ODE to start in a statein I and reach a state in U . Or equivalently: Every solution of theODE with an initial state in I never reaches a state in U .

This verification task can be reduced [3] to a constraint solvingproblem containing logical quantifiers (∀, ∃). Such problems have beentraditionally solved by computer algebra techniques [2], but methodsbased on interval computation have been shown to be competitive [4].However, the constraint solving problem arising in safety verificationhas a specific structure. In the talk we present an interval based algo-rithm that exploits this structure.

This work was supported by GACR grant 15-14484S and by the long-term strategic develop-

ment financing of the Institute of Computer Science (RVO:67985807).

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References

[1] B. F. Caviness and J. R. Johnson, eds., Quantifier Eliminationand Cylindrical Algebraic Decomposition, Springer, Wien, 1998.

[2] G. E. Collins, Quantifier elimination for the elementary theoryof real closed fields by cylindrical algebraic decomposition, in Sec-ond GI Conf. Automata Theory and Formal Languages, vol. 33 ofLNCS, Springer Verlag, 1975, pp. 134–183. Also in [1].

[3] S. Prajna and A. Jadbabaie, Safety verification of hybrid sys-tems using barrier certificates, in HSCC’04, R. Alur and G. J. Pap-pas, eds., no. 2993 in LNCS, Springer, 2004.

[4] S. Ratschan, Efficient solving of quantified inequality constraintsover the real numbers, ACM Transactions on Computational Logic,7 (2006), pp. 723–748.

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An Interval-Based Algorithm for FeatureExtraction from Speech Signals

Andreas Rauh1, Susann Tiede2 and Cornelia Klenke2

1University of Rostock, Chair of MechatronicsD-18059 Rostock, Germany

[email protected]

2Speech TherapistsBahnhofstraße 35, D-17087 Altentreptow, Germany

[email protected],

logopaedie [email protected]

Keywords: Branch-and-bound algorithm, Speech analysis, Phonemeand pattern recognition

To develop a computer-based assistance system for speech therapy,it is essential to distinguish between the linguistic levels of lexicon,grammar, and pronunciation [1] and to deal with (i) the automatictranscription and preprocessing of speech involving erroneous pronun-ciation, (ii) the classification of pronunciation disorders, and (iii) agrammatical analysis. For the tasks (i) and (ii), this contribution ex-ploits an online frequency analysis of speech signals based on stochasticfiltering approaches and the identification of points of time at whichtransitions between subsequent phonemes can be expected (cf. [3,4]).

In general, phonemes can be classified into voiced and unvoicedsounds [2,3]. Voiced sounds (e.g. normal vowels) are characterized byseveral relatively sharp formant frequencies produced by vibrations ofthe vocal folds representing a fluidic resistance against the outflow ofair expelled from the lungs. In contrast, unvoiced sounds (e.g. whis-pered vowels and fricatives such as ch, ss, sh, f) are caused by a turbu-lent, partially irregular, air flow with negligible vibrations of the vocalfolds. To some extent, they are produced by fizzing sounds originat-ing between teeth and lips as well as between tongue and hard or softpalate. Here, sharp formant frequencies are characteristic for voiced

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phonemes, whereas wide frequency bands are typical for unvoiced ones.For both cases, a stochastic filtering approach [3] can be employed toestimate expected values of the formant frequencies and their associ-ated covariances, where the broad-band nature of unvoiced speech isreflected by (co-)variance estimates that are significantly larger thanfor the voiced case. Transitions between subsequent phonemes are in-dicated by rapid changes in the above-mentioned estimation results [4].

In this contribution, an interval algorithm is presented for the ex-traction of speech features from the estimated formant frequencies andsignal amplitudes on the level of individual phonemes. Besides a pat-tern recognition for correctly pronounced sounds, this procedure helpsto identify and classify disorders (together with expert knowledge ofthe speech therapist if these features are not yet included a phonemedatabase). Moreover, stuttering disorders (those characterized by rep-etitions of individual/ multiple phonemes/ syllables) can be detectedfrom the extracted features. Numerical results for speech sequenceswithout and with pronunciation disorders conclude this contribution.

References:

[1] J.S. Damico, N. Muller, and M.J. Ball, The Handbook of Languageand Speech Disorders, ser. Blackwell Handbooks in Linguistics.Chichester, West Sussex, UK: Wiley, 2010.

[2] P. Ladefoged and I. Maddieson, The Sounds of the World’s Lan-guages, ser. Phonological Theory. Chichester, West Sussex, UK:Wiley, 1996.

[3] A. Rauh, S. Tiede, and C. Klenke, Observer and Filter Approachesfor the Frequency Analysis of Speech Signals, Proc. of 21st IEEEIntl. Conf. on Methods and Models in Automation and Robotics,Poland, 2016, under review.

[4] A. Rauh, S. Tiede, and C. Klenke, Stochastic Filter Approachesfor a Phoneme-Based Segmentation of Speech Signals, Proc. of21st IEEE Intl. Conf. on Methods and Models in Automation andRobotics, Miedzyzdroje, Poland, 2016, under review.

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Interval-Based Identification of Frictionand Hysteresis Models

Andreas Rauh and Harald Aschemann

University of Rostock, Chair of MechatronicsD-18059 Rostock, Germany

Andreas.Rauh,[email protected]

Keywords: Interval-based identification, Friction, Hysteresis, Verifiedsolution of non-smooth ODEs

Dynamic models with state- and input-dependent transitions be-tween different continuous-time representations are common for themathematical modeling of a large variety of technical systems. Amongthese, especially the description of friction and hysteresis is widely usedin both mechanical and control engineering [1, 3, 4, 5].

In previous work, interval-based identification routines have beenderived where state- and input-dependent transitions were included inpiecewise-defined ordinary differential equations (ODEs) [2].

In this framework, the experimental identification of a mathemat-ical model for a drive train test rig was considered. The correspond-ing non-smooth set of ODEs was characterized by an automaton rep-resentation for all possible transitions between static friction and apiecewise linear, velocity-proportional model for sliding friction. Inthis work, the coefficients for the rotary mass moment of inertia, thevelocity-proportional sliding friction, and the initial breakaway torquewere identified in terms of interval parameters on the basis of measureddata with an additive bounded uncertainty [3, 4].

To make the identification procedure applicable for further — es-pecially more complex — system models, algorithmic extensions aredeveloped which allow for preventing an excessive growth of the com-putational effort if a larger number of uncertain parameters are consid-ered. During that process, the friction model used in [3, 4] is extendedto a polynomial representation of the sliding friction behavior, to an

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inclusion of Stribeck characteristics near the breakaway point, to com-binations of friction and Bouc-Wen hysteresis models [1, 5], and tomore detailed representations for the actuator dynamics. The Bouc-Wen model introduces further points into the set of non-smooth ODEsat which state-dependent transitions between individual system com-ponents occur. A numerical case study for the experimental parameteridentification of a drive train test rig available at the Chair of Mecha-tronics at the University of Rostock concludes this paper with a focuson a verified exclusion of friction and hysteresis models that are defi-nitely not compatible with the given interval-bounded measured data.

References:

[1] F. Ikhouane and J. Rodellar, Systems with Hysteresis: Analysis,Identification and Control using the Bouc-Wen Model, Chichester,West Sussex, UK: John Wiley&Sons, Ltd., 2007.

[2] N.S. Nedialkov and M. v. Mohrenschildt, Rigorous Simulation ofHybrid Dynamic Systems with Symbolic and Interval Methods,Proc. of American Control Conf. ACC, Anchorage, USA, 2002.

[3] A. Rauh, L. Senkel, and H. Aschemann, Verified Parameter Identi-fication for Dynamic Systems with Non-Smooth Right-Hand Sides,Proc. of 16th GAMM-IMACS International Symposium on Scien-tific Computing, Computer Arithmetic, and Validated NumericsSCAN 2014, Wurzburg, Germany, Vol. 9553 of Lecture Notes inComputer Science, Heidelberg: Springer, 2016.

[4] A. Rauh, L. Senkel, and H. Aschemann, Experimental Compari-son of Interval-Based Parameter Identification Procedures for Un-certain ODEs with Non-Smooth Right-Hand Sides, Proc. of 20thIEEE Intl. Conf. on Methods and Models in Automation and Ro-botics, Miedzyzdroje, Poland, 2015.

[5] A. Rauh, Ch. Siebert, and H. Aschemann, Verified Simulation andOptimization of Dynamic Systems with Friction and Hysteresis,Proc. of ENOC 2011, Rome, Italy, 2009.

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Toward the Optimal Parameterization ofInterval-Based Variable-Structure State

Estimation Procedures

Andreas Rauh and Harald Aschemann

University of Rostock, Chair of MechatronicsD-18059 Rostock, Germany

Andreas.Rauh,[email protected]

Keywords: Sliding mode observers, Lyapunov functions, Ito differen-tial operator, Time discretization

In previous work, interval-based extensions of sliding mode stateobserves [4] have been presented by the authors which allow for aguaranteed stabilization of the associated error dynamics [2, 3].

The parametrization of these observers is so far based on the on-line evaluation of a suitable candidate for a Lyapunov function andits related time derivative. The variable-structure gain of this type ofobserver is determined in such a way that the time derivative of theLyapunov function candidate can be guaranteed to be negative defi-nite despite bounded uncertainty in system parameters as well as inthe measured output variables. Besides the treatment of interval uncer-tainty, extensions were developed which guarantee asymptotic stabilityin cases in which stochastic measurement noise influences the systemdynamics. These stochastic disturbances are taken into account by areplacement of the “classical” time derivative of the Lyapunov functioncandidate by the so-called Ito differential operator [1, 2].

The limitations of this estimation approach are

• the assumption of negligibly small time discretization errors in aquasi-continuous implementation of the interval-based estimationprocedure and

• the use of the stability requirement of the error dynamics (with ana-priori fixed underlying observer gain for all linear system com-ponents) as the only design requirement.

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In this contribution, the following extensions of the interval-basedvariable-structure state and parameter estimator are developed anddemonstrated for suitable application scenarios: Firstly, time discretiza-tion errors are taken into account in a discrete-time implementationof the stability proof for the state estimation procedure which needsto be performed in real time. Secondly, a first attempt toward theoptimal parameterization of the underlying linear state observer aswell as extensions of the variable-structure gain computation are pre-sented. These extensions aim at choosing the gain values in such away that the rate of convergence of the estimated states toward theirtrue values (which are, however, in practical applications unknown)can be optimized. There, it is necessary to find a trade-off betweenthe maximum rate of convergence, on the one hand, and the robustnessof the estimation as well as the achievable tightness of the computedinterval bounds, on the other hand. Simulation results for prototypicalapplications in control engineering conclude this contribution.

References:

[1] H. Kushner, Stochastic Stability and Control, New York: AcademicPress, 1967.

[2] L. Senkel, A. Rauh, and H. Aschemann, Sliding Mode Techniquesfor Robust Trajectory Tracking as well as State and ParameterEstimation, Mathematics in Computer Science, Vol. 8 (2014), Is-sue 3–4, pp. 543–561.

[3] L. Senkel, A. Rauh, and H. Aschemann, Sliding Mode ApproachesConsidering Uncertainty for Reliable Control and Computation ofConfidence Regions in State and Parameter Estimation, Proc. of16th GAMM-IMACS International Symposium on Scientific Com-puting, Computer Arithmetic, and Validated Numerics SCAN 2014,Wurzburg, Germany, Vol. 9553 of Lecture Notes in Computer Sci-ence, Heidelberg: Springer, 2016.

[4] V. Utkin, Sliding Modes in Control and Optimization, Berlin, Hei-delberg: Springer, 1992.

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HPC and interval computations

Nathalie Revol

Inria - University of Lyon, LIP (UMR 5668 CNRS - ENS de Lyon -INRIA - UCBL)

ENS de Lyon, 46 allee d’Italie, 69364 Lyon Cedex 07, [email protected]

Keywords: interval arithmetic, high-performance computing

Interval computations can benefit from high-performance environ-ments: larger problems can be solved, execution time can be reduced.Furthermore, as each numerical value is used several times for eachoperation in interval arithmetic, the ratio between computations anddata moves is favourable. However, it is not totally clear whetherHPC environments offer the required facilities, in particular whetherdirected rounding modes are properly set by these environments.

Reciprocally, high-performance computations may need interval com-putations. Typically, fault-tolerant or resilient algorithms, which per-form large amounts of computations, also perform quick computationsto detect whether a fault has occured. With floating-point arithmetic,these quick computations frequently return results which differ fromthe results of the long computations, because of roundoff errors, cf.[1]. An issue is to determine whether this difference is due to floating-point computations being non reproducible or to a failure. Intervalarithmetic could be useful to compute these certificates: it would re-turn a value with bounds and thus the comparison between the resultof the long computation and the certificate would boil down to an in-clusion test. Although much slower than floating-point arithmetic, theoverhead induced by interval arithmetic would apply only to a smallcomputation. Current resilient algorithms such as the ones in [2] applyan equality test up to a threshold, and this threshold can be difficultto choose. We will detail if, how and when interval arithmetic cansuppress the use of this threshold.

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References:

[1] Revol, Nathalie, and Philippe Theveny, Numerical repro-ducibility and parallel computations: Issues for interval algorithms,IEEE Transactions on Computers, 63 (2014), No. 3, pp. 1915–1924.

[2] S. Cools, E.F. Yetkin, E. Agullo, L. Giraud, W. Van-roose, Analysis of rounding error accumulation in conjugate gra-dients to improve the maximal attainable accuracy of pipelinedCG, arXiv:1601.07068, (2016), 27 pages.

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Tube ProgrammingApplied to State Estimation

Simon Rohou, Luc Jaulin, Lyudmila Mihaylova,Fabrice Le Bars, Sandor M. Veres

Lab-STICC, ENSTA Bretagne, Brest, FranceUniversity of Sheffield, Western Bank Sheffield S10 2TN, UK

[email protected]

Problem. We consider the guaranteed non-linear state estimationof a robot described by its state x ∈ Rn and measuring distances tobeacons, see Fig. 1. xᵀ = x, y, θ, v. System’s description is given by:

R

x = f(x,u)ri(ti) = gj(x(ti))

(1)

Figure 1: Localization ofrobot R among beacons

With f : Rn × Rm → Rn, the continu-ous evolution function based on robot’sstate x and input u ∈ Rm, and gj :Rn → R, a sporadic observation func-tion giving a range value ri ∈ R betweenthe robot and the j -th beacon at time ti.

Main approach. In a set member-ship approach, x and x respectively be-long to bounded signals evolving withtime: ∀t , x(t) ∈ [x](t) , x(t) ∈ [x](t).These trajectories are represented withtubes [1], see Fig. 2. The more uncer-tain a trajectory is, the thicker the tubecontaining it will be. When trajectory’sestimation is improved, the tube needs to be contracted, thus becom-ing thinner. To this end, we propose to break down the problem intoseveral elementary constraints involving contractors on tubes.

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t

[f ]

Figure 2: a tube: guaranteed representation of a set of trajectories (in light gray). A signal s(·) (in orange) belongsto an interval function so that: ∀t, s(t) ∈ [f ](t) = [f−(t), f+(t)]. This ensures a guaranteed representation of signals.Functions f−(t) and f+(t) are represented with a set of boxes (in blue).

To this end, we implemented a new C++ library for tubes representations. Code is freely available on theGitHub software development platform:

IBEX-Robotics libraryA complementary C++ library of IBEX for robotics purposes.https://github.com/ibex-team/ibex-robotics

Future related paper:• Tube contractions applied to guaranteed non-linear state estimation with time-uncertainties

Simon Rohou, Luc Jaulin, Lyudmila Mihaylova, Fabrice Le Bars, Sandor M. Veres

3.3 Guaranteed integrationThis research on tubes and their related tools – such as contractors, Section 3.2 – brings new ideas and enable usto consider the problem of guaranteed integration of differential equations [3, 19, 11, 8, 17, 12]. From an initialbox [x] (0), guaranteed integration [20] provides a set of techniques based on interval arithmetic to compute a box-valued function [x] (t) (a tube) containing true values for the initial value problem. These methods provide intervalcounterparts of Euler [16], Runge-Kutta [7] or Taylor [18] integration and validate results using the Picard Theorem.

In robotics, guaranteed integration provides a reliable knowledge about robots trajectories. In this work, we pro-pose a new approach based on tube arithmetic. Current experiments show interesting results when computing robottrajectories. This has been compared with existing methods, namely the CAPD DynSys library, well recognizedin the community. Our approach is in fact particularly suitable for robots trajectories and seems to offer betterresults in this context. We plan to open discussions about this work during next conferences SWIM’16 and SCAN’16.

Future related paper:• Guaranteed integration of robot trajectories

Simon Rohou, Luc Jaulin, Lyudmila Mihaylova, Fabrice Le Bars, Sandor M. Veres

4 Issues/Problems preventing from achieving PhD TargetsNone, except time.

3

Figure 2: A tube [f ](·) implemented with a set of slices

Break down. Let us consider the given equations: x = v · cos(θ),√x(t1)2 + y(t1)2 = r1, the following constraints can be established:

continuous constraints

a = cos(θ)b = v · ax =

∫b

sporadic constraintx(t1) =

√r21 − y(t1)2

Tube contraction. Our contribution is to perform a reliable stateestimation with a simple and general method involving continuous orfleeting [2] constraints on tubes. In the continuous case, constraintsare managed with tube arithmetic: variables a and b are tubes too.[x](·) = [x](·) ∩

∫[b](·). In the sporadic case, we will show that to

be compliant with tube’s representation [x](·), such a contraction canonly be done considering the derivative tube [x](·).References:

[1] A. Bethencourt, L. Jaulin, Solving non-linear constraint satis-faction problems involving time-dependant functions, Mathematicsin Computer Science, 2014.

[2] F. Le Bars, J. Sliwka, O. Reynet, L. Jaulin, State estima-tion with fleeting data, Automatica, 2012.

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Sharp error bounds for the Gammafunction over the whole floating-point

range

Siegfried M. Rump

Institute for Reliable ComputingHamburg University of Technology

Schwarzenberg-Campus 1, 21071 Hamburg, Germanyand

Waseda UniversityFaculty of Science and Engineering

3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, [email protected]

Keywords: Gamma function, rigorous error bounds, floating-point

The Matlab built-in Gamma function delivers approximations withrelative error up to 100%.

We present an algorithm for computing rigorous error bounds forthe Gamma function over the whole floating-point range. More pre-cisely, for a given floating-point number x, an inclusion of Γ(x) iscomputed with relative error less than 6 · 10−16.

If the input argument is an interval X ∈ IF, then an inclusionof Γ(x) : x ∈ X of similar quality is computed. The presentedfunction is part of INTLAB, the Matlab/Octave toolbox for ReliableComputing.

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The origin of interval arithmetic

Siegfried M. Rump

Institute for Reliable ComputingHamburg University of Technology

Schwarzenberg-Campus 1, 21071 Hamburg, Germanyand

Waseda UniversityFaculty of Science and Engineering

3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, [email protected]

Keywords: error bounds, intervals, reliable computing

Much has been said and published on the origin of interval arith-metic.

In this talk we give a brief overview and concentrate on the seminalMaster Thesis by Sunaga, handwritten in Japanese, submitted at theUniversity of Tokyo 60 years ago on February 29, 1956.

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A method of verified computation forconvex programming

Ryo Kobayashi1, Takuma Kimura2 and Shin’ichi Oishi1

1Waseda University, Tokyo 169-8555, Japan2Saga University, Saga 840-8502, Japan

[email protected]

Keywords: numerical verification, convex programming, Kantorovich’stheorem, continuous Newton method

In this paper, we are concerned with a convex programming

minimize f(x), subject to g(x) ≤ 0 and x ≥ 0, (1)

where g(x) = (g1(x), g2(x), . . . , gm(x))T , and f, gi (i = 1, . . . ,m) :Rn → R are sufficiently smooth and convex. We assume that problem(1) satisfies Slater’s constraint qualification.

In [1], Oishi and Tanabe have studied a numerical method of in-cluding an optimal solution to a linear programming provided that anapproximation of an optimal solution is given. In this talk, we pro-pose a numerical method of including an optimal solution to the aboveconvex programming.

The KKT conditions of (1) are represented as a complementarityproblem

φ(x, y) :=

(x (∇f(x) +

∑mi=1 yi∇gi(x))

y g(x)

)= 0, (2a)

subject to

∇f(x) +m∑

i=1

yi∇gi(x) ≥ 0, g(x) ≤ 0, x ≥ 0 and y ≥ 0, (2b)

where y ∈ Rm is a Lagrange multiplier. Here, for u, v ∈ Rn, u vdenotes an element-wise product (u v)i = uivi (i = 1, . . . , n).

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Hereafter, we employ the notation z = (xT , yT )T . The purposeof this talk is to verify the accuracy of a numerically computed ap-proximation of an optimal solution to (2). We propose the followingtheorem:Theorem 1. Let z be an interior point such that

gi(x) < 0 and x > 0.

Suppose that φ′(z) is nonsingular and there exists positive numbers α, ωsuch that

∥∥φ′(z)−1∥∥∞ ‖φ(z)‖∞ ≤ α and

∥∥φ′(z)−1 (φ′(z′)− φ′(z′′))∥∥∞ ≤

ω ‖z′ − z′′‖∞ , ∀z′, z′′ ∈ Rn+m. If

αω ≤ 1

4, (3)

there exists an optimal point z∗ ∈ B(z, ρ) := z′ ∈ Rn+m | ‖z′ − z‖∞ ≤ ρsatisfying (2), which is in fact unique in B(z, ρ), where

ρ =1−√

1− 3αω

ω. (4)

Theorem 1 enables us to verify the existence of an optimal solution to(2) around a given approximation and to obtain the error bound forthe approximation. Moreover, Theorem 1 guarantees feasibility of theoptimal solution. Theorem 1 is proved by using Kantorovich’s theoremand continuous Newton method [2].

Some numerical examples will be presented, which show that ourmethod is useful for verifying solutions to convex programmings.

References:

[1] S. Oishi, K. Tanabe, Numerical Inclusion of optimal solution forLinear, JSIAM Letters, Vol. 1 (2009), pp. 5–8.

[2] K. Tanabe, Continuous Newton-Raphson method for solving anunderdeterminded system of nonlinear equations, Nonlinear Anal.T.M.A., Vol. 3 No. 4 (1979), pp. 495–503.

[3] S. Boyd, L. Vandenberghe, Convex Optimization, CambridgeUniversity Press, New York, 2004.

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Numerical computation of invariantobjects with wavelets

Lluıs Alseda i Soler and David Romero i SanchezUniversitat Autonoma de Barcelona

Departament de MatematiquesEdifici Cc, 08913 Cerdanyola del Valles, Barcelona, Catalonia

[email protected]

Keywords: Wavelets, regularity, quasiperiodically forced system.

In certain classes of dynamical systems invariant sets with a strangegeometry appear. For example the iteration of two-dimensional quasi-periodically forced skew product, under certain conditions, gives usStrange Non-Chaotic Attractors.

On the other side, when approximating dynamical invariant ob-jects, a usual standard approach is to use Fourier expansions (ratherthan Wavelet ones). However, to obtain an analytical approximationof the aforesaid objects it seems more natural to use wavelets instead ofthe Fourier approach. The aim of the talk is to describe an algorithmfor the semi-analytical computation of the invariant object (numericalcomputations of the wavelet coefficients) using both Daubechies andHaar wavelets.

One of the advantages of this approach is that wavelets also de-fine certain regularity spaces that provide a natural framework for theapproximations that one gets. Indeed, the computation of the regu-larity (depending on parameters) can give some insight in the study ofthe fractalization or other routes of creation of Strange Non-ChaoticAttractors and help in detecting this bifurcation.

Thus, the aim of the above exercise is to be able to detect, by meansof the estimation of the regularity of the object, how a non-chaotic at-tractor “becomes” strange. The study of this regularity (dependingon parameters) may give another point of view to the “fractalizationroutes” described in the literature and that are currently under dis-cussion.

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The Julia package ValidatedNumerics.jland its application to the rigorous

characterization of open billiard models

David P. Sanders, Luis Benet and Nikolay Kryukov

Department of Physics, Faculty of SciencesUniversidad Nacional Autonoma de Mexico (UNAM)

& Computer Science and Artificial Intelligence Laboratory,Massachusetts Institute of Technology, USA

Ciudad Universitaria, Del. Coyoacan, Ciudad de Mexico 04510,Mexico

[email protected]

Keywords: validated software, billiard models, decorations

We firstly present ValidatedNumerics.jl [1], a new software pack-age for validated numerics using the Julia programming language, writ-ten from scratch. Julia provides a unique combination of mathematical-style syntax, high-level interactive usability, and C-like performance,making it an excellent option for technical computing, in particularfor interval arithmetic. The package is well on the way to being con-formant with the IEEE Standard 1788-2015, including decorations. Itis free/libre open-source software (FLOSS), with a permissive MIT li-cense, and a simple code base that allows working with both doubleand arbitrary precision. Due to the excellent composability of Julia,it interacts seamlessly with other Julia packages, which provide, forexample, generic linear algebra and automatic differentiation capabili-ties that immediately work with intervals, with (almost) no additionalcoding.

We then apply this to study open billiard models, which are deter-ministic dynamical systems in which point particles collide with fixedobstacles. Despite the relevance of billiard models, there seems to havebeen little previous work on applying interval methods to them. Westudy open billiard models, in which particles may escape to infinity,

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so that the billiard map, which finds the next collision with an ob-stacle, has a non-trivial domain; thus the use of decorated intervalsarises naturally in these systems. We find enclosures of the domain ofdefinition for the n-collision billiard map, as well as periodic orbits.

References:

[1] David P. Sanders and Luis Benet, ValidatedNumerics.jlJulia package,https://github.com/dpsanders/ValidatedNumerics.jl

[2] Nikolay Kryukov, David P. Sanders and Luis Benet, Rig-orous calculation of periodic trajectories in open billiard models,In preparation

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124

A norm estimation for an inverse of linearoperator using a minimal eigenvalue

Kouta Sekine, Kazuaki Tanaka and Shin’ichi Oishi

Waseda university3-4-1 Okubo, Tokyo, [email protected]

Keywords: Numerical verification, Differential equations

Let A : H2(Ω)∩H10(Ω)→ L2(Ω) be an elliptic operator with Dirich-

let boundary conditions and N : H10(Ω)→ L2(Ω) be a linear operator

which is given. We define a linear operator L := A + N . The aim ofthis talk is to estimate a constant K satisfying

‖φ‖H10≤ K‖Lφ‖L2, φ ∈ H2(Ω) ∩H1

0(Ω) (1)

using an eigenvalue problem. The evaluation of the constant K playsan important role for a verified numerical proof to solutions of semi-linear partial differential equations, and have been studied [1-4].

In this talk, we first proof the following theorem:

Theorem 1. Let λ1 be an minimal eigenvalue satisfying an eigenvalueproblem

Find u ∈ H2(Ω) ∩H10(Ω), λ ∈ R s.t. LA−1L∗u = λu, (2)

where L∗ is a dual operator of L. If λ1 6= 0 holds, then L is nonsingular,and the constant

K =1√λ1

satisfies the inequality (1).

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Next, we show how to compute the minimal eigenvalue λ1. Let σbe a positive constant satisfying (Du, u)L2 ≥ 0, where D := A+ σI −(N +N ∗) +N ∗A−1N . We put µ = λ+ σ, and the eigenvalue problem(2) is transformed into

Du = µu. (3)

Then, we can compute the minimal eigenvalue µ1 using Liu-Oishi’stheorem[5], and obtain the constant K.

References:

[1] S. Oishi, Numerical verification of existence and inclusion of solu-tions for nonlinear operator equations, Journal of Computationaland Applied Mathematics, 60.1 (1995), pp. 171-185.

[2] M.T. Nakao, K. Hashimoto, and Y. Watanabe, A numer-ical method to verify the invertibility of linear elliptic operatorswith applications to nonlinear problems, Computing, 75.1 (2005),pp. 1-14.

[3] K. Tanaka, A. Takayasu, X. Liu, and S. Oishi, Verified normestimation for the inverse of linear elliptic operators using eigen-value evaluation, Japan Journal of Industrial and Applied Mathe-matics, 31:3 (2014), pp. 665-679.

[4] Y. Watanabe, K. Nagatou, M. Plum, and M.T. Nakao,Norm bound computation for inverses of linear operators in Hilbertspaces, Journal of Differential Equations, 260.7 (2016), pp. 6363-6374.

[5] X. Liu, and S. Oishi, Verified eigenvalue evaluation for the Lapla-cian over polygonal domains of arbitrary shape, SIAM Journal onNumerical Analysis, 51.3 (2013), pp. 1634-1654.

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Interval computations in the metrology

Semenov K.K.1, Solopchenko G.N.1 and Kreinovich V.Ya.2

1 Peter the Great St. Petersburg Polytechnic University29, Polytechnicheskaya str., St. Petersburg, 195251, Russia

[email protected],

[email protected] University of Texas at El Paso

500 W. University, El Paso, TX 79968, [email protected]

Keywords: interval computations, metrology, heat measuring

The modern informational world is antagonistic to the absolute pre-cision [1]. The using of the concept ’the absolute accuracy’ is possibleonly for mental theoretical constructs, but it isn’t possible for the real-life applications because we need to take into account the used datauncertainty. The human’s reasoning in the engineering practice andthe decision-making in the automatic control systems are based on themeasurements results that are distorted by the errors and that are of-ten used as the initial data. Moreover, in practice, we operate with theinaccurate results of the uncertain data mathematical processing, soour decisions should take this circumstance into account, otherwise wemay be wrong in our conclusions. We often obtain the empirical dataabout object under observation in the interval form from the directmeasurements results, the metrological assurance and the accuracy ofwhich are provided by the metrology. When this derived uncertain in-formation is mathematically processed, the accuracy estimation of itsresults is also assigned to the metrology. The solution of this problemfor the big amount of heterogeneous initial data is difficult becauseof the big number of their possible values combinations. However,at present time, this problem turns out to be solvable in the area ofmetrology of computer programs because of using the theory of fuzzysets and the interval calculus.

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The interval age in computational techniques began in the 60-iesof the XX century with the IFIP’s (that was headed by Wilkinson)propositions on the accompanying the data processing software withthe interval-valued estimation of the results inaccuracy. In 1962, Mooresystematically considered the interval arithmetic [2] and offered thefirst practical manuals for its applications. In the 70-ies of the XXcentury, the first software shells (Linpack, for example) and librariesappeared for the linear computations. They allowed users to get thefinal results as the intervals of their possible values.

L. Zadeh expanded the scope of the term ’uncertainty’ from onlyinstrumental causes (rounding errors) to the initial data inaccuracyand the expert a priori information by introducing the fuzzy variables.To represent the measurement results and a priori information aboutthem and their uncertainty as the intervals, L. Reznik introduced thefuzzy intervals [3]. V. Kreinovich and H. T. Nguyen considered thenested intervals as the measure of the fuzzy variables uncertainty [4].This development trend expressed the real-life applications’ necessityto operate with the inaccurate initial data.

The expanding using of the computational techniques in the mea-surement systems and the data processing performed by them causedthe problem of the metrological software tests organization. This cir-cumstance induced the metrological community to take into accountthe success of the interval computations and the fuzzy variables andto include them into the metrological practice. The first step in thisdirection was made in 1985 with the special issue of the magazine”Measurement Techniques” that was dedicated to the problem of themetrological support of the data processing in the metrology and themeasuring.

To date, the different interval approaches and methods were pro-posed and used in the various metrological applications, particularlythe method [5,6] that corresponds with the requirements of the metro-logical norms and standards.

To illustrate how deeply the interval computations and their expan-sions can be spread in the real life, we consider the example from oureveryday practice: the calculations in the systems of heat metering.

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We cannot measure the consumed heat quantity directly, so its valueis calculated from the indirect measurement results by the computa-tional block of the meter (that is called calculator). The initial datafor these calculations are received from two sensors signals of the heatconveying liquid amount in the forward and return directions of theflow, two sensors signals of the temperature in the forward and returnflows of heat-transfer agent, the signal of the temperature differencebetween these two flows and the sensor signal of the pressure of the liq-uid. The instrumental inaccuracy of all used sensors is standardized bythe special European normative document [7]. The uncertainty of themeasured heat amount value results to the inaccuracy in determininghow much the heat was really consumed and how much the consumershould pay.

For the heat calculator, only the uncertainty of temperature per-ception is normalized, but not of any other signals mentioned above.This circumstance draws some criticism. That’s why the real commer-cial task is to estimate uncertainty of the heat calculator results withtaking into consideration the uncertainties of all initial data, not onlythe temperatures difference. In full, this problem can be solved onlyusing the interval computations tools in the sense of the works [5,6].In the report, the particular examples are presented for the intervalcomputations that provide the reliable intervals of the uncertainty ofthe consumed and paid heat.

References:

[1] V. Knorring, History and mthodoloy of science and technics: in-formarmational scope of human’s agency from from aincient timesto the beginning of XVI cent., St.Petersburg State Polytechnic Uni-versity, St.Petersburg, 2013 (in Russian).

[2] R. Moore, Interval arithmetic and automatic error analysis indigital computing, a thesis submitted for the degree of Doctor ofPhilosophy, Stanford University, 1962.

[3] L. Reznik, Mathematical toolkit for processing fuzzy informa-tion of experimenter in data processing systems, Proceedings of

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All-Russian Scientific Research Institute on Electrical Instrument-Making, 1983, pp. 45–55 (in Russian).

[4] H. Nguyen, V. Kreinovich, Nested intervals and sets: concepts,relations to fuzzy sets, and applications, Applications of intervalcomputations, 1996, pp. 245–290.

[5] K. Semenov, G. Solopchenko, Combined method of metrolog-ical self-tracking of measurement data processing programs, Mea-surement Techniques, 54 (2011), No. 4, pp. 378–386.

[6] K. Semenov, L. Reznik, G. Solopchenko, Fuzzy IntervalsApplication for Software Metrological Certification in Measure-ment and Information Systems, Int. Journal of Uncertainty, Fuzzi-ness and Knowledge-Based Systems, 23 (2015), Suppl. 1, pp. 95–104.

[7] European Standard prEN 1434, Heat meters, In 6 parts. Euro-pean Committee for Standardization, Brussels, 2006.

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Verified numerical computations forblow-up solutions of ODEs

Akitoshi Takayasu1, Kaname Matsue2, Takiko Sasaki3,Kazuaki Tanaka3, Makoto Mizuguchi3 and Shin’ichi Oishi3

1University of Tsukuba,1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan

2The Institute of Statistical Mathematics3Waseda University

[email protected]

Keywords: ordinary differential equations, blow-up solutions, com-pactifications, Lyapunov functions, verified numerical computations

This talk is concerned with blow-up solutions of ordinary differentialequations (ODEs) in Rm (m ∈ N):

dy(t)

dt= f (y(t)) , y(0) = y0, (1)

where t ∈ [0, T ) with 0 < T ≤ ∞, f : Rm → Rm is a C1 function andy0 ∈ Rm. Unless otherwise noted, f is assumed to be a polynomial,whose coefficients are real numbers. The blow-up solutions are a classof solutions of (1).

Definition. Define tmax > 0 as

tmax := supt : a solution y ∈ C1([0, t)) of (1) exists

.

We say that the solution y of (1) blows up if tmax <∞. In such a casetmax is called the blow-up time of (1).

We provide a method for verifying blow-up solutions with theirblow-up times on the basis of compactifications and verification of theLyapunov function.

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Blow-up solutions are not compatible with numerical computationswithout any ideas, since their norm goes to infinity in finite time.To overcome this difficulty, we analyze blow-up solutions with twoinstruments. The first one is compactification of base spaces and ofdynamical systems on them, following Poincare ([1] see also [2,3]). Thesecond one is verification of Lyapunov functions. The monotonicity ofthe Lyapunov function enables us to derive a re-parameterization oftrajectories, which is called Lyapunov tracing proposed in [4]. This re-parameterization yields explicit estimates of blow-up times. The abovetwo instruments with standard methodologies of numerical verificationmethods [5,6] and dynamical systems lead to verification of blow-upsolutions with their blow-up times.

References:

[1] H. Poincare, Memoire sur les Courbes Definies par une EquationDifferentielle, Oeuvres., 1881.

[2] U. Elias, H. Gingold, Critical points at infinity and blow up ofsolutions of autonomous polynomial differential systems via com-pactification, J. Math. Anal. Appl., 318 (2006), pp. 305–322.

[3] J. Hell, Conley index at infinity, Ph.D. Thesis in Freie UniversitatBerlin, 2010.

[4] K. Matsue, T. Hiwaki, and N. Yamamoto, On the con-struction of Lyapunov functions with computer assistance, arXivpreprint arXiv:1604.05953, 2016.

[5] K. Kashiwagi, M. Kashiwagi, Numerical Verification of ordi-nary differential equations using Affine Arithmetic and Mean ValueForm, Transactions of the Japan Society for Industrial and AppliedMathematics, 21 (2011), No. 1, pp. 37–58 (in Japanese).

[6] M. Kashiwagi, kv - C++ Numerical Verification Libraries.http://verifiedby.me/kv/

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On verification methods for parabolicpartial differential equations using the

evolution operator

Akitoshi Takayasu1, Makoto Mizuguchi2, Takayuki Kubo1

and Shin’ichi Oishi2

1University of Tsukuba,1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan

2Waseda [email protected]

Keywords: evolution operator, parabolic PDEs, wrapping effect

In this talk, we provide verification methods for an initial-boundaryvalue problem of parabolic partial differential equations:

∂tu(t, x)−∆u(t, x) = f(u(t, x)) 0 < t ≤ T, x ∈ Ω,u(t, x) = 0 0 < t ≤ T, x ∈ ∂Ω,u(0, x) = u0(x) x ∈ Ω,

(1)

where T > 0, Ω = (0, 1)d ⊂ Rd (d = 1, 2, 3), u0 ∈ L2(Ω), and the map fsatisfies suitable assumptions. Such a method is based on the evolutionoperator U(t, s), which is the solution operator of the homogeneousinitial value problem:

∂tu+ A(t)u = 0, 0 ≤ s < t ≤ T,u(s) = φ,

(2)

where A(t)0≤t≤T is a certain family of unbounded operators andφ ∈ L2(Ω). The evolution operator gives the formula u(t) = U(t, s)φfor representing a solution of (2). From the classical theory of theevolution operator [1, 2, 3, 4], the criteria of generating the evolutionoperator are that A(t) for each t ∈ (s, T ] is sectorial and the familyA(t) is Holder continous with respect to t.

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In the talk, we introduce the precise assumptions for providingthe verification method and give several estimates associated with theevolution operator. By using the evolution operator, an fixed-pointformulation is obtained in order to prove the existence of the midsolution of (1). A sufficient condition for a solution to be enclosedwithin a neighborhood of a numerical solution is also derived fromBanach’s fixed point theorem.

Furthermore, if the sufficient condition is satisfied, another fixed-point formulation is used for estimating the error at a fixed time be-tween the enclosed mid solution and the numerical solution. This for-mulation is based on the generalized semigroup property of the evolu-tion operator:

U(t, s) = U(t, r)U(r, s), 0 ≤ s ≤ r ≤ t ≤ T,U(s, s) = I, 0 ≤ s ≤ T.

Consequently, this formulation is similar to the techniques for reducingthe wrapping effect when solving ordinary differential equations byinterval methods.

References

[1] H. Tanabe, On the equations of evolution in a Banach space, Os-aka Mathematical Journal, 12:2 (1960), pp. 363–376.

[2] P. E. Sobolevskii, On equations of parabolic type in Banach spacewith unbounded variable operator having a constant domain, Akad.Nauk Azerbaidzan. SSR Doki, 17:6 (1961) (in Russian).

[3] A. Pazy, Semigroups of linear operators and applications to partialdifferential equations, Springer, New York, 1983.

[4] A. Yagi, Abstract Parabolic Evolution Equations and their Appli-cations, Springer-Verlag, Berlin Heidelberg, 2010.

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On verified numerical computation forpositive solutions to elliptic boundary

value problems

Kazuaki Tanaka1, Kouta Sekine1, 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.

[email protected]

Keywords: elliptic equation, positive solution, verified numerical com-putation

We are concerned with verified numerical computations for solu-tions to the following elliptic problem:

−Lu = f (u) (1a)

and

u > 0 in Ω (1b)

with an appropriate boundary condition. Here, Ω is a bounded domain(i.e., an open connected bounded set) in Rn (n = 1, 2, 3, · · · ), f isa given nonlinear operator from V (an appropriate solution space)to L2 (Ω), and L is a uniformly elliptic self-adjoint operator from itsdomain D(L) to L2 (Ω) (the domain D(L) depends on the smoothnessof the boundary ∂Ω).

Verified numerical computation methods for elliptic equations havebeen developed by many researchers (see, e.g., [1,2,3]). These methodsenable us to numerically obtain a concrete ball containing a solutionto target equations, typically in the sense of the norms ‖∇·‖L2(Ω) and

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‖·‖L∞(Ω). No matter how small the radius of the ball is, however, somesolutions have the possibility not to be positive. For example, in thehomogeneous Dirichlet case, it is possible for a verified solution not tobe positive near the boundary ∂Ω.

To verify solutions to (1), we propose two numerical methods forproving the positiveness of a function u satisfying (1a). One is anextended result of [4, Theorem 2], which provides a sufficient conditionfor the positiveness on the basis of the strong maximum principle.The other method computes the upper bound of the number of thenodal domains of a verified solution u∗ to (1a), by regarding u∗ asan eigenfunction of a self-adjoint elliptic operator; if the number is atmost one, we can ensure the positiveness of u∗.

Numerical examples for some concrete elliptic boundary value prob-lems will be presented.

References:

[1] M.T. Nakao, Numerical verification methods for solutions of ordi-nary and partial differential equations, Numerical Functional Anal-ysis and Optimization, 1 (2001), pp. 321–356.

[2] M. Plum, Computer-assisted enclosure methods for elliptic differ-ential equations, Linear Algebra and its Applications, 324 (2001),pp. 147–187.

[3] A. Takayasu, X. Liu, and S. Oishi, Remarks on computable apriori error estimates for finite element solutions of elliptic prob-lems, Nonlinear Theory and Its Applications, IEICE 5 (2014),pp. 53–63.

[4] K. Tanaka, K. Sekine, M. Mizuguchi, and S. Oishi, Numer-ical verification of positiveness for solutions to semilinear ellipticproblems, JSIAM Letters 7 (2015), pp. 73–76.

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Error Analysis of Lagrange Interpolationon Tetrahedrons

Kenta Kobayashi Takuya Tsuchiya

Hitotsubashi University Ehime UniversityKunitachi, 186-8601 Japan Matsuyama, 790-8577 Japan

[email protected]

Keywords: Lagrange interpolation, tetrahedrons, generalized circum-radius, finite elements

Let K ⊂ R3 be a tetrahedron with vertices xi, i = 1, · · · , 4. Letλi be its barycentric coordinates with respect to xi. By definition,we have 0 ≤ λi ≤ 1,

∑4i=1 λi = 1. Let N0 be the set of nonnegative

integers, and γ = (a1, · · · , a4) ∈ N40 be a multi-index. Let k be a

positive integer. If |γ| :=∑4

i=1 ai = k, then γ/k := (a1/k, · · · , a4/k)can be regarded as a barycentric coordinate in K. The set Σk(K) ofpoints on K is defined by

Σk(K) :=γk∈ K

∣∣∣ |γ| = k, γ ∈ N40

.

For k, Pk is the set of all polynomials of three variables whose degreeis at most k. For a continuous function v ∈ C0(K), the Lagrangeinterpolation IkKv ∈ Pk of degree k is defined as

v(x) = (IkKv)(x), ∀x ∈ Σk(K).

In this talk We present an error estimation of |v − IkKv|m,p,K in termsof hK := diamK and the generalized circumradius GK of K, which isdefined in the following.

Let B be a facet of K and hB := diamB. Let RB be the circumra-dius of B. Consider a projection δθ from R3 into a 2-dimensional affinelinear space such that δθ(B) is a segment. Then, δθ(K) is always a tri-angle on the plane. Let RP be the maximum value of the circumradiusof δθ(K). We define the generalized circumradius GK by

GK := minB

RBRP

hB,

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where the minimum is taken for all facets of K. The following is themain theorem.

Theorem 1 Let K be an arbitrary tetrahedron. Let hK := diamKand GK be the generalized circumradius of K. Let 1 ≤ p ≤ ∞ andk, m be integers with k ≥ 1 and 0 ≤ m ≤ k. Then, there existsa constant C = C(k,m, p) independent of K such that, for arbitraryv ∈ W k+1,p(K),

|v − IkKv|m,p,K ≤ CGmKh

k+1−2mK |v|k+1,p,K

= C

(GK

hK

)mhk+1−mK |v|k+1,p,K .

Note that no geometric condition is imposed in the main theorem.

References:

[1] K. Kobayashi, T. Tsuchiya, A Babuska-Aziz type proof of thecircumradius condition, Japan J. Indust. Appl. Math., 31 (2014),193-210.

[2] K. Kobayashi, T. Tsuchiya, On the circumradius condition forpiecewise linear triangular elements, Japan J. Indust. Appl. Math.,32 (2015) 65–76.

[3] K. Kobayashi, T. Tsuchiya, A priori error estimates for La-grange interpolation on triangles. Appl. Math., Praha, 60 (2015),485–499.

[4] K. Kobayashi, T. Tsuchiya, Extending Babuska-Aziz theoremto higher order Lagrange interpolation. Appl. Math., Praha, 61(2016), 121–133.

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An implicit algorithm for validatedenclosures of the solutions to variational

equations for ODEs.

Irmina Walawska and Daniel Wilczak

Institute Of Computer Science and Computational Mathematics,Jagiellonian University

Lojasiewicza 630-348 Krakow

[email protected]

Keywords: validated numerics, variational equations, initial valueproblem

We propose a new algorithm for computing validated bounds for thesolutions to the first order variational equations associated to ODEs:

x(t) = f(x(t)),

V (t) = Df(x(t)) · V (t),x(0) ∈ [x0],V (0) ∈ [V0],

(1)

where f : Rn → Rn is a smooth function and [x0] ⊂ Rn, [V0] ⊂ Rn2

aresets of initial conditions. Solutions the variational equation V (t) giveus an information about sensitivities of trajectories with respect to ini-tial conditions. They proved to be useful in finding periodic solutions,proving their existence and analysis of their stability. They are usedto estimate invariant manifolds of periodic orbits. Derivatives withrespect to initial conditions are used to prove the existence of connect-ing orbits. The method that we propose for computation of validatedsolutions to (1), uses a high-order Taylor method as a predictor stepand an implicit method based on Hermite-Obreshkov interpolation asa corrector step. The proposed algorithm is an improvement of theC1-Lohner algorithm proposed by Zgliczynski and it provides sharper

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bounds. Our algorithm, cannot produce worse estimations than theC1-Lohner algorithm. Complexity analysis shows that, in low dimen-sions, it is slower than the C1-Lohner algorithm by the factor 9/8 only.This lack of performance is compensated by a significantly smallertruncation error of the method. This allows to take larger time stepswhen computing the trajectories and thus our algorithm appears to beslightly faster than the C1-Lohner in real applications.

References:

[1] Alefeld, G., Inclusion methods for systems of nonlinear equations—the interval Newton method and modifications. Topics in validatedcomputations (Oldenburg, 1993). Vol. 5 of Stud. Comput. Math.North-Holland, Amsterdam, pp. 7–26.

[2] Zgliczynski, P., C1-Lohner algorithm, Foundations of Computa-tional Mathematics, (2002), 2 (4), 429–465.

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The properties of negation and zero inringoids as defined by Kulisch

Ronald van Nooijen and Alla Kolechkina

Delft University of [email protected]

Keywords: ringoid, algebra, computer arithmetic

In [1,2] Kulisch defines (ordered) ringoids and vectoids to providea theoretical basis for computer arithmetic and interval arithmetic.One interesting aspect of his treatment is the search for necessary andsufficient conditions for a meaningful notion of negation and zero. Inthis paper we consider this both from the point of view of functionson the underlying set and from a category theoretical standpoint. Itturns out that the conditions provided by Kulisch can be restated inother forms, but that the original form is probably both necessary andsufficient for the intended purpose.

References:

[1] U.W. Kulisch and W.L. Miranker, The arithmetic of thedigital computer: a new approach, SIAM Rev., 28 (1986), No 1,pp. 1–40.

[2] U.W. Kulisch, Computer arithmetic and validity: Theory, imple-mentation, and applications, de Gruyter, Berlin, 2nd revised andextended edition, 2013.

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Validated constructive error estimatationsfor bi-harmonic problems

Yoshitaka Watanabe, Takehiko Kinoshita and Mitsuhiro T.Nakao

Kyushu University6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8518, Japan

[email protected]

Keywords: PDEs, bi-harmonic problem, error estimation

This talk presents some constructive error estimates for the approx-imate solutions of two-dimensional bi-harmonic equations:

∆2u = f in Ω,

u =∂u

∂n= 0 in ∂Ω

(1)

by using verified computational techniques. Here Ω is a bounded con-vex polygonal domain, f ∈ L2(Ω), and ∂u/∂n stands for the outernormal derivative of u. The error estimations for (1) are expected toprovide invaluable information for computer-assisted proofs of nonlin-ear bi-harmonic problems [1]. Several numerical examples based onLegendre polynomials [2] confirm the effectiveness of our proposed ap-proach.

References:

[1] K. Nagatou, K. Hashimoto, M.T. Nakao, Numerical verifi-cation of stationary solutions for Navier-Stokes problems, Journalof Computational and Applied Mathematics, 199 (2007), pp. 445–451.

[2] T. Kinoshita, M.T. Nakao, On very accurate enclosure of theoptimal constant in the a priori error estimates for H2

0 -projection,Journal of Computational and Applied Mathematics, 234 (2010),pp. 526–537.

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Computing the Worst-Case Peak Gain ofDigital Filter in Interval Arithmetic

Anastasia Volkova, Christoph Lauter and Thibault Hilaire

Sorbonne Universites, UPMC Univ Paris 06, UMR 7606, LIP64, place Jussieu 75005 Paris, Francefirst name.last [email protected]

Keywords:digital filters, interval arithmetic, worst-case peak gain

The Worst-Case Peak Gain (WCPG) of a Linear Time Invariant(LTI) filter is used to determine the output interval of a filter and inerror propagation analysis [5].

Consider a stable LTI filter H in state-space representation:

H

x(k + 1) = Ax(k) + Bu(k)y(k) = Cx(k) + Du(k)

(1)

where u(k) is the input vector, y(k) is the output vector, x(k) is thestate vector and matrices A, B, C, D contain the filter coefficients.

The WCPG of a linear filter can be computed [1] as the infinitesum W := |D| +∑∞k=0

∣∣CAkB∣∣ . In [6] the authors have proposed an

algorithm for the reliable evaluation of the WCPG matrix in multipleprecision.

However, usually the filter coefficients are rounded prior to imple-mentation, changing A, B, C and D by rounding. To provide a re-liable filter implementation, these rounding errors must be taken intoaccount in the WCPG computation. We represent the rounded coef-ficients as interval [2] matrices with small radii. Let MI := 〈Mc,Mr〉to be an interval matrix centered in Mc with radius Mr. Then,the WCPG matrix of a filter H =

(AI ,BI ,CI ,DI

)is an interval

WI :=∣∣DI

∣∣+∑∞

k=0

∣∣∣CIAIkBI∣∣∣.

In this work we adapt the algorithm presented in [6] to obtain a reli-able evaluation of the WCPG interval. The WCPG is computed in twostages: the reliable truncation of the infinite sum and the summation.

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We determine the truncation order only for the center matricesbut add a correction term after the final step. This step requires toperform an eigenvalue decomposition. To obtain trusted error boundson the computed eigenvalues we use the Theory of Verified Inclusionsdeveloped by S. Rump [4].

The summation is done using Interval Arithmetic in midpoint-radius form. However, powering a dense interval matrix can lead to aninterval explosion. Instead of powering AI we power an almost diag-onal matrix TI , for which ||TI ||2 < 1 is true. We use an analogue ofGershgorin circle theorem [3] to verify a spectral norm condition thatneeds to be satisfied for the WCPG sum to converge.

It is obvious that we cannot guarantee an a priori given bound onthe WCPG matrix radius Wr because the radii of the input matricesare the limiting factors. However, when given point coefficient matrices(intervals with zero radii) and an absolute error bound ε we guaranteethat the output WCPG interval in not larger than ε in width.

References[1] V. Balakrishnan and S. Boyd. On computing the worst-case peak gain of linear

systems. Systems & Control Letters, 19:265–269, 1992.

[2] H. Dawood. Theories of Interval Arithmetic: Mathematical Foundations andApplications. LAP Lambert Academic Publishing, 2011.

[3] S. Gershgorin. Uber die Abgrenzung der Eigenwerte einer Matrix. Bull. Acad.Sci. URSS, 1931(6):749–754, 1931.

[4] S. M. Rump. New results on verified inclusions. In Accurate Scientific Computa-tions, Symposium, Proceedings, 1985.

[5] A. Volkova, T. Hilaire, and C. Lauter. Determining fixed-point formats for adigital filter implementation using the worst-case peak gain measure. In 201549th Asilomar Conference on Signals, Systems and Computers, Nov 2015.

[6] A. Volkova, T. Hilaire, and C. Lauter. Reliable evaluation of the worst-case peakgain matrix in multiple precision. In Computer Arithmetic (ARITH), 2015 IEEE22nd Symposium on, pages 96–103, June 2015.

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Computer-assisted existence proofs forone-dimensional Schrodinger-Poisson

systems

Jonathan Wunderlich

Karlsruhe Institute of TechnologyDepartment of Mathematics

Englerstraße 2, 76131 Karlsruhe, [email protected]

Keywords: Computer-assisted proof, Schrodinger-Poisson, existence,enclosure

We are aiming at non-trivial solutions of the one-dimensional time-independent Schrodinger-Poisson system

−u′′ + V u+ Φuu = f(u)

−Φ′′u + cΦu = u2

on R,

limx→±∞

Φu = 0

where c > 0 is an additional parameter needed in the one-dimensionalcase, V ∈ L∞(R) is a positve potential and f ∈ C1(R).

This one-dimensional Schrodinger-Poisson system is a simplifiedmodel for the three-dimensional time-dependent version

−i~∂tψ −~2

2m∆ψ + qeWeψ = f(u)

−ε∆We = qe|ψ|2

on [0,∞)× R3

lim|x|→∞

We = 0

which (for f ≡ 0) plays an important role in today’s semiconductortechnology. ψ represents the wavefunction of a particle, in our caseof an electron, and m is its mass. We describes the electric potential

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which depends on the wavefunction ψ by the above Poisson equationwith Dirichlet boundary conditions.

If the nonlinearity satisfies the covariance property

f(eiϕz) = eiϕf(z) (z ∈ C, ϕ ∈ R),

we can eliminate the time dependency by the standing wave ansatz

ψ(x, t) = u(x)eiωt (t ∈ [0,∞), x ∈ R)

with a parameter ω > 0.To prove non-trivial solutions of the one-dimensional Schrodinger-

Poisson system we first “solve” the second equation using the corre-sponding Green’s function Γ, and insert the result into the first one:

−u′′ +(V +

∫ ∞

−∞Γ(·, t)u(t)2 dt

)u = f(u) on R.

Furthermore u should be a solution with finite energy level for physicalreasons, i.e. we look for a solution u ∈ H1(R).

Applying computer-assistance to the above equation, we are ableto prove the existence of a non-trivial solution of the one-dimensionalSchrodinger-Poisson system for the case c = 50, constant potentialV ≡ 1, and the nonlinearity f chosen as f(y) = y3 (y ∈ R).

Starting from a numerical approximate solution, we compute abound for its defect, and a norm bound for the inverse of the lineariza-tion at the approximate solution. For the latter, eigenvalue boundsplay a crucial role, especially the eigenvalues “close to” zero. There-for we use the Rayleigh-Ritz method and a corollary of the Temple-Lehmann theorem to get enclosures of the eigenvalues of the lineariza-tion below the essential spectrum.

With these data in hand, we can use a fixed-point argument to ob-tain the desired existence of a non-trivial solution “nearby” the approx-imate one. In addition to the pure existence result, the used methodsalso provide an enclosure of the exact solution.

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Numerical verification of existence ofhomoclinic orbits in dynamical systems

Nobito Yamamoto1, Kaname Matsue2 and Shun Yamano1

1The University of Electro-Communications,Chofu-ga-oka, Chofu, 182-8585, Tokyo, Japan

2The Institute of Statistical Mathematics,10-3, Midori-Cho, Tachikawa, Tokyo, 190-8562, Japan

[email protected]

Keywords: verified numerics, dynamical systems, ODEs, homoclinicorbits

We propose numerical verification methods to prove existence ofhomoclinic orbits in dynamical systems described by ODEs. We treatan example problem with a saddle equilibrium in R3 which has 1-dimensional stable manifold and 2-dimensional unstable manifold.

Whereas a previous research [1] adopts time-reverse computationfor homoclinic orbits of hyperbolic equilibria with 1-dimensional un-stable manifolds, it is so hard for us to follow the flows (time dependentsolutions of ODEs) by time-reverse computation because of numericalinstability of our problem. This means that we cannot use naive meth-ods, e.g. intermediate value theorem, in order to capture homoclinicorbits of hyperbolic equilibria with 2-dimensional unstable manifolds.

Our methods are based on theories of mapping degree as well asLyapunov functions constructed explicitly by verified computation.Brouwer coincidence theorem, which leads to Brouwer fixed point the-orem, is used for specifying parameters to have a homoclinic orbit.

Consider an autonomous system of ODEs :

d

dtx = f(x; p), x, f ∈ Rn, (1)

where p ∈ R2 is a pair of parameters p = (a, b). We suppose thatthe system (1) may have a hyperbolic equilibrium with 2-dimensional

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unstable manifold admitting a homoclinic orbit, and our aim is to provethe existence of the homoclinic orbit and to specify a narrow area whichcontains the parameters attaining the existence of the homoclinic orbit.

Computing an approximate flow to be homoclinic-like, we constructa mapping from a small rectangular domain of the parameters (a, b) toa plane including a family of equilibria along with flows starting fromspecified points on the unstable manifolds in a neighborhood of theequilibria with respect to p = (a, b). The starting points are chosenfrom a small area where the approximate flow runs through, and theflows are expected to come back to a neighborhood of the equilibriawhich contains the stable manifolds. Lyapunov functions are com-posed by verified computation in order to specify the starting pointson the unstable manifolds and the neighborhood of the stable mani-folds, where we use our previous works [2]. The degree of this mappingis estimated by our new theorem so called Interval Simplex Theorem,which counts the degree using interval arithmetic and verified compu-tation. Finally, we apply Brouwer coincidence theorem to proving thatthere is an equilibrium as an image of the mapping, which means thatthe rectangular domain of p contains a pair of parameters that admitsa homoclinic orbit.

References:

[1] D. Wilczak, The Existence of Shilnikov Homoclinic Orbits intheMichelson System: A Computer Assisted Proof, Foundationsof Computational Mathematics, 6(4), 2006, pp. 495–535.

[2] K. Matsue, T. Hiwaki and N. Yamamoto, On the construc-tion of Lyapunov functions with computer assistance,, arXiv preprint,arXiv:1604.05953

[3] S. Yamano, Numerical verification methods for homoclinic or-bits in continuous dynamical systems, Master’s thesis for graduateschool of The University of Electro-Communications, in Japanese,2016.

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Verification method for system of linearequations by QR factorization

Yuka Yanagisawa†, Shin’ichi Oishi‡ and Fumi Noda‡1

† Research Institute for Science and Engineering, Waseda University‡ Department of Applied Mathematics, Faculty of Science and

Engineering, Waseda University3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

[email protected]

Keywords: linear equation, QR factorization, accurate numerical al-gorithm

We are concerned with the matrix equation Ax = b where A is ann×n real matrix and x and b be n-vectors. If A = QR is a QR factor-ization, then we can write QRx = b where Q is n×n with orthonormalcolumns and R is n × n and upper triangular. This equation is easysolve because R is triangular. The principal method for computingQR factorization is Householder triangularization, which is excellentnumerical stability. Nevertheless, QR factorization is not the standardmethod for computing the approximate solution to Ax = b in prac-tice, since it requires two times computational cost of LU factorizationwith partial pivoting, which is unstable for matrices with large growthfactors2 [1].

Assume that an approximate solution x is given with an approxi-mate QR factorization. In this talk, we will present an accurate andfast method using QR factorization for proving nonsingularity of Aand for calculating rigorous error bounds for ‖A−1b − x‖2. Specifi-cally, we focus a method of calculating an upper bound of ‖I − BA‖2where B is an approximate inverse of A and I is the n × n iden-tity matrix. It is well-know that If ‖I − BA‖2 is satisfied, then A is

proved to be nonsingular and ‖A−1b− x‖2 ≤ ‖B(Ax−b)‖21−‖I−BA‖2 . In a previous

1NTT Communications Corporation2In practice, such matrices are very rare in applications.

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work, Oishi-Rump proposed a method utilizing an approximate LUfactors [2]. This method satisfies ‖I−BA‖∞ ≈ nu ·κ2(A) where κ2(A)is the 2-norm condition number of A and u is the unit roundoff. Itrequires 16

3 n3 flops (43n

3 flops for calculating an approximate inverse Busing the LU factors and 4n3 flops for calculating the upper bound of|I−BA| ). Our method based on the Yomoda’s approach [3] utilize anapproximate QR factors. Our method which requires 17

3 n3 flops (43n

3

flops for calculating the QR factors Q, R such that A ≈ QR, 13n

3 flopsfor calculating an approximate inverse of R, 2n3 flops for calculatingthe upper bound of |I −Q>Q| and 2n3 flops for calculating the upperbound of |Q>−XA| ) gives more accurate result than the Oishi-Rumpmethod [2]. Moreover, our algorithm can treat the case where A isill-conditioned such that κ2(A) ≈ u−1, if we use an accurate dot prod-uct algorithm [4]. We also present detailed analysis of our method andsome numerical results.

References:

[1] Lloyd N. Trefethen, David Bau III, Numerical Linear alge-bra, SIAM, Philadelphia, 1997.

[2] S. Oishi and S. M. Rump, Fast Verification of Solutions of matrixequations, Numer. Math, 90-4 (2002), pp. 755–773.

[3] E. Yomoda, Studies on the numerical verification of regularityof matrices, Unpublished thesis for master degree, Tokyo Woman’sChristian University, (2012), (in Japanese).

[4] K. Ozaki, T. Ogita, S. Oishi, Tight and Efficient Enclosureof Matrix Multiplication by Using Optimized BLAS, NumericalLinear Algebra with Applications, 18-2 (2011), pp. 237–248.

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An Accurate and Efficient Solution ofIll-conditioned Linear Systems by

Preconditioning Methods

Yuka Kobayashi1,∗, Takeshi Ogita1 and Katsuhisa Ozaki2

1Tokyo Woman’s Christian University167–8585 Tokyo, Japan

2Shibaura Institute of Technology337–8570 Saitama, Japan∗ [email protected]

Keywords: accurate numerical algorithm, solution of linear systems,preconditioning technique

We are concerned with an accurate numerical solution of linearsystems

Ax = b, A ∈ Rn×n, b ∈ Rn (1)

by using floating-point arithmetic. There are two standard meth-ods using an LU factorization to solve (1) accurately. One of thestandard methods is using an LU factorization with the iterative re-finement method. If the problem is not ill-conditioned, it can workwell. However, this method is not effective for ill-conditioned problems.The other method is using an LU factorization with multiple-precisionarithmetic [1, 2]. Although it can work for ill-conditioned problems,it takes significant computing time regardless of the condition numberof A. To remedy the defects of the standard methods, we propose analgorithm based on the preconditioning method [3] using a result of anLU factorization by floating-point arithmetic. The proposed algorithmcan provide accurate numerical solutions for ill-conditioned problemsbeyond the limit of the working precision. Moreover, it requires lesscomputational cost than the previous preconditioning method [4, 5]using an approximate inverse of A as a preconditioner. We conductednumerical experiments using the proposed algorithm on MATLAB. Forn = 10000, we can obtain approximate solutions of Ax = b with the

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condition number up to 1030. If we use accurate computations usingBLAS [6], it takes only 7 ∼ 9.5 times as long as an LU factoriza-tion. Results of numerical experiments are presented for confirmingthe effectiveness of the proposed algorithm.

References:

[1] The GNU MPFR Library, http://www.mpfr.org

[2] The GNU Multiple Precision Arithmetic Library, https://gmplib.org

[3] T. Ogita, Accurate matrix factorization: inverse LU and inverseQR factorizations, SIAM J. Matrix Anal. Appl., 31:5 (2010), pp. 2477–2497.

[4] S. M. Rump, Inversion of extremely ill-conditioned matrices infloating-point, Japan J. Indust. Appl. Math., 26 (2009), pp. 249–277.

[5] S. M. Rump, Accurate solution of dense linear systems, part I:Algorithms in rounding to nearest, J. Comp. Appl. Math., 242(2013), pp. 157–184.

[6] K. Ozaki, T. Ogita, S. Oishi, S. M. Rump, Error-free transforma-tions of matrix multiplication by using fast routines of matrix mul-tiplication and its applications, Numerical Algorithms, 59:1 (2012),pp. 95–118.

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Author index

Afravi Mahdokht, 73Alseda i Soler Lluıs, 122Aschemann Harald, 110, 112Awala Hussein, 37

Banhelyi Balazs, 21Bagoczki Zsolt , 21Baker Kearfott Ralph, 69Barragan Pedro, 38Benet Luis, 123Bouwmeester Henricus, 31Brehard Florent, 40Brisebarre Nicolas, 40

Capinski iMaciej, 23Ceberio Martine, 24Chapoutot Alexandre, 42Costa Tiago M., 31Csendes Tibor, 21

Defour David, 59Desrochers Benoit, 44Domes Ferenc, 92Du Peibing, 65

Fevotte Francois, 46Franek Peter, 48, 106Fevotte Francois, 97

Gaidashev Denis, 50Garloff Jurgen, 51Graillat Stef, 59, 63, 97

Grodet Aymeric, 53

Hilaire Thibault, 143Hladık Milan, 55Husken Matthias, 57

Iakymchuk Roman, 59

Jaulin Luc, 44, 61, 116Jeangoudoux Clothilde, 63Jeannerod Claude-Pierre, 98Jiang Hao, 65Joldes Mioara, 26, 40, 100Jezekiel Fabienne, 97

Kashiwagi Masahide, 67, 90Kimura Takuma, 71, 120Kinoshita Takehiko, 142Klenke Cornelia, 108Kobayashi Kenta, 137Kobayashi Ryo, 120Kobayashi Yuka, 151Kokubu Hiroshi, 28Kolechkina Alla, 141Krcal Marek, 48Kreinovich V.Ya., 127Kreinovich Vladik, 38, 73Kryukov Nikolay, 123Kubica Bart lomiej, 75Kubo Takayuki, 133Kur atko Jan, 106

Lathuiliere Bruno, 46

153

SCAN 2016

Lathuiliere Bruno, 97Lauter Christoph, 63, 143Lavor Carlile, 31Le Bars Fabrice, 116Lessard Jean-Philippe, 30Li Kuan, 65List Ivo, 77LIU Xuefeng, 79Lodwick Weldon, 31Lodwick Weldon A., 81Louvet Nicolas, 98

Markot Mihaly, 92Matsuda Nozomu, 82Matsue Kaname, 84, 131, 147McKenna P. Joseph, 33Mihaylova Lyudmila, 116Minamoto Teruya, 71Miyajima Shinya, 86, 88Mizuguchi Makoto, 131, 133Montanher Tiago, 92Morikura Yusuke, 90Mukkula Gottu, 95Muller Jean-Michel, 98, 100

Nagatou Kaori, 33Nakao Mitsuhiro T., 71, 142Nataraj P.S.V., 104Neumaier Arnold, 92Noda Fumi, 149Nozawa Yusuke, 90

Ogita Takeshi, 93, 151Oishi Shi’nichi, 90Oishi Shin’ichi, 120, 125, 131,

133, 135, 149

Ozaki Katsuhisa, 93, 151

Paulen Radoslav, 95Peng Lin, 65Picot Romain, 97Plet Antoine, 98Plum Michael, 33Popescu Valentina, 100Popova Evgenija D., 102Purohit Harsh, 104

Ratschan Stefan, 106Rauh Andreas, 108, 110, 112Reddy Anwesh, 95Revol Nathalie, 114Rohou Simon, 116Romero i Sanchez David, 122Rump Siegfried M., 118, 119

Sanders David P., 123Sandretto Julien Alexandre dit,

42Sasaki Takiko, 131Schichl Hermann, 92Sekine Kouta, 90, 125, 135Semenov K.K., 127Solopchenko G.N., 127Stadtherr Mark A., 35

Taher Lotfi, 81Takayasu Akitoshi, 131, 133Tanaka Kazuaki, 125, 131, 135Tiede Susann, 108Titi Jihad, 51Tsuchiya Takuya, 53, 137

154

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van Nooijen Ronald, 141Veiseh Hana, 81Veres Sandor M., 116Volkova Anastasia, 143

Wagner Hubert, 48Walawska Irmina, 139Watanabe Yoshitaka, 142Wilczak Daniel, 139Wunderlich Jonathan, 145

Yamamoto Nobito, 82, 147Yamano Shun, 147Yampolsky Michael, 50Yanagisawa Yuka, 149

Zerr Benoıt, 61Zhao Yao, 35

155

scan 2016 - math.uu.se€¦ · schedule – scan 2016 as of september 12, 2016. sunday, september...

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