The International Conference

Mathematical Modeling with Applications(M2A19)

Dedicated to Gérard Meurant

Mohammed V University Faculté des Sciences--Institut Scientifique

Rabat Morocco, 1--4 April 2019https://m2a19.univ-littoral.fr/

Organizing Committee (Chairs):

- Souad El Bernoussi, Rabat Morocco - Khalide Jbilou, Calais France - Abderrahim Messaoudi, Rabat Morocco

Preface

The conference « Mathematical Modeling with Applications » (M2A19), co-organized by'Laboratoire de Mathématiques Informatique et Applications- Sécurité de l'Information' (LabMIA-SI), Faculté des Sciences of Mohammed V University in Rabat Morocco and Laboratoire deMathématiques Pures et Appliquées (LMPA) of Université du Littoral Côte d'Opale in CalaisFrance, will be held at 'Facultés des Sciences-Institut Scientifique' in Rabat Morocco, April 1–4,2019. The conference is celebrating the 70th birthday of Gérard Meurant for his many contributions tonumerical linear algebra. The aim of this conference is to bring together researchers working indifferent topics from mathematical modeling and applications. There will be more than 130participants from Morocco and other countries around the world (17 different nationalities). The main topics of the conference are : numerical linear algebra, scientific computation,approximation, optimization, numerical methods for partial differential equations, inverse and ill-posed problems- image processing and computational statistics.

The Organizers (Chairs)

Souad El Bernoussi Khalide Jbilou Abderrahim Messaoudi

Mat h ematical Modeling with Applications (M2A19) Dedicated to Gérard Meurant for his 70th birthday

Invited Plenary Speakers

Claude Brezinski, FranceFroilan Dopico, SpainGérard Meurant, France

Hassane Sadok, FranceYousef Saad, USA Mohammed Seaid, UK

Invi t ed Session Speakers

Belkacem Abdous, MoroccoMohamed Farhloul, Canada

Christos Koukouvinos, GreeceMarilena Mitrouli, Greece

Organizing Committee

Jilali Abouir, MoroccoSouad Elbernoussi (Chair), Morocco Mohammed Errachid, MoroccoMustapha Hached, France

Khalide Jbilou (Chair), France Abderrahim Messaoudi (Chair), MoroccoJilali Mikram, MoroccoRachid Sadaka, Morocco

Scientific Committee

Belkacem Abdous, MoroccoMohammed Benjelloun, FranceAbdesslem Bentbib, MoroccoFayssal Benkhaldoun, FranceAbderrahman Bouhamidi, France Claude Brezinski, FranceFroilan Dopico, SpainVladimir Druskin, USASouad El Bernoussi, MoroccoSaid El Hajji, Morocco Jocelyne Erhel, France Mohamed Farhloul, Canada Khalide Jbilou, France Christos Koukouvinos, Greece

Abderrahim Messaoudi, MoroccoGérard Meurant, France Marilena Mitrouli, GreeceDriss Ouazar, MoroccoMarc Prévost, FranceMichela Redivo Zaglia, ItalyLothar Reichel, USACarole Rosier, FranceYousef Saad, USARachid Sadaka, MoroccoHassane Sadok, France Mohammed Seaid, UKPaul Van Dooren, BelgiumMarc Van Barel, Belgium

Local Organizing Committee from Morocco

Said AgoujilAbderrahim BenazzouzIsmail El Moudden Mustapha Esghir

Ahmed HajjiMohammed HeyouniBouchta Rhanizar

Sponsors of the Conference :

• Université Mohammed V de Rabat

• Office Chérifien des Phosphates (OCP)

• Institut Scientifique de Rabat

• Ministère de l'éducation nationale, de la formation professionnelle, de l’enseignement supérieur et de la recherche scientifique, Maroc

• Université du Littoral Côte d'Opale, France

• Faculté des Sciences de Rabat

• Laboratoire de Mathématiques Informatique et Applications-Sécurité de l'Information, Rabat

• Ecole d'Ingénieurs du Littoral Côte d'Opale, Calais France

• Ecole Normale Supérieure de Rabat

• Conseil de la Ville de Rabat

• Association Marocaine de Modélisation et Ingénierie Mathématiques (A2MIM)

• Institut National de Statistique et d'Economie Appliquée, Rabat

• Centre National de la Recherche Scientifique et Technique (CNRST), Rabat

• Académie Hassan II des Sciences et Techniques, Rabat

• GASUP, le Groupement d’Assurances du SUPérieur, Maroc.

• Professor Constantin M. Pedriti, Athens, Greece.

Nawel AbdesselamMemory and Nonlinear Boundary Feedback Stabilization

for Schrödinger Equations with Variable Coefficientsp. 1

Belkacem AbdousTime scales concepts and smoothing techniques for

discrete and mixed-type datap. 2

Karima Adel-AissanouMarkov chain to model energy consumption in wireless

sensor networksp. 3

Imane AgmourPreserve the marine wealth of Moroccan coastal zones

by applying mathematical modelingp. 4

Imane AgnaouGene selection using a specific genetic algorithm for

cancer classificationp. 5

Ahmed Ahmed MahdiNon-existence of global solutions for fractional systems

with nonlocal power-type sourcesp. 6

Nabila AianeA stochastic procedure to solve linear ill-posed

problems with $\alpha$-mixing errorsp. 7

Rim AkhrifSmoothing variational splines of the Economic statistical

spliting data of Moroccop. 8

Fares Alazemi

Parameter estimation for Gaussian mean-reverting

Ornstein-Uhlenbeck processes of the second kind: non-

ergodic case

p. 9

Mohammad W. Alomari Generalized Taylor formula and its Applications p. 10

Sultan Alyobi Modelling the Spread of Tree Disease Through Forests p. 11

Yassine AmallasNumerical approach for predicting shape of geotextile

tubes based on degree of fillingp. 12

Mohamed

RidouanAmattouch

A new splitting method for a non linear reaction

diffusion equationp. 13

Ilham AsmouhA time-domain unstructured finite volume approach for

acoustic wave equationsp. 14

Bassem S. AttiliNumerical Solution of a Third Order Falkner-Skan Like

Equation Arising in Boundary Layer Theoryp. 15

LIST OF ABSTRACTS

Adnane AzzouziThe convergence of iterative methods for Solving the

Cauchy Problem with FV methodp. 16

Achraf BadahmaneThe regularized preconditioned GMRES and the

regularized iteration methodp. 17

Fadoua BadaouiCopulas for modeling the relationship between inflation

and the exchange ratep. 18

Khouya BassouA semi-Lagrangian finite element conjugate gradient

method for three-dimensional incompressible flowsp. 19

Zineb Belhallaj Solving fuzzy linear system by Jacobi method p. 20

Walid Ben Aribi

Compartmental model of the transmission of the

Hepatitis A virus (HAV). Global stability of the endemic

equilibrium

p. 21

Lahbib BenahmadiMathematical Modeling of New Moroccan Immigration

Policies in Controlling Infectious Diseasesp. 22

El Houcine BergouStochastic Three Points Method for Unconstrained

Smooth Minimizationp. 23

Francisco BernalSolving nonlinear systems of elliptic PDEs with the PU-

RBF methodp. 24

Ahmed Bir-JmelGene selection as maximum weight independent set for

cancer classificationp. 25

Feriel BouhadjeraNonparametric relative error estimation of the

regression function for censored datap. 26

Khalifa BoumzoughNumerical methods for the resolution of large linear

systems. Sparse matrix in CSR formatp. 27

Claude Brezinski Prescribed behavior of extrapolation methods p. 28

Mikhail BulatovOn regularization properties of collocation-variation

difference schemes for differential-algebraic equationsp. 29

Moulay Abdallah ChkifaLeja and $\Re$-Leja points for computing and

preconditioning linear systemsp. 30

Anna Concas On bipartization of networks p. 31

Dries DeSamblanx Multivariate vector polynomial interpolation p. 32

Patricia Díaz de Alba A Matlab library for EM data inversion p. 33

Froilan M. DopicoLocal Linearizations of Rational Matrices with

application to Nonlinear Eigenvalue Problemsp. 34

Jurjen Duintjer TebbensOn admissible eigenvalue approximations from solving

linear systems with the GMRES methodp. 35

Safaa El GharbiGrey-Markov Model for the prediction of the electricity

production and consumption in Moroccop. 36

Mohamed El GhomariExtended Nonsymmetric Global Lanczos Method for

Matrix Function Approximationp. 37

Mounia El Hafyani Analysis of stochastic models for daily precipitation p. 38

Smahane El Halouy Bi-Lanczos method for solving Sylvester tensor equation p. 39

Raghia El HanineConstruction of a Preconditioner for indefinite saddle

point problems using Riesz Mapp. 40

Khalid El HimdiArtificial neural network for forecasting one day ahead

of global solar irradiancep. 41

Atmane El HouchOutput feedback stabilization of distributed bilinear

time delay systemsp. 42

A. El Maliki

Efficient preconditioning techniques and accelerated

primal-dual active set method for large-scale

constrained optimization problems

p. 43

Nabil El MoçaydPolynomial chaos decomposition for sensitivity analysis

of the OCP slurry rheologyp. 44

Mustapha El MouddenA multiple reduced gradient algorithm for linearly

constrained multiobjective optimizationp. 45

Abdenasser Elidrissi

Extending the multivariate homogeneous qd-algorithm

to two-point case for the estimation of parametrized

eigenvalues

p. 46

Fevzi ErdoganEfficient Numerical Methods for Singularly Perturbed

Differential Equationsp. 47

Mohamed ErrachidNew algorithm for computing Lagrange multivariate-

interpolation polynomialsp. 48

Mouna EssaouiniNumerical method for solving the two-dimensional

Schrodinger equation with Irregular Singularitiesp. 49

Mohamed Farhloul Mixed finite element methods for the Oseen problem p. 50

Caterina Fenu Photometric Stereo under unknown lights position. p. 51

Paraskevi Fika

Numerical methods for the computation of the

absorption inverse in graphs with absorption with

applications

p. 52

Anass GuitanouNumerical approximation of an inverse problem of

image reconstructionp. 53

Mustapha Hached

Approximate solutions to large nonsymmetric

differential Riccati problems with applications to

transport theory

p. 54

Riane HajjamiStability and bifurcation analysis of non-linear epidemic

models with vaccination and distributed delaysp. 55

M.A. HamadiThe extended rational block Arnoldi method for model

reduction in large-scale dynamical systemsp. 56

Farida HamraniEstimation of the relative error in regression analysis

under random left-truncation modelp. 57

Mohamed Heyouni The BCMRH method for multiple linear systems p. 58

El Houcine HibbaBayesian Inference For a Continuous Time Hybrid

Switching Diffusion Processp. 59

M'barek IaousseProperties of the Correlation Matrix Implied by a

Recursive Path Modelp. 60

Bouchta JouilikNumerical approximation of an inverse boundary

coefficient problemp. 61

Yassine KaouaneA block tangential Lanczos method for model reduction

of large-scale first and second order dynamical systemsp. 62

Imad Kissami A multi-level adaptive mesh refinement method p. 63

Abdelouahed KouibiaA variational method for solving two-dimensional

Bratu's problemp. 64

Ekaterina Kruglova

Design of numerical method with adaptively

transformed Chebyshev nodes for sloving 1D problem

of non-isothermal polymeric fluid flow

p. 65

Radek KuceraAlgorithms for the Stokes flow with stick-slip boundary

conditionsp. 66

Fatima Zahra LamzouriFrom space frequency representation to the analysis of

2D-imagesp. 67

Aziz Madrane

Entropy Stable Discontinuous Galerkin Finite Element

Methods with Multidimensional Slope Limitation for

Hyperbolic Systems of Conservative Laws

p. 68

Mohamed MandariThe Generalized Finite Volume SUSHI Scheme for the

Discretization of the peaceman modep. 69

Gérard Meurant On prescribing CG convergence p. 70

Marilena Mitrouli Fast and stable estimates for matrix functionals p. 71

Bechir NaffetiNew trisection method to solve a bi-objective Lipschitz

and Holder optimization problemsp. 72

Nabila NagidA new gmres for schur complement domain

decomposition methodp. 73

Nihad Nouri Stein’s Unbaised Risk Estimate : Image processing p. 74

Abdelouahed OuardghiHigh-order implicit time integration for the enriched

finite element solution of transient diffusion problemsp. 75

Elias Ould SaidRobust regression analysis for a censored response and

functional regressorsp. 76

Abdessamad Ousaadane

Robin-Robin domain decomposition algorithm coupled

with boundary elements method for solving an inverse

problem in elasticity equation

p. 77

Fatemeh Panjeh Ali Beik

The tensor Golub-Kahan bidiagonalization for ill-posed

tensor equations with applications to color image

restoration

p. 78

Constantin Petridi Mathematical Structures defined by Identities p. 79

Constantin Petridi Mathematical Structures defined by Identities III p. 80

Harshana Rajakaruna

A data-driven modeling approach to compute killing

dynamics and clearance of a pathogen using backward

birth-death-immigration clustering processes of killer

p. 81

Michela Redivo-Zaglia Matrix Shanks transformations p. 82

Oussama RidaStochastic diffusion process based on Goel-Okumoto

curvep. 83

Yousef Saad Numerical Linear Algebra for data-related applications p. 84

El Mostafa SadekMinimal residual method for solving symmetric positive

Sylvester tensor equationsp. 85

Lakhlifa SadekBlock extended Krylov method for solving large-scale

generalized differential Sylvester equationsp. 86

Hassane Sadok

Review of the convergence of some Krylov subspace

methods for solving linear systems of equations with

one or several right hand sides

p. 87

Said Salim

An extended conjugate duality for generalized semi-

infinite programming problems via a convex

decomposition

p. 88

Mohammed Seaid

Efficient Preconditioning of Linear Systems Arising from

the Enriched Finite Element Discretization of

Conduction-Radiation Equations

p. 89

Abdellatif SemmouriMarkov Decision Processes with Discounted Cost : New

action elimination procedurep. 90

Sarah Sidki Recursive orthogonal polynomials p. 91

Soumaia SlimaniIdentification of the source function for a seawater

intrusion problem in unconfined aquiferp. 92

Laurent Smoch New implementation of the Block GMRES method p. 93

Ouafa SoualhiFinite volume method in a general mesh for a coupled

system of parabolic-parabolic equationsp. 94

Yahyeh Souleiman

Numerical Analysis of a Sliding frictional contact

problem with Unilateral Contact and Normal

Compliance

p. 95

Weiwei Sun New Perturbation Bounds for Matrix Decompositions p. 96

Mostafa Tahiri Numerical solution of a new fractional epidemic model p. 97

Khadija TalaliA fully implicit finite volume scheme for seawater

intrusion models in coastal aquifersp. 98

Abderrahim TazdayteA seventh order bi-quintic B-spline collocation method

applied to biharmonic Dirichlet problemsp. 99

Petr Tichy Approximating the extreme Ritz values in CG p. 100

Halima Torbi The erosion of a sand bed by the flow effect p. 101

Niel Van Buggenhout Biorthogonal Rational Krylov Subspace Methods p. 102

Abdelhak ZoglatDimensionality reduction and class prediction algorithm

with application to microarray Big Datap. 103

Stefania Maria Zoi Imaging of an elastic ellipsoid using near-field data p. 104

Memory and nonlinear boundary feedback sta-bilization for Schrodinger equations with vari-able coefficients.

Nawel Abdesselam1, Khaled Melkemi2

1nawelabedess@gmail.com2k.melkemi@univ-batna2.dz

AbstractFirst we consider the boundary stabilization of Schrodinger equations with variable coefficientmemory and nonlinear feedback. This is done by using Riemannian geometry methods and themultipliers tech- nique. Then we explore the stabilization limits of Schrodinger equations whoseelliptical part has a variable coefficient. We established the exponential decay of solutions usingthe multipliers tech- niques. The introduction of dissipative boundary conditions of memory typeallowed usto obtain an accurateestimateon theuniform rateof decay of theenergy for Schrodingerequations.

References

[1] M. AASSILA , M. M. CAVALCANTI , J. A. SORIANO, Asymptotic stability and energydecay rate for solution of wave equation with memory in a star shaped domain, SIAMJ. Control Optim., 38 (2000), 1581-1602.

[2] S. CHAI , Y. GUO, Boundary stabilization of wave equationswith variable coefficientsand memory, Differential and Integral Equations, 17 (2004), 669-980.

[3] R. CIPOLOTTI , E. MACHTYNGIER, SAN PERDO SIQUEIRA, Nonlinear boundary feed-back stabilization for Schrodinger equations, Differrential and Integral Equations, 9(1996), 137-148.

[4] V. KOMORNIK, Exact controlability and stabilization. Themultiplier method, Masson,Paris, 1994.

[5] I . LASIECKA , R. TRIGGIANI, Optimal regularity, exact controllability and uniformstabilization of Schrodinger equations with Dirichlet control, Differential and IntegralEquations, 5 (1992), 521-535.

1

Time scales concepts and smoothing tech-niques for discrete and mixed-type data

Belkacem Abdous1

1Instiut National de Statistique et d’Economie Appliquee

AbstractTime scales is a relatively recent and exciting mathematical theory introduced by the Germanmathematician Stefan Hilger in hisdissertation [3], seealso [4]. It providesageneral unifying toolthat builds the missing bridges between discrete and continuous analysis. Since then, time scalesconceptshavefound applications in several theoretical and applied problems. For acomprehensiveoverview on time scales calculus see, for instance, Bohner and Peterson [2], Agarwal et al. [1].

In this talk, basic concepts of time scales calculus together with various applications willbe presented. A particular emphasis will be put on nonparametric functional estimation. Moreprecisely, we will show how time scales concepts might be used to cast continuous, discrete andmixed-type smoothing estimation problems under the same framework.

References

[1] Agarwal, R., Bohner, M., ORegan, D., and Peterson, A., (2002). Dynamic equationsontime scales: asurvey, Journal of Computational and Applied Mathematics141, 126.

[2] Bohner M. & Peterson, A. (2001), Dynamic Equationson Time Scales, Birkhauser.

[3] Hilger, S., (1988), Ein Maβkettenkalkul mit Anwendung auf Zentrumsmannig-faltigkeiten. (German). Universitat Wurzburgn, .

[4] Hilger, S., (1990), Analysis on measure chains a unified approach to continuous anddiscrete calculus. Results in Mathematics, 18:1856.

2

Markov chain to model energy consumptionin wireless sensor networks

Karima Adel-Aissanou1, Djamila Boukredera2, Ouiza Lekadir3

1LaMOS research unit (Modeling and Optimization of Systems), University of Bejaia, Bejaia 06000, Algeria2LMA Laboratory, University of Bejaia, Bejaia 06000, Algeria3LaMOS research unit (Modeling and Optimization of Systems), University of Bejaia, Bejaia 06000, Algeria

AbstractWireless sensor networks (WSNs) have many important use such as surveillance, environmentalmonitoring, inventory tracking and localization. Since these node operate on batteries and thesesbatteriescan not bereplaced in most practical situations, it isnecessary to manage theenergy con-sumption. In this paper, we model the node operation using Markov chain to evaluate the energyconsumption for wireless sensor node. A sensor node can sense, measure and gather informationfrom its environments, then, it transmit theses information to the users. A sensor node can alsoforward information about itsneighbors. Theenergy consumed in each application isnot thesame.For this reason, our model take into account all the different states and we obtain the energy con-sumed. We used petri net modeling to evaluate the effectiveness of our model. This mathematicalmodel can beused to dressamap of energy of thenetwork and contribute to thethebest utilizationof routing protocols.

Keywords: Wireless sensor networks, Mathematical modeling, Marcov chain, energy con-sumption, Petri net simulations.

References

[1] H. Y. Zhou, D. Y. Luo, Y. Gao and D.C. Zuo, Modeling of Node Energy Consumptionfor WirelessSensor Networks, WirelessSensor Network, 2011, 3, 18-23.

[2] Y. W. Chung and H. Y. Hwang, Modelingand Analysisof Energy Conservation SchemeBased on Duty Cycling in Wireless Ad Hoc Sensor Network, Sensors2010, 10, 5569 -5589; doi:10.3390/s100605569.

[3] S. Liu, R. Srivastava, C. E. Koksal and P. Sinha, Pushback: A Hidden Markov ModelBased Scheme for Energy Ef?cient Data Transmission in Sensor Networks, Ad HocNetworks2009, 7(5):973-986, DOI: 10.1016/j.adhoc.2008.09.001.

[4] A. Shareef and Y. Zhu, Energy Modeling of Wireless Sensor Nodes Based on PetriNets, in 39th International Conference on Parallel Processing, ICPP 2010, San Diego,California, USA, 13-16 September 2010, DOI: 10.1109/ICPP.2010.19.

[5] G. A. NunezandC. B. Margi, Energy MapModel for Software-DefinedWirelessSensorNetworks, in XXXV Brazilian Symposiumon Telecommunicationsand Signal Process-ing, 03 to 06 deSeptember, 2017, Brazil.

3

Preserve the marine wealth of Moroccan coastalzones by applying mathematical modeling

I. Agmour1, M. Bentounsi2, N. Achtaich3, Y. El Foutayeni4

1Hassan II University,Laboratory of Analysis, Modeling and Simulation, Faculty of science Ben M’siik2Hassan II University,Laboratory of Analysis, Modeling and Simulation, Faculty of science Ben M’siik3Hassan II University,Laboratory of Analysis, Modeling and Simulation, Faculty of science Ben M’siik4Hassan II University,Laboratory of Analysis, Modeling and Simulation, Faculty of science Ben M’siik

AbstractIn thiswork weproposeabio economic model of thefivesmall pelagic speciesexploited by seinersin the maritime zones of Morocco. We search to determine the fishing effort that maximizes theprofit of each seiner exploit thesefish populations, but all of them have to respect two constraints :the first one is thesustainable management of the resources and the second one is thepreservationof the biodiversity. A comparison is made between the fishing effort that we determine using thegeneralized Nash equilibrium problem and the actual fishing effort given by the National Instituteof Fisheries Research. As a result, we find that master data of the NationalNInstitute of FisheriesResearch support the outcomes shown in our work.

References

[1] Y. EL Foutayeni, M. Khaladi, A Bio-economic model of fishery where prices dependon harvest, AMO Advanced Modeling and Optimisation, 14 (2012) 543-555.

[2] Y. EL Foutayeni, M. Khaladi, A generalized bio-economic model for competingmultiple-fish populations where prices depend on harvest, AMO Advanced Modelingand Optimisation, 14 (2012) 531-542.

[3] M. Bentounsi, I. Agmour, N. Achtaich, and Y. EL Foutayeni, TheImpact of Priceon theProfitsof FishermenExploitingTritrophicPrey-Predator Fish Populations, InternationalJournal of Differential Equations, Vol. 2018, Article ID 2381483.

[4] I. Agmour, M. Bentounsi, N. Achtaich, and Y. EL Foutayeni, Optimization of the TwoFishermen’sProfitsExploiting ThreeCompeting SpeciesWherePricesDepend on Har-vest, International Journal of Differential Equations, Vol. 2017, Article ID 3157294.

[5] I. Agmour, N. Achtaich, Y. EL Foutayeni, M. Khaladi and A. Zegzouti. Addendumto ”A generalized Nash equilibrium for a bio-economic problem of fishing, Journal ofApplied Research on Industrial Engineering, Vol 4 (2) (2017) 75-76.

[6] I. Agmour, M. Bentounsi, N. Achtaich, and Y. EL Foutayeni, Catchability coefficientinfluenceon the fishermen’snet economic revenues, Communications in MathematicalBiology and NeuroscienceVol 2018 (2018), Article ID 2.

4

Gene selection using a specific geneticalgorithm for cancer classification

Imane Agnaou1, Souad El Bernoussi2, Sidi Mohamed Douiri3

1 imaneagnaou92@gmail.com2souad.elbernoussi@gmail.com3douirisidimohamed@gmail.com

AbstractDNA microarray is an efficient new technology that allows to analyze, at the same time, the ex-pression level of millions of genes. The gene expression level indicates the synthesis of differentmessenger ribonucleic acid (mRNA) molecule in a cell. Using this gene expression level, it ispossible to diagnosediseases, identify tumors, select thebest treatment to resist illness. In order toachieve that purpose, several computational techniques such as pattern classification approachescan be applied. The classification problem consists in identifying different classes or groups as-sociated with a particular disease. However, the enormous quantity of genes and few samplesavailable, make difficult the processes of learning and recognition of any classification technique.Hence, we require the selection of a subset of relevant genes so as to achieve good classificationperfomance. The aim of this research is to propose a methodology for classifying cancer. In thispaper, we propose a three-stage selection algorithm which uses at first a filter method in the pre-selection step, in the second step we apply our specific genetic algorithm on the achieved subsetto determine an even smaller set of genes that gives the highest classification accuracy and finallyin the last step we evalute the selected subsets by using a linear SVM classifier. According to fourdatasets, theexperiment resultsprove that our proposed approach isvery effectivefor selecting themost relevant genes with high classification accuracy.

References

[1] Hernandez, J. C., Duval, B., Hao, J. A genetic embedded approach for gene selectionand classification. EVOBIO, 90101, 2007.

[2] L. Zhang, Z. Li, and H. Chen. An effective gene selection method based on relevanceanalysisand discernibility matrix. In PAKDD, 10881095, 2007.

5

Non-existence of global solutions for fractionalsystems with nonlocal power-type sources

Ahmed Mahdi Ahmed1

1Prep Year Math Program, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

AbstractIn this paper, we study the nonexistence of (nontrivial) global solutions for a fractional integro-differential problem in an appropriate underlying space. Theproblem contains a fractional deriva-tive of Caputo type and a first order derivative. In the right-hand side, the source consists of aconvolution of a (possibly singular) kernel with a polynomial of the state. We establish somecriteria under which there are no (nontrivial) global solutions. It is shown that a dissipation oforder between zero and one or even a (frictional) dissipation of order one does not help providingglobal (nontrivial) solutions. Weusetheweak formulation for theproblem with an appropriate testfunction and several appropriate estimations. Examples with numerical computations are given toillustrate the results.

References

[1] R. P. Agarwal, M. Benchohra, and S. Hamani, A survey on existence results for bound-ary value problems of nonlinear fractional differential equations and inclusions, ActaAppl. Math., (2010), 109(3), 973-1033.

[2] Bai, Y. Chen, H. Lian, S. Sun, On the existence of blow up solutions for a class offractional differential equations, Fract. Calc. Appl. Anal. 17.4 (2014), 1175-1187.

[3] M. D. Kassim, K. M. Furati , N.-e. Tatar, Non-existence for fractionally damped frac-tional differential problems, Acta Math. Sci., 37.1 (2017), 119-130.

[4] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applicationsof fractionaldifferential equations, Elsevier B. V., Amsterdam, Netherlands, 2006.

[5] M. Kirane, M. Medved, N. E. Tatar; On the nonexistence of blowing-up solutions to afractional functional differential equations, Georgian Math. J. 19 (2012), 127-144.

[6] J. Ma, Blow-up solutions of nonlinear Volterra integro-differential equations, Math.Comput. Model, 54 (2011), pp. 2551–2559.

6

A stochastic procedure to solve linear ill-posedproblems with α-mixing errors

Nabila Aiane1, Abdelnasser Dahmani2

1Universit de Bejaia, Laboratoire de Mathematiques Appliquees, Bejaia, Algerie;2Centre universitaire de Tamanrasset, Laboratoire de Mathematiques Appliquees, Tamanrasset, Algerie

AbstractThemain objectiveof thiswork istoestimatethesolution of thefollowing linear operator equation:Ax = u, in which the second member is measured with α-mixing errors, by using the Robbins-Monro algorithm [3] defined by

Xn+1 = Xn − an [AXn − un] (1)

with arbitrary X1 and (an)n∈N∗ is a sequence of positive numbers satisfies the following condi-tions,

+∞∑

n=1

an = ∞ and+∞∑

n=1

a2n < ∞. (2)

Sinceui = uex + ξi, then (1) can be written

Xn+1 = Xn − an [AXn − uex − ξn] . (3)

In this work we take an = a

nwith a > 0, for which we show the convergence almost com-

pletely (a.co) to element Xex and specifying its convergence rate:

Xn+1 −Xex = O(

√

n−2s2n lnn)

a.co. (4)

To check validity of our results, wegiveasimulation application into alinear operator of Fredholmintegral equation of first kind.

References

[1] Azadivar, F. (1999). Simulation Optimization Methodologies. In proceedings of the1999 Winter Simulation Conference. 93-100.

[2] Dahmani, A. Zerouati, H. Bouhmila, F. (2012). TheL-pseudo-solution using stochasticalgorithm of Landweber. Meccanica. 47(8):1935-1943.

[3] H. Robbins, S. Monro, A Stochastic Approximation Method, Ann. Math. Stat: 22,(1951), 400-407.

7

Smoothing variational splines of the economicstatistical spliting data of Morocco

R. Akhrif1, E. Delgado-Marquez2, A. Kouibia3, M. Pasadas4

1Faculte des Sciences Juridiques et Economiques et Sociales de Tetouan, Maroc.2University of Leon, Department of Economics and Statistics, Spain.3Dept. Applied Mathematics, Faculty of Sciences, University of Granada, Spain.4Dept. Applied Mathematics, Faculty of Sciences, University of Granada, Spain.

AbstractIn this work we present a new method to solve the statistical difficult when economic data arespliced. Therefore, wediscussthescopeof somesplicing toolsin theliterature, namely thesplicingby variation and linear interpolation methods. These methods carry on some problems of non-linearity. In this way, certain internal inconsistency or structural incongruity can appear whenthe economic data are spliced. This is a serious problem for the System of National Accounts.We introduce an approximation method for statistical splicing of economic data by smoothingquadratic splines. The proposed technique is linear and thus structurally congruent. Finally, weshow the effectiveness of our method by the results of the splicing of the GDP (Gross DomesticProduct) of Morocco between 1970 and 2015, and the corresponding economics activities.

8

Parameter estimation for Gaussian mean-reverting Ornstein-Uhlenbeck processes ofthe second kind: non-ergodic case

Fares Alazemi, Abdulaziz Alsenafi, Khalifa Es-Sebaiy

Department of Mathematics, Faculty of Science, Kuwait University, Kuwait.

AbstractWe consider a least square-type method to estimate the drift parameters for the mean-revertingOrnstein-Uhlenbeck process of the second kind Xt, t ≥ 0 defined as dXt = θ(µ + Xt)dt +

dY(1)t,G , t ≥ 0, with unknown parameters θ > 0 and µ ∈ R, where Y

(1)t,G :=

∫ t

0e−sdGas

with

at = Het

H , and Gt, t ≥ 0 is a Gaussian process. In order to establish the consistency and theasymptotic distribution of least square-type estimators of θ and µ based on the continuous-timeobservations Xt, t ∈ [0, T ] as T → ∞ we impose some technical conditions on the processG, which are satisfied, for instance, if G is a fractional Brownian motion with Hurst parameterH ∈ (0, 1), G is a subfractional Brownian motion with Hurst parameter H ∈ (0, 1) or G is abifractional Brownian motion with Hurst parameters (H,K) ∈ (0, 1) × (0, 1]. Our method isbased on pathwise properties of Xt, t ≥ 0 and Y

(1)t,G , t ≥ 0 proved in the sequel.

References

[1] Es-Sebaiy, K., Alazemi, F. and Al-Foraih, M. (2019). Least squares typeestimation fordiscretely observed non-ergodic Gaussian Ornstein-Uhlenbeck processes. Acta Mathe-maticaScientia, to appear.

[2] Azmoodeh, E., Morlanes, I. (2013). Drift parameter estimation for fractional Ornstein-Uhlenbeck processof theSecond Kind. Statistics: A Journal of Theoretical and AppliedStatistics. DOI: 10.1080/02331888.2013.863888

[3] Azmoodeh, E., Viitasaari, L., (2015). Parameter estimation based on discrete observa-tionsof fractional Ornstein-Uhlenbeck processof the second kind. Statistical Inferencefor Stochastic Processes, Volume18, Issue 3, pp 205-227.

[4] Bajja, s., Es-Sebaiy, K. and Viitasaari, L. (2017). Least squares estimator of fractionalOrnstein-Uhlenbeck processes with periodic mean. J. Korean Statist. Soc. 46, no. 4,608-622.

[5] Belfadli, R. Es-Sebaiy, K. and Ouknine, Y. (2011). Parameter Estimation for FractionalOrnstein-Uhlenbeck Processes: Non-Ergodic Case. Frontiers in Science and Engineer-ing (An International Journal Edited by Hassan II Academy of Scienceand Technology)1, no. 1, 1-16.

9

Generalized Taylor formulaand its applications

Mohammad W. Alomari

Department of Mathematics, Faculty of Science and Information Technology,Irbid National University, 2600 Irbid 21110, Jordan.

AbstractA sequenceof polynomialsPk (t, ·)

∞

k=0 satisfiestheAppell condition if ∂

∂tPk (t, ·) = Pk−1 (t, ·)

(∀k ≥ 1) withP0 (t, ·) = 1, for all well-defined order pair (t, ·). Thiswork providesaperspectivenew expansion of functions of real variables defined on real intervals. Namely, we proved that forany Harmonic sequence of polynomialsHk (t), the expansion of f(x)

f (x) = f (y) +∞∑

k=1

(−1)k[

Hk (y) f(k) (y)−Hk (x) f

(k) (x)]

isvalid, wheref isassumed to besuch that f is real analytic. Clearly, by settingHn (t) = (t−x)k

k!,

which reduces to the Taylor formula. Some applications regarding quadrarure rules are provided.Thoughts to generalize and extend the above expansion in terms of Laurent series are pointed outand indicated.

References

[1] A.M. Fink, Boundsof thedeviation of afunction from itsavereges. Czechoslovak Math.J., 42 (117) (1992), 289–310.

[2] A. Guessab and G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamardtype, J. Approx. Th., 115 (2002), 260–288.

[3] P.K. Kythe and M.R. Schaferkotter, Handbook of computational methods for integra-tion, Chapman & HallL/CRC, A CRC PressCompany, London, 2005.

[4] M. Matic, J. Pecaric and N. Ujevic, On new estimation of the remainder in generalizedTaylor’s formula, Math. Ineq. Appl., 2 (3) (1999), 343–361.

[5] F. Qi, Q.-M. Luo and B.-N. Guo, Darbouxsformulawith integral remainder of functionswith two independent variables, Appl. Math. comp., 199 (2008) 691–703.

10

Modelling the spread of tree disease throughForests

S. Alyobi1, N. G. Parker1, A. W. Baggaley1

1School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK

AbstractInfectious disease outbreaks caused by non-native forest insects, pests or different pathogens cancause substantial damage to trees and agriculture crops. These diseases present key challenges toboth the ecosystem and economy. Thus, it is imperative to understand how such pathogens andpests propagate through a system of trees in attempts to manage, control and minimise tree loss.In arecent paper, Orozco-Fuenteset al. have investigated adiseasepropagation in aforest throughmeasuring the spreading velocity of the pathogen on a lattice model of susceptible trees. Theirfindings illustrate that thereexistsacritical transition that separates two phases: local confinementand disease outbreak, that depends on thedensity of the susceptible individuals [1]. Here the treesare populated randomly so it is by chance if we have segregation or clustering. In reality, treesare not distributed randomly, but are clustered [2, 3]. This motivates us to develop an algorithmto generate lattice configurations where susceptible trees exhibit a controllable level of clustering.Subsequently, weuseBesag’s function, widely used in thefield of spacial statistics, to characterisethe generated tree distributions as a function of the algorithm’s parameters, paying particularlyattention to the lengthscale of the clustering [4]. We simulate the disease spread through theclustered trees, with the aim of establishing how the clustering affects the spread of tree disease.In thefutureweshall train our clustering algorithm on observational imagesof real foreststo makeit more applicable to the real world.

References

[1] Orozco-Fuentes, S., et al. ”Early warning signalsin plant diseaseoutbreaks.” EcologicalModelling 393 (2019): 12-19.

[2] Manabe, T., et al. ”Population structureand spatial patternsfor trees in a temperateold-growth evergreen broad-leaved forest in Japan.” Plant Ecology 151.2 (2000): 181-197.

[3] Li, Lin, et al. ”Spatial distributions of tree species in a subtropical forest of China.”Oikos118.4 (2009): 495-502.

[4] Besag, J. E. ”Commentson Ripleyspaper: Royal Statistical Society.” Journal 39 (1977):193-195.

11

Numerical approach for predicting shape ofgeotextile tubes based on degree of fi lling

Yassine Amallas1, Youssef Ajdor2

1,2Laboratory of Civil Engineering, Mohammedia School of Engineers, Av Ibn Sina B.P 765, Agdal -Rabat. Morocco

AbstractGeotextile tubes consist of a water permeable, sand-sealed geotextile filled with sand or othergranular materials. The tube diameter can vary from 0.5 m to 5.0 m and the length may vary from25 to 100 m. They are severally more and more used in maritime field as systems of beachesprotection against coastal erosion. The predicting of their dimensions is very important, becausethese dimensions govern the stability against several site conditions of wave and current. Thereare many studies to calculate dimensions of geotextile tubes, but they require the running of acomputer program and this limit their access to large public designers. The most popular is theapproach presented by Leshchinsky and the computer program GeoCoPS2.0. In this paper, anewnumerical approach is presented to calculate dimensions of geotextile tubes regarding the fillingrateFA .The results are presented in a computer program established from an algorithm. At theconclusion formulas, depending only on FA , to calculate dimensions of filled geotextile tube areproposed.

12

A new splitting method for a non linear reac-tion diffusion equation

Amattouch Mohamed Ridouan1, Hassan Belhadj2, Maaquol Salaheddine3,Nabila Nagid4

1,2,3,4University of Abdelmalek Essaadi, Faculty of sciences and techniques, department of mathematics, BP.416, Tangier,Morocco

AbstractThis work is devoted to an robust splitting method applied to solve the time dependent nonlinearreaction diffusion equation. The proposed method is based on an analytical step of computing, astep of solving linear ordinary equation and a Poisson equation step. To accelerate the algebraicresolution of the Poisson equation we use a krylov preeconditionnerSeveral test-cases of analytical problems illustrate this approach and show the efficiency of theproposed new method. Key words: Non linear reaction diffusion equation, Splitting Operator,

Krylov preeconditionner.

13

A time-domain unstructured finite volumeapproach for acoustic wave equations

I. Asmouh1, M. El Amrani2, A. El Kacimi3, M. Seaid1,4

1Laboratory of CSEHS, University Mohammed VI Polytechnic, Benguerir, Morocco.2Abdelmalek Essaadi University, Faculty of Sciences and Techniques, Tangier, Morocco.3Laboratory of Modeling and Combinatorial, FP Safi, Cadi Ayyad University, Morocco.4School of Engineering, University of Durham, South Road, Durham DH1 3LE, UK.

AbstractMany problems in physics and engineering involve in their modelling acoustic wave equations forboth frequency and timedomainssuch asoceanography, seismology, medical imaging, geophysicsfor oil exploration and electromagnetic. Finite difference and finite element methods have beenwidely used to solve this class of problems using implicit time stepping schemes in their imple-mentations. The drawback of these techniques is mainly related to conservation property and tofailure of highly resolving sharp gradients in the numerical solution. The focus in this study is onthedevelopment of afinitevolumeapproach for thenumerical solution of acoustic waveequationsin time domains using unstructured grids. A cell-centered finite volume discretization is used forthe space to build mass and stiffness matrices associated with the wave equation under study. Tointegrate the obtained semi-discrete form, we consider an implicit Newmark scheme allowing forlarge stability ranges compared to itsexplicit counterparts. Hence, the proposed method isuncon-ditionally stable, positive preserving and fully conservative by reconstruction. The performanceof the new time-domain unstructured finite volume method is assessed using several test exam-ples as those studied in [1, 2] among others. Some results are presented for comparative study toother well-established numerical methods widely used in the literature for solving acoustic waveproblems. Other numerical results are presented for the first time and show the accuracy and theefficiency of the proposed finitevolume method.

References

[1] M. ADDAM , A. BOUHAMIDI , M. HEYOUNI. On Solving an Acoustic Wave ProblemVia Frequency-Domain Approach and Tensorial Spline Galerkin Method. J. Sci. Com-put. 74:1193-1220 (2018)

[2] M.J. GROTE, M. MEHLIN, T. M ITKOVA. Runge-Kutta-Based explicit local time-stepping methods for wave propagation. SIAM J. Sci. Comput. 37:A747-A775 (2015)

14

Numerical solution of a third order Falkner-Skan like equation arising in boundary layertheory

Basem S. Attili1

1University of Sharjah - Department of Mathematics, Sharjah - United Arab Emirates

AbstractWe consider the third order nonlinear boundary value problem, equivalent to the well knownFalkner-Skan equation[1, 3], arising in boundary layer flow with stream-wise pressure gradientof the form

f′′′(η) + (1 + λ) f(η)f ′′(η) + 2λ

(

1− f′(η)

)

f′(η) = 0; 0 ≤ η < ∞

subject tof(0) = 0, f

′(0) = γ, f′(∞) = 1,

where f is the dimensionless stream function, η is the dimensionless normal coordinate, γ ≥ 0 isthe movement velocity ratio of the plate to the mainstream and f ′(∞) := lim

η→∞

f ′(η).

Numerically, theapproach wearegoing to follow is to first transform thethird order boundaryvalue problem on the semi-finite domain into a second order nonlinear boundary value problemon a finite domain through introducing an auxiliary equation g (η) = f ′ (η) . Then we use theshooting method to approximate the solution to the original third order problem and the reducedsecond order problem[2]. Numerical resultsof both approaches will be presented and comparisonwith the work of others will also be done.

References

[1] Aly EH, Elliott L, Ingham DB (2003) Mixed Convection Boundary-Layer Flow Over aVertical SurfaceEmbedded in a PorousMedium. Eur JMech B Fluid, 22(6):529–543

[2] Attili, B. (1993) ”On the Numerical Implementation of the Shooting Methods to theOne-Dimensional Singular Boundary Value Problems” , Intern. J. Computer Math., 47,65-75.

[3] Falkner V. M. and Skan, S. W., LXXXV. Solutions of the Boundary-Layer Equations.The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science,12(80):865896, 1931. https://doi.org/10.1080/14786443109461870.

15

The convergence of iterative methods for solv-ing the Cauchy problem with FV method

A. Azzouzi1, A. Nachaoui2

1Polydisciplinary faculty of Taroudant, Ibn Zohr University, Morrocco.2Jean Leray Laboratory of mathematics, Nantes University, France.

AbstractIn this paper, the Finitevolume method [1] isused in order to implement numerically the iterativealgorithms to solvenumerically aclassof inverseboundary problems [2, 3, 4]. In comparison withboundary element method and finite element method used usually for this type of problems, thechoice of finitevolume method in thispaper is mentioned and explained. Wediscuss in detail thisimplementation [5] and we present some results of numerical experiments with this method. AFortran code has been developed based on the finitevolume cell vertex formulation.

References

[1] R. Eymard, T. Gallou et, and R. Herbin A new finite volume scheme for anisotropicdiffusionproblemson general grids: convergenceanalysis. C. R. Math. Acad. Sci. Paris,6(344):403406, 2007.

[2] V. A. Kozlov, V. G. Mazya, A. V. Fomin An iterative method for solving the Cauchyproblem for elliptic equations, Comput. Math. Phys. 31 (1991) 45-52.

[3] M. Jourhmane, A. Nachaoui Convergence of an alternating method to solve Cauchyproblem for Poissonsequation, Appl. Analysis81 (2002) 1065-1083.

[4] C. H. Huang, W. Y. Shih An inverse problem in estimating interfacial cracks in bima-terials by boundary element technique, International Journal for Numerical Methods inEngineering 45 11 (1999) 1547-1567.

[5] Azzouzi A., Coudiere Y., Turpault R., Zemzemi Z. A mathematical model of thePurkinje- Muscle Junctions. Mathematical Biosciences and Engineering, Vol. 8, No.4, 915-930, 2011.

16

The regularized preconditioned GMRES andthe regularized iteration method

Badahmane Achraf1

1FST-Marrakech, Laboratoire de Mathematiques Appliquees et Informatique, Marrakech, Morocco. LMPA, Universite duLittoral Cote d’Opale, 50 Rue F. Buisson, BP 699 - 62228 Calais cedex, France, email: badahmane.achraf@gmail.com.

AbstractIn the present paper we propose a new preconditioner for solving the saddle point problem. Thepreconditioner isobtained by replacing the(1, 2) and (2, 2) blocks in theoriginal saddle-point ma-trix A by another well chosen block. Theproposed preconditioner can beused asapreconditionercorresponding to the stationary itearative method or to accelerate the convergence of the gener-alized minimal residual method (GMRES). The convergent and semi-convergent analyses of theregularized iteration method are presented. Meanwhile, we analyzed the eigenvalue-distributionand the forms of the eigenvectors of the preconditioned matrix are analyzed in detail. Finally,numerical results show the effectiveness of the proposed preconditioner as compared to other pre-conditioners.

References

[1] Arnoldi ED, The principle of minimized iterations in the solution of the matrix eigen-valueproblem, Quart. Appl. Math., 9 (1951), pp. 1729.

[2] Bai ZZ, Golub HG, Lu ZL, Yin FJ. Block triangular and skew-Hermitian splitting meth-ods for positive-definite linear systems. Siam J. Sci. Comput., 26 (2004), pp. 844-863.

[3] Bai ZZ, Golub HG, Pan YJ. Preconditioned Hermitian and skew-Hermitian splittingmethods for non-Hermitian positive semidefinite linear systems. Numer. Math., 98(2004), pp. 1-32.

17

Copulas for modeling the relationship betweeninflation and the exchange rate.

Laila Ait Hassou1, Fadoua Badaoui2, Okou Guei Cyrille3, Amine Amar4,Abdelhak Zoglat5, Elhadj Ezzahid6

1,5Laboratoire de Mathematiques, Statistique et Applications, Faculte des Sciences, Centre de Recherche Mathematiquesde Rabat, Universite Mohammed V de Rabat, Rabat, Morocco2Departement Statistique, Demographie et Actuariat, Institut National de Statistique et dEconomie Appliquee, Rabat,Morocco3Unite de Formation et Recherche (UFR) Environnement, Universite Jean Lorougnon Guede de Daloa, Daloa, Cte dIvoire4Moroccan Agency for Sustainable Energy, Rabat, Morocco6Faculty of Law and Economics, Laboratory of Applied Economics, Mohammed V University in Rabat, Rabat, Morocco.

AbstractCopulas are useful tools for formalizing the dependence structure between variables. They haveproven to be very valuable in economics, where the dependence plays a key role. In this chapter,we use copulas to analyze the dependence between inflation and US/Euro exchange rates in theEuro area, during different periods. We first explore the dependence between the variables usinga nonparametric approach. Then, we select an appropriate parametric copula for each period.Results confirm the sensibility of copulas to macroeconomic fluctuations that occur during theanalyzed periods.

References

Chollete, L., Ning, C.: The Dependence structure of macroeconomic variables in theUS. Working PapersSeries0204, University of Ryerson (2009).Xiongtoua, T., Sriboonchitta, S.: Analysis of volatility and dependence betweenexchangerateand inflation rate in Lao peoplesdemocratic republic using copula-basedGARCH approach. Model. Depend. Econ. 251, 201214 (2014).Kole, E. et al.: Selecting copulas for risk management. J. Bank. Fin.https://doi.org/10.1016/j. jbankfin.2006.09.010.(2007).Gregoire, V., Genest, C., Gendron, M.: Using copulas to model price dependence inenergy markets. Energy Risk 5, 5864 (2008).Genest, C., Gendron, M., Bourdeau-Brien, M.: The advent of copulas in finance. Eur.J. Fin. 15(7/8), 609618 (2009).

18

Solving fuzzy linear system by Jacobi method

Zineb Belhallaj1, Abdellatif Semmouri2

1dep. of Mathematics, Faculty of Sciences and Techniques, Sultan Moulay Slimane University, Beni Mellal, Morocco2dep. of Mathematics, Laboratory TIAD, Faculty of Siences and Techniques, Sultan Moulay Slimane University, BeniMellal, Morocco

AbstractFuzzy linear systemsof equationsplay amajor rolein several areasof sciencesuch asengineering,physics and economics when the information is imprecisely.

In this paper we apply the Jacobi method to solve the fuzzy linear equation Ax=b, where A isa real matrix and b is a fuzzy vector after converting the fuzzy system to a crisp system of linearequations by using parametric form of fuzzy numbers to obtain the solution. of the used iterativemethod and we illustrate itsefficiency by solving some numerical examples

Finally, wediscuss theconvergence using triangular and trapezoidal convex normalized fuzzysets for the present analysis.

19

Compartmental model of the transmission ofthe Hepatitis A virus (HAV). Global stability ofthe endemic equilibrium

Walid Ben aribi1, Bechir Naffati2, Kaouther Ayouni3, Hamadi Ammar4,Slimane Ben Miled5, Henda Triki6

1,2Faculty of Sciences of Bizerte, University of Carthage, Tunis, Tunisia3Faculty of Sciences of Tunis, University of Tunis El Manar, Tunis, Tunisia, Campus Universitaire 2092 - El Manar Tunis.4Faculty of Economic Sciences and Management of Nabeul, University of Carthage, Tunis, Tunisia5Laboratory of Bioinformatics, biomathematics and biostatistics, Pasteur Institute of Tunis, University of Tunis El Manar,Tunis, Tunisia6Laboratory of Clinical Virology Pasteur Institute of Tunis, University of TunisElManar, Tunis, Tunisia

AbstractThis paper deals with the transmission and the propagation of the hepatitis A virus in the cen-tral west of Tunisia (the city of Thala). The target of this framework is to estimate the basicreproduction number to study the global stability. We model this situation by means of a SEIRDepidemiological model where the infectious compartment is divided into symptomatic and asymp-tomatic ones. We study the global stability of the endemic equilibrium stability using the Lyapunovfunction that depends on the value of basic reproduction number R0. Hence, we prove that whenR0 < 1, the disease-free equilibrium (DFE) is globally asymptotically stable, but when R0 > 1

the DFE is unstable and the endemic equilibrium is globally stable.

20

Mathematical modeling of new Moroccan im-migration policies in controlling infectious diseases

Lahbib Benahmadi1, Khadija Niri2, Abdessamad Tridane3

1Department of mathematics and Computer Science Faculty of Science, University of Hassan II Ain Chock, Casablanca,Morocco2Department of mathematics and Computer Science Faculty of Science, University of Hassan II Ain Chock, Casablanca,Morocco3Department of Mathematical Sciences, United Arab Emirates University

AbstractMathematical modeling allows us to assess the intervention or policies adopted to counteract anepidemic outbreak and predict possiblefutureprogressabout diseaseoutbreaks. In thistalk, weuseSEIR model to show how New Moroccan policieshavehelped control thegeneral public infectiousdiseaseoutbreaks. Wewill usetheWell-posednessof themodel specifically thebasic reproductionnumber to show the policies are effective in controlling the disease outbreaks.

References

[1] Babangida, B. G., Nazziwa, A., Noor, K., Adiukwu, R. N., Ngaloru, S. N., Mafuyai, Y.M., Obi, E. N., Onwunali, M. C. and Obanny, A., ”Analysisof non-linear transmissionof EBOLA virusdisease and the impact of public health control interventionsin hospi-tal: the case of Guinea (West Africa) outbreak 2014,” Islamic University Journal, pp.10-28, Vol. 5, No. 1, June2015.

[2] Talley, J., Davis, L. B., Morin, B. and Liu, L., ”A vector-borne contamination modelto assess food-borneoutbreak intervention strategies” Journal of Applied MathematicalModelling, pp. 383-403, Vol. 66, No. 1, February 2019.

[3] Wang, X., Peng, H., Shi, B., Jiang, D., Zhang, S. and Chen, Biaosong, ”Optimal vacci-nation strategy of a constrained time-varying SEIR epidemic model” Communicationsin Nonlinear Science and Numerical Simulation, pp. 37-48, Vol. 67, No. 1, February2019.

21

Stochastic three points method for unconstrainedsmooth minimization

El Houcine Bergou1, Eduard Gorbunov2, Peter Richtarik3

1King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia. MaIAGE, INRA, Universite Paris-Saclay, 78350 Jouy-en-Josas, France2Moscow Institute of Physics and Technology3King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia. University of Edinburgh, Edin-burgh, United Kingdom. Moscow Institute of Physics and Technology

AbstractIn thispaper weconsider theunconstrained minimization problem of asmooth function inR

n in asetting where only function evaluations are possible. We design a novel randomized direct searchmethod based on stochastic three points (STP) and analyze itscomplexity. At each iteration, STPgenerates a random search direction according to a certain fixed probability law. Our assumptionson this law are very mild: roughly speaking, all laws which do not concentrate all measure on anyhalfspace passing through theorigin will work. For instance, weallow for theuniform distributionon the sphere and also distributions that concentrate all measure on a positive spanning set.

Although our approach is designed to not use explicitly derivatives, it covers some first ordermethods. For instance if the probability law is chosen to be the Dirac distribution concentrated atthe sign of thegradient then STP recovers theSigned Gradient Descent method. If the probabilitylaw istheuniform distribution on thecoordinatesof thegradient thenSTP recovers theCoordinateDescent Method.

Given a current iteratex, STP compares the objective function at three points: x, x+ αs andx − αs, whereα > 0 is a stepsize parameter and s is the random search direction. The best ofthese three points is the next iterate. We analyze the method STP under several stepsize selectionschemes (fixed, decreasing, estimated through finitedifferences, etc).

The complexity of STP depends on the probability law via a simple characteristic closelyrelated to the cosine measure which is used in the analysis of deterministic direct search (DDS)methods. Unlike in DDS, whereO(n) (n is the dimension of x) function evaluations must be per-formed in each iteration in the worst case, our method only requires two new function evaluationsper iteration. Consequently, whileDDS depends quadratically on n, our method depends linearlyon n. In particular, in the nonconvex case, STP needs O(nε−2) function evaluations to find apoint at which the gradient of the objective function is below ε, in expectation. In the convexcase, the complexity is O(nε−1). In the strongly convex case STP converges linearly, meaningthat the complexity isO

(

n log(

ε−1

))

. Wealso propose aparallel version for STP, with iterationcomplexity bounds which do not depend on the dimension n.

22

Solving nonlinear systems of elliptic PDEs withthe PU-RBF method

Francisco Bernal1, Elisabeth Larsson2, Alfa R.H. Heryudono3

1Department of Integrated Sciences, Universidad de Huelva (Spain)2Department of Information Technology, Uppsala University (Sweden)3Department of Mathematics, University of Massachusetts Dartmouth (USA)

AbstractThe partition of unity - radial basis function method (PU-RBF) is a recent meshless algorithmfor solving PDEs with smooth solution [1]. It retains the features of RBF collocation methods(geometrical flexibility, ease of coding, spectral convergence), while potentially overcoming themain drawback of being restricted to small discretisations (owing to ill conditioning). PU-RBF isthus a topical method for PDEs, being currently investigated along several directions.

In this work, we focus on nonlinear elliptic PDEs. The resulting nonlinear algebraic systemsare tackled with a PU-RBF version of the trust-region algorithm introduced in [2]. For all butmildly nonlinear PDEs, curvature information about the residual landscape must be incorporatedinto the trust-region model, in order to enable convergence towards a system root to take place inan efficient way (or at all). In particular, two-dimensional projection of theHessian (as introducedin [3]) is discussed. Highly nonlinear problems, such as the Monge-Ampere PDE and the steadyNavier-Stokes equations are satisfactorily worked out using the new method.

References

[1] A. Safdari-Vaighani, A. Heryudono and E. Larsson: A radial basis function partition ofunity collocation method for convection-diffusion equations arising in financial appli-cations. J. Sci. Comput. 64, 341–367 (2015).

[2] Bernal, F.: Trust-region methodsfor nonlinear elliptic equationswith radial basis func-tions. Comput. Math. Appl. 72(7), 1743–1763 (2016).

[3] R. H. Byrd, R. B. Schnabel and M. H. Schultz: Approximate solution of the trust-region problem by minimization over two-dimensional subspaces. Math. Program. 40,247–263 (1988).

23

Gene selection as maximum weight indepen-dent set for cancer classification.

Ahmed Bir-jmel1, Sidi Mohamed Douiri1, Souad Elbernoussi1

1Laboratory of Mathematics, Computing & Applications-Information Security, Faculty of Sciences, Mohammed V Univer-sity in Rabat, Morocco.

AbstractGene expression data (DNA microarray) enable researchers to simultaneously measure the levelsof expression of several thousand genes. These levelsof expression arevery important in the clas-sification of different types of tumors.In this Poster, we are interested in gene selection, which is an essential step in the data pre-processing for cancer classification. Thisselection makes it possible to represent asmall subset ofgenes from a large set, and to eliminate the redundant, irrelevant or noisy genes. The combinato-rial nature of the selection problem requires the development of specific techniques such as filtersand Wrappers, or hybrids combining several optimization processes. In this context, we proposea filter stage method (MWIS) for the gene selection problem using a new graph-based approachfor gene selection (MWIS), in which we seek to minimize the redundancy between genes by con-sidering the correlation between the latter, and maximize gene ranking scores, finally we use the1NN classifier for the evaluation of the relevance of the output subset of genes. In order to verifythe performance of our methods, we have tested them on some well-known microarray datasetsofhigh dimensions.The experiments carried out on the different datasets show that our method improves the classifi-cation accuracy, we talk about a selection that ensures better accuracy.

References

[1] Bir-Jmel, A., Douiri, S. M., Elbernoussi, S. (2019). Geneselection viaBPSO and Back-ward generation for cancer classification. RAIRO-OperationsResearch, 53(1), 269-288.

[2] Sharbaf, F. V., Mosafer, S., Moattar, M. H. (2016). A hybrid gene selection approachfor microarray data classification using cellular learning automata and ant colony opti-mization. Genomics, 107(6), 231-238.

[3] Tabakhi, S., Najafi, A., Ranjbar, R., Moradi, P. (2015). Gene selection for microarraydata classification using a novel ant colony optimization. Neurocomputing, 168, 1024-103

[4] Zhang, Z., Hancock, E. R. (2012). Hypergraph based information-theoretic feature se-lection. Pattern Recognition Letters, 33(15), 1991-1999.

24

Nonparametric relative error estimation of theregression function for censored data

Bouhadjera Feriel1, Ould Said Elias2, Remita Mohamed Riad.3

1,2Universite du Littoral Cote d’Opale.1,3Universite Badji-Mokhtar Annaba.

AbstractLet (Ti)i be a sequence of independent identically distributed (i.i.d.) random variables (r.v.) ofinterest distributed asT and (Xi)i beacorresponding vector of covariates taking valuesonR

d. Incensorship models the r.v. T issubject to random censoring by another r.v. C. In thiscontributionwe built a new kernel estimator based on the so-called synthetic data of the mean squared relativeerror for the regression function. We establish the uniform almost sure convergence with rateover a compact set and its asymptotic normality. The asymptotic variance is explicitly given andas product we give a confidence bands. A simulation study has been conducted to comfort ourtheoretical results.Keywords: Asymptotic normality. Censored data. Kernel estimate. Relative regression error. Uni-form almost sure convergence. V-C classes.

References

[1] DeheuvelsP., Einmahl J.HJ. (2000). Functional limit lawsfor theincrementsof Kaplan-Meier product limit processesand applications. Ann Probab., 28, 1301-1335.

[2] Guessoum Z., Ould Saıd E. (2008). On nonparametricestimation of theregression func-tion under random censorship model. Stat. and Decisions, 26, 1001-1020.

[3] Gine E., Guillou A. (1999). Law of the iterated logarithm for censored data. Ann. ofProbab., 27, 2042- 2067.

[4] GineE., Guillou A. (2002). Ratesof strong uniform consistency for multivariatekerneldensity estimators. Ann. I. H. Poincare. 38, 907- 921.

[5] Kaplan, E.L., Meier, P. (1958). Nonparametric estimation from incomplete observa-tions. J. Amer. Stat. Assoc., 53, 458-481.

[6] Park H., Shin K.I., Jones M.C., Vines S.K. (2008). Relative error prediction via kernelregression smoothers. Journal of Stat. Plann. and Infer., 138, 2887-2898.

[7] Park H., Stefanski L.A. (1998). Relative error prediction. Stat .& Probab. Lett., 40,227-236.

25

Numerical methods for the resolution of largelinear systems. Sparse matrix in CSR format

Khalifa Boumzough1, Abderrahim Bouidi2, Adnane Azzouzi3

1Laboratory LabSI, Faculty of Sciences University of IBN ZOHR, Morocco.2Laboratory LabSI, Faculty of Sciences University of IBN ZOHR, Morocco.3Laboratory LabSI, Faculty of Sciences University of IBN ZOHR, Morocco.

AbstractA vast majority of problems in computational science and engineering can be described by PDEs.The discritization these equations by numerical methods (finite element, finite difference, finitevolume,. . . ) leods large sparse linear systems. The solving large sparse linear systems can lead toextremely long computation times. Although iterative methods for solving linear systems requireless memory space than direct methods, the success of an iterative method depends firstly onthe choice of the iterative method to accelerate the convergence and then depends on the storagemethod to obtain a better solution. In this work, we will focus on the study of conjugate gradientmethod and thepreconditioned conjugategradient methodsusing CSR format storageto acceleratethe convergence.

References

[1] Lin Du, Shujian Liu, Study on PreconditionedConjugateGradient Methodsfor SolvingLargeSparseMatrix in CSR Format, 1-22.

[2] Yousef Saad. IterativeMethodsfor SparseLinear Systems. Second Edition 2003

[3] MagnusHestenesand Eduard Stiefel. Methodsof conjugategradientsfor solving linearsystems. Journal of Research of theNational Bureau of Standards, 49, 409-436.

26

Prescribed behavior of extrapolation methods

Claude Brezinski1, Michela Redivo-Zaglia2

1Universite de Lille, France2Universita di Padova, Italy

AbstractWhen a sequence (sn) is slowly converging, it can be transformed, by a linear or a nonlinearsequence transformationT , into another sequence(tn) which, under someassumptions, convergesfaster to the same limit. Conditions for the convergence of (tn), and its faster converge have beengiven for many transformations and many classes of sequences (see, for example, [1, 3, 4, 5, 6]).

In this talk, wediscuss the inverse problem which seemsto havenever been treated. Westudyif it is possible to find the sequence (sn) such that (tn) has a prescribed behavior tn = an for alln, where (an) is an arbitrary sequence. The study uses two approaches namely the mathematicaldefinition of the transformation and the recursive algorithm for its implementation.

These results show that almost any convergence behavior for a sequence transformation canbe prescribed by a particular choice of the sequence (sn) to be transformed. Some examples aregiven. See [2] for more details.

References

[1] C. Brezinski, M. Redivo–Zaglia, Extrapolation Methods. Theory and Practice, North-Holland, Amsterdam, 1991.

[2] C. Brezinski, M. Redivo-Zaglia, Some unusual results on extrapolation methods, inprogress.

[3] J.P. Delahaye, Sequence Transformations, Springer–Verlag, Berlin, 1988.

[4] A. Sidi, Practical Extrapolation Methods. Theory and Applications, Cambridge Uni-versity Press, Cambridge, 2003.

[5] E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergenceand thesummation of divergent series, Comput. Phys. Rep., 10 (1989) 189–371.

[6] J. Wimp, Sequence Transformations and their Applications, Academic Press, NewYork, 1981.

27

On regularization properties of collocation-variationdifference schemes for differential-algebraicequations

M. V. Bulatov1, L. S. Solovarova1

1Matrosov Institute for System Dynamics and Control Theory of SB RAS, Lermontov st., 134, Irkutsk, Russia

AbstractConsider

A(t)x′

(t) +B(t)x(t) = f(t), t ∈ [0, 1], x(0) = x0, (1)

whereA(t), B(t) are (n× n)-matrices, f(t) and x(t) are the given and unknown n-dimensionalvector-functions, respectively. It is assumed that detA(t) ≡ 0 and an initial condition is given sothat (1) has a solution. Systems of the form (1) are called differential-algebraic equations (DAEs).

Consider a perturbed version of (1):

A(t)x′

(t) + B(t)x(t) = f(t), t ∈ [0, 1], x(0) = a, (2)

where elements of the input data are given with the known level of perturbation

‖alm(t)− alm(t)‖C[0,1]

≤ δ,∥

∥

∥blm(t)− blm(t)

∥

∥

∥

C[0,1]

≤ δ, (3)

∥

∥

∥fl(t)− fl(t)

∥

∥

∥

C[0,1]

≤ δ, ‖x0 − x0‖ ≤ δ, l, m = 1, 2, ..., n.

Herealm, alm, blm, blm, fl, fl are elements of A(t), A(t), B(t), B(t), f(t), f(t).DAEs belong to the class of ill-posed problems, i.e. the solution to (3) may not exist. For

specific class of (1), it was shown [1] that a number of difference schemes generate a regulariza-tion algorithm. In this talk, we consider collocation-variation difference schemes for DAEs. Theintegration step ischosen using the discrepancy principle for these algorithms.

This work was partially supported by the Russian Foundation for Basic Research, projects18-51-540001, 18-29-10019.

References

[1] M. Bulatov, L. Solovarova, On self-regularization propertiesof a differenceschemeforlinear differentialalgebraic equations, Appl. Numer. Math. 130 (2018) 86–94.

28

Leja and ℜ-Leja points for computing and pre-conditioning linear systems

Moulay Abdellah Chkifa1

1MSDA, Mohammed VI Polytechnic University, lot 660, Hay Moulay Rachid, Ben Guerir, Maroc

AbstractWe introduce Leja sequences on the unit disk U and ℜ-Leja sequences on the interval [−1, 1]and discuss the growth of associated Newton polynomials and their derivatives and the Lebesgueconstant for polynomial interpolation. These growths show that such sequences provide nestedalternatives to roots of unity on U and Tchebychev abscissas on [−1, 1] for polynomial interpola-tion and polynomial preconditioning. In light of theseresults, given amatrix A with spectrum in ageneral compact connected domain K of C or R, we discuss the convergence, stability and accu-racy properties of operating with the matrix f(A) on a vector v, i.e. computing w = f(A)v, andof Tchebychev-type methods for acceleration and preconditioning Richardson’s iteration method.Such problem are discussed e.g. in [1, 2].

Wehave studied Lejaandℜ-Lejasequences in the framework of uni-variate and multi-variatehierarchical polynomial interpolation, see [3, 4]. The particular ordering of such sequences con-veysthem remarkablestructural propertieswhich imply minimal cost, high stability and controlledaccuracy in the aforementioned framework. We present here the ramifications of such results inthe separate framework of acceleration techniques of linear systems.

References

[1] Tal-Ezer, Hillel, Polynomial approximation of functions of matrices and applications.Journal of Scientific Computing4(1), 25–60 (1989)

[2] Bernd Fischer and Lothar Reichel, A Stable Richardson Iteration Method for ComplexLinear SystemsNumer. Math 54, 225–242 (1988)

[3] Moulay Abdellah Chkifa On the Lebesgue constant of Leja sequences for the complexunit disk and of their real projection. Journal of Approximation Theory 166, 176–200(2013)

[4] Moulay Abdellah Chkifa, On thestability of theNewton formula in multivariatehierar-chical polynomial interpolation. In preparation

29

On bipartization of networks

Anna Concas1, Silvia Noschese2, Lothar Reichel3, Giuseppe Rodriguez1

1Department of Mathematics and Computer Science, University of Cagliari,Viale Merello, 92, 09123 Cagliari, Italy2Dipartimento di Matematica, SAPIENZA Universita di Roma,P.le Aldo Moro 5, 00185 Roma, Italy3Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA

AbstractNetwork analysis aims to identify important nodes in a network and to uncover structural proper-tiesof anetwork such asbipartivity. A network issaid to bebipartite if itsnodescan besubdividedinto two nonempty sets such that there are no edges between nodes in the same set. It isa compu-tationally difficult problem to determine the closest bipartite network to a given network. The aimof thiswork is to describe how a given network can beapproximated by abipartiteone by solvinga sequence of fairly simple optimization problems. We also show, with numerical examples, howthe same bipartization procedure can be used to detect the presence of large anti-communities in anetwork and to identify them.

References

[1] L. Chen, Q. Yu, andB. Chen, Anti-modularity andanti-community detecting in complexnetworks, Inf. Sci., 275 (2014), pp. 293–313.

[2] A, Concas, S. Noschese, L. Reichel and G. Rodriguez, On bipartization of networks,arXiv:1812.08408 [math.NA], 2018.

[3] E. Estrada and J. Gomez–Gardenes, Network bipartivity and the transportation effi-ciency of European passenger airlines, PhysicaD, 323-324 (2016), pp. 57–63.

[4] D. Fasino and F. Tudisco, A modularity based spectral method for simultaneous com-munity and anti-community detection, Linear AlgebraAppl., 542 (2017), pp. 605–623.

[5] N. Higham, Matrix nearness problemsand applications, in M. J. C. Gover and S. Bar-nett, eds., Applications of Matrix Theory, Oxford University Press, Oxford, 1989, pp.1–27.

30

Solving multivariate vector polynomial inter-polation problems

Dries De Samblanx1, Marc Van Barel1, Raf Vandebril1

1KU Leuven, Department of Computer Science, B-3001 Leuven(Heverlee), Belgium

AbstractAn algorithm is discussed for computing a generating set for all multivariate polynomial vectors(g1(z), g2(z), . . . , gm(z))T that satisfy the following homogeneous interpolation conditions:

pk1 g1(ωk) + pk2 g2(ωk) + · · ·+ pkm gm(ωk) = 0 for all 1 ≤ k ≤ ℓ,

whereωk are multivariate interpolation points with corresponding interpolation data pkj . More-over, the generating set is constructed in such a way that it is easy to extract all polynomial vectorsolutions having a specific degree structure. For that purpose a matrix representation for a partialor total multivariate monomial order is generalised to the vector case. The algorithm works in aniterative way, i.e., the generating set is updated using sparse transformation matrices when addingan additional interpolation condition.

Wegive someapplications and thenumerical resultsof multiple representations for thegener-ating set are studied and compared.

31

Local Linearizations of Rational Matrices withapplication to Nonlinear Eigenvalue Problems

F.M. Dopico1, S. Marcaida2, M.C. Quintana1, P. Van Dooren3

1Departamento de Matematicas, Universidad Carlos III de Madrid, Leganes, Spain.2Departamento de Matematica Aplicada y EIO, Universidad del Paıs Vasco UPV/EHU, Bilbao, Spain.3Department of Mathematical Engineering, Universite catholique de Louvain, Louvain-la-Neuve, Belgium.

AbstractThe numerical solution of nonlinear eigenvalue problems (NLEP) has attracted considerable at-tention since 2004, mainly as a consequence of the influential reference [5]. A variety of methodshave been developed for these problems and the most successful ones can be found in the recentsurvey [2]. In the large-scale setting the preferred methods consist of three steps [3, 4]: (1) to ap-proximate theNLEPby arational eigenvalue problem (REP) in acertain regionΣ; (2) to constructa linear eigenvalue problem (LEP) that has the same eigenvalues of the REP in Σ; (3) to computevia structured rational Krylov methods the eigenvalues of the LEP. The purpose of this talk is todevelop a local theory of linearizations of REPs that allows us, among other things, to establishrigurously the properties of the LEPs that have been used in [3, 4] and to extend and simplify theresults published recently in [1].

References

[1] A. Amparan, F. M. Dopico, S. Marcaida, and I. Zaballa, Strong linearizations of rationalmatrices, SIAM J. Matrix Anal. Appl., 39(4) (2018) 1670–1700.

[2] S. Guttel and F. Tisseur, The nonlinear eigenvalue problem, Acta Numer., 26 (2017)1–94.

[3] S. Guttel, R. Van Beeumen, K. Meerbergen, and W. Michiels, NLEIGS: a class of fullyrational Krylov methods for nonlinear eigenvalue problems, SIAM J. Sci. Comput., 36(2014) A2842–A2864.

[4] P. Lietaert, J. Perez, B. Vandereycken, and K. Meerbergen, Automatic rational approxi-mation and linearization of nonlinear eigenvalue problems, arXiv:1801.08622v2.

[5] V. Mehrmann and H. Voss, Nonlinear eigenvalue problems: a challenge for moderneigenvalue methods, GAMM Mitt. Ges. Angew. Math. Mech., 27 (2004) 121–152.

32

On admissible eigenvalue approximations fromsolving linear systems with the GMRES method

Jurjen Duintjer Tebbens1, Gerard Meurant2

1Institute of Computer Science, Czech Academy of Sciences, Pod Vodarenskou vezı 2, 18 207 Praha 8 - Liben andCharles University, Faculty of Pharmacy, Heyrovskeho 1203, 500 05 Hradec Kralove, Czech Republic. (duintjertebbens@cs.cas.cz)230 rue du sergent Bauchat, 75012 Paris, France. (gerard.meurant@gmail.com)

AbstractWhen solving non-normal linear systems of the form

Ax = b

with the GMRES method, eigenvalue approximations for A can be obtained from the eigenvaluesof the generated Hessenberg matrices, which represent the orthogonal restrictions of A to thesubsequent Krylov subspaces. Often the eigenvalues of the Hessenberg matrices themselves areused. These so-called Ritz values are the zeros of the polynomials in the parallel FOM process.But perhaps morepopular areso-called harmonic Ritz values, which arethezerosof thegeneratedGMRESpolynomials and the eigenvalues of a rank-one modification of the Hessenberg matrices.

In our talk weaddress therelation between thegenerated GMRESresidual normsand thegen-erated (harmonic) Ritz values. We explain that they can be chosen independently except for whenresidual norms stagnate [2], [1]. In other words, matricesA and right-hand sides b exist that gen-erate prescribed (harmonic) Ritz values in every iteration and prescribed (non-increasing) residualnorms as well. This is more surprising for harmonic Ritz values than for ordinary Ritz values,because for the former this implies that the size of a residual vector and the zeros of the polyno-mial inA generating that same vector, can be chosen independently. We elaborate on this fact [3]and plan to discuss closed-form expressions for (harmonic) Ritz values in terms of componentsof the right hand side, eigenvalues and eigenvectors of A. We also hope to address the relationwith deflation and augmentation techniques for restarted GMRES, which rely on eigenspace ap-proximations obtained from (harmonic) Ritz values. A related research area are inverse eigen- orRitz-value problems, where for a given matrix A, the admissible Ritz value sets are characterizedwhen b can be chosen.

References

[1] K. DU, J. DUINTJER TEBBENS, AND G. MEURANT, Any admissible harmonic Ritzvalueset ispossible for GMRES, ETNA, 47 (2017), pp. 37–56.

[2] J. DUINTJER TEBBENS AND G. MEURANT, Any Ritz value behavior is possible forArnoldi and for GMRES, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 958–978.

[3] G. MEURANT, The coefficients of the FOM and GMRESresidual polynomials, SIAMJ. Matrix Anal. Appl., 38 (2017), pp. 96–117.

33

A Matlab library for EM data inversion

Gian Piero Deidda1, Patricia Dıaz de Alba2, Caterina Fenu2, GabrieleLovicu3, Giuseppe Rodriguez2

1Department of Civil, Environmental, and Architectural Engineering, University of Cagliari, Piazza d’Armi 1, 09123Cagliari, Italy.2Department of Mathematics and Computer Science, University of Cagliari, viale Merello 92, 09123 Cagliari, Italy.3Department of Electrical and Electronic Engineering, University of Cagliari, via Marengo 2, 09123 Cagliari, Italy.

AbstractThis work introduces a Matlab library for the inversion of electromagnetic data collected by aground conductivity meter. Based on a nonlinear forward model used to describe the interactionbetween an electromagnetic field and the soil, the software reconstructs either the electrical con-ductivity or the magnetic permeability of the soil with respect to depth, by a regularized dampedGauss-Newton method. The package contains a graphical user interface which allows the user toexperiment with synthetic and real datasetsand different numerical strategies, in order to comparethem and draw conclusions.

References

[1] G.P. Deidda, P. Dıaz DeAlba, and G. Rodriguez, Identifying themagnetic permeabilityin multi-frequency EM data inversion, Electron. Trans. Numer. Anal., 47 (2017), pp.1-17.

[2] G. P. Deidda, P. Dıaz de Alba, G. Rodriguez, and G. Vignoli, Smooth and sparse in-version of EMI data from multi-configuration measurements, in 2018 IEEE 4th Inter-national Forum on Research and Technology for Society and Industry (RTSI) pages213-218, Palermo, Italy, September 2018. ISBN: 978-1-5386-6282-3.

[3] G. P. Deidda, C. Fenu, and G. Rodriguez, Regularized solution of a nonlinear problemin electromagnetic sounding, InverseProblems, 30:125014 (27 pages), 2014.

[4] P. Dıaz deAlbaand G. Rodriguez, Regularized inversion of multifrequency EM data inGeophysical applications. Book Trendsin Differential Equationsand Applications, vol-ume8 of SEMA SIMAI Springer Series, pages357-369. Springer, Switzerland, 2016.

34

Grey-Markov model for the prediction of theelectricity production and consumption in Mo-rocco

S. Elgharbi1, M. Esghir1, O. Ibrihich2, A. Abarda3, S. El Hajji1, S. Elbernoussi1

1Mohammed V University in Rabat, Faculty of Sciences, Laboratory of Mathematics, Computing and Applications -Information Security, Rabat, Morocco.2Sultan Moulay Slimane University in Beni Mellal, Ecole Nationale des Sciences Appliquees de Khouribga. Departementde genie reseaux et telecommunications, Morocco.3Hassan 1st University in Settat, FSJES. Laboratoire de Modelisation, Mathematiques et de Calculs economiques, Mo-rocco.

AbstractThe electricity planning of a country requires accurate models for the prediction of the electricityproduction and consumption. It is an important process which guarantees a better future of theenergy sector. In the field of predictive modeling and probabilistic forecasting, the Grey-Markovmodel is considered desirable since it gathers the GM(1,1) prediction model with the ability ofthe Markov chain to correct the errors related to the prediction. As an application, the historicaldata of the electricity production and consumption in Morocco from 1990 to 2017 was selected,and the predicted results from 2018 to 2030 show the efficiency of the proposed approach. Therelative error was reported to assure the great accuracy of the Grey-Markov model comparing tothe GM(1,1) model.

References

[1] L IU S. AND L IN Y. Grey Information : Theory and Practical Applications, AdvancedInformation and KnowledgeProcessing, Springer (2006).

[2] SUO R. The energy consumption prediction model based on Grey-Markov model,International Journal of Academic Research and Reflection Vol. 5, No. 5, (2017) 2309-0405.

[3] HU S. Prediction of city traffic accidentsbased on grey Markov chain model, Revistade laFacultad de IngenieriaU.C.V. Vol. 32, No. 4, pp. 144-151, 2017.

35

Extended nonsymmetric Global Lanczosfor matrix function approximation

Mohamed El Ghomari1, Abdeslem Hafid Bentbib2, Khalide Jbilou3

1Laboratory LAMAI, Faculty of Sciences and Technologies, Cadi Ayyad University, Marrakech, Morocco.E-mail: m.elghomari10@gmail.com2Laboratory LAMAI, Faculty of Sciences and Technologies, Cadi Ayyad University, Marrakech, Morocco.E-mail: a.bentbib@uca.ac.ma3Laboratory LMPA, 50 Rue F. Buisson, ULCO Calais cedex, France. E-mail: Khalide.Jbilou@univ-littoral.fr

AbstractThe extended Krylov subspace method is an attractive scheme for computing approximations ofmatrix functions and other problems producing a large scale matrices. In this work, we proposethe extended global bi-Lanczos method which uses short recursion relations for generating a bi-orthonormal bases for the extended Krylov subspaces Ke

m(A, V ) and Ke

m(AT ,W ). HereA is alarge nonsymmetric matrix and V,W ∈ R

n×s are block vectors. New algebraic properties of theproposed approach are developed. The proposed method is applied to compute the approximationof the trace(W T f(A)V ). Application to the approximation of the transfer function associatedto multi-input, multi-output linear dynamical system. Numerical examples are given to show theperformance of the extended global Lanczos.

References

[1] O. Abidi, M. Heyouni, and K. Jbilou, On some properties of the extended block andglobal Arnoldi methods with applications to model reduction, Numer. Algorithms, 75(2017), pp. 285–304.

[2] M. Bellalij, L. Reichel, G. Rodriguez, and H. Sadok, Bounding matrix functionals viapartial global block Lanczosdecomposition, Appl. Numer. Math., 94 (2015), pp. 127–139.

[3] V. Druskin and L. Knizhnerman, Extended Krylov subspaces: Approximation of thematrix square root and related functions, SIAM J. Matrix Anal. Appl., 19 (1998), pp.755–771.

[4] K. Jbilou, A. Messaoudi, and H. Sadok, Global FOM andGMRESalgorithmsfor matrixequations, Appl. Numer. Math., 31 (1999), pp. 49–63.

36

Analysis of stochastic models for daily pre-cipitation

Mounia El Hafyani1

1NR 334 Lot Ouled Mtaa, Temara, Morocco

AbstractPrecipitation is one of the key variables in water resource planning. The objective of this study isto provide an overview of published research in thedomain of stochastic modeling and simulationof daily precipitation.

We start by describing a variety of parametric and non-parametric models for single and mul-tisite daily rainfall data, which are widely developed; a well-chosen combination of both of theseapproaches is frequently used for complex system. Furthermore, space-time modeling works hasenhanced both of parametric and nonparametric approaches with promising results.

The strength and the weakness of each model are explored and directions for more develop-ment and advancement are provided.

References

1] D.S. Wilks, Multisite generalization of a daily stochastic precipitation generationmodel, J.Hydrol. 1998, 210, 178191.[2] Z. Li, F. Brissette, J. Chen, Finding the most appropriate precipitation probabilitydistribution for stochastic weather generation and hydrological modelling in Nordic wa-tersheds. Hydrol. Process. 2013, 27, 37183729.[3] E.M. Furrer, R.W. Katz, Generalized linear modeling approach to stochastic weathergenerators. Clim. Res. 2007, 34, 129144.[4] A. Verdin, B. Rajagopalan, W. Kleiber, R.W. Katz, Coupled stochastic weather gen-eration using spatial and generalized linear models. Stoch. Environ. Res. Risk Assess.2015, 29, 347356.[12] V. Nourani, M.T. Alami, M.H. Aminfar, A combined neural-wavelet model for pre-diction of Ligvanchai watershed precipitation, Eng. Appl. Artif. Intell. 22 (3), 2009,466-472.[15] J. P. Hughes, P. Guttorp, S.P. Charles, A non-homogeneoushidden Markov modelfor precipitation occurrence, Applied Statistics, in press,1999.[17] T .Lee, JD .Salas, Copula-based stochastic simulation of hydrological data appliedto Nile River flows. Hydrol Res 42:318330, 2011

37

Bi-Lanczos method for solving Sylvester ten-sor equation.

Smahane El-Halouy1, Abdeslem Hafid Bentbib1, El Mostafa Sadek2

1Laboratory LAMAI, University Cadi Ayyad, Marrakesh, Morocco2Laboratory labSIPE, ENSA, d’EL Jadida, University Chouaib Doukkali, Morocco

AbstractWe present the Bi-Lanczos method in tensor format to solve the following Sylvester tensor equa-tion:

X ×1 A1 + X ×2 A2 + · · ·+ X ×N AN = B (1)

whereAi ∈ RIi×Ii , i = 1, 2, · · · , N, arethecoefficient matrices, and thetensor B ∈ R

I1×I2×···×IN

is known and X ∈ RI1×I2×···×IN is the unknown tensor.

Bi-Lanczos method is proposed here on tensor format (Bi-Lanczos-BTF). Our goal is to combineBi-Lanczos-BTF with a low rank decomposition for tensors in order to reduce the dimension.Numerical experiments are presented.

References

[1] A. Beik, F. Panjeh, F. S. Movahed, On the Krylov subspace methods based on tensorformat for positivedefiniteSylvester tensor equations, Numer. Linear AlgebraAppl, 23,pp. 444–466. (2016).

[2] Kolda, T. G., Bader, B.W, Tensor Decompositionsand Applications. SIAM Rev. 51, pp.455–500 (2009).

38

Construction of a preconditoner for indefinitesaddle point problems using Riesz Map

Raghia El Hanine1, Said Raghay1, Hassane Sadok2

1 Laboratory LAMAI, Faculty of Sciences and Technologies University of Cadi Ayyad Marrakech, Morocco.2 LMPA Universite du Littoral Cote d’Opale, batiment H. Poincarre, 50 rue F. Buisson, 62280 Calais Cedex, France

AbstractTheobjectiveof thiswork isto describetheeffect of thedependenceof an algebraic preconditionerwith all parts of the resolution of a partial differential equations (PDEs) on the final numerical so-lution and present some numerical results related to Josef Malek and Zdenek Strakos’s work[1],using FEM. The algebraic preconditioner of a linear system is developed independently from dis-crete linear problem however, according to the work of Josef Malek and Zdenek Strakos we reacha conclusion that the algebraic preconditioner is related to the PDE’s analysis, and the infinite-dimensional formulation of the conjugate gradient method, thanks to the Riesz map.

References

[1] Malek, Josef, and Zdenek Strakos. Preconditioning and the conjugate gradient methodin the context of solving PDEs., Vol. 1. Siam, 2014.

[2] Liesen, Jorg, and Zdenek Strakos. Krylov subspace methods: principles and analysis.Oxford University Press, 2013.

39

Artificial neural network for forecasting oneday ahead of global solar irradiance

Khalid El Himdi1, Hamid Ettayyebi2

1Laboratory of Mathematics, Statistics and Applications, Department de Mathematics, Faculty of Sciences, MohammedV University in Rabat. Rabat, Morocco.2Laboratory of Mathematics, Statistics and Applications, Department de Mathematics, Faculty of Sciences, MohammedV University in Rabat. Rabat, Morocco.

RésuméDue to global warming, the world is seeking to use more renewable energy. In this study, we

focus on solar energy, which has been receiving increased amounts of attention in the last fewdecades. The integration of solar energy into electricity networks requires reliable forecast infor-mation of solar resources enabling it to quantify the available energy and allowing it to optimallymanage the transition between intermittent and conventional energies. Throughout our research,we investigated different forecasting techniques in order to find which one is appropriate for fore-casting the daily global solar irradiance for the region of Rabat.

The first-tested approach is linear modeling based on classical ARIMA-GARCH and expo-nential smoothing models. Thesecond approach proposes non-linear modeling based on ArtificialNeural Networks (ANNs) models. Numerous research has demonstrated the ability of ANNs topredict time series of weather data. In this study, we will examine a particular structure of ANNs,Multilayer Perceptron (MLP), which has been used the most among ANN structures in renewableenergy and timeseries forecasting broadly. Weused some statistical feature parameters to find theoptimal structure of MLP in the univariate case and the multivariate case. The results showed thatthe PMC with exogenous variables performed better than the other models.

40

Output stabilization and polynomial decay es-timate for time delay bilinear systems

A. El Houch1, A. Tsouli2, A. Attioui1

1Laboratory of Mathematics and Applications, ENS, Hassan II University, Casablanca, Morocco2Laboratory of Analysis Modeling and Simulation, ENSAM, Hassan II University, Casablanca, Morocco

AbstractIn this work, we discusses the output stabilization of distributed bilinear system with time delayr > 0 described as follows:

dz(t)

dt= Az(t) + v(t)Bz(t− r), t ≥ 0,

z(t) = ϕ(t), t ∈ [−r, 0],(1)

where z(t) is the state on a Hilbert space H endowed with the inner product 〈., .〉 and its corre-sponding norm ‖.‖. Furthermore, the linear operator A : D(A) ⊂ H −→ H generates a stronglycontinuous semigroup of contractions S(t) on H and ϕ ∈ C([−r, 0], H) is a given initial func-tion. While B is a bounded linear operator defined from H into itself, whereas t 7−→ v(t) is ascalar function which represents the control. Moreover, we assume that the system (1) is coupledwith the following observability equation:

y(t) = Cz(t), t ≥ 0, (2)

where y(t) is the output on a Hilbert space Y endowed with the inner product 〈., .〉Y and its cor-responding norm ‖.‖Y and C ∈ L(H,Y ) is a fixed output operator. Our objective is to propose acontinuous and bounded feedback control that guarantee both strong and weak output stabilizationof the system (1). In the case of output strong stabilization, a polynomial decay rate of the stabi-lized part isestimated. Theobtained resultsare illustrated by examplesand numerical simulations.

References

[1] OuzahraM (2013) Partial stabilization of semilinear systemsusing bounded controls. IntenationalJournal of Control 86:2253-2262

[2] Wu J (1996) Theory and Applications of Partial Functional Differential Equations. Springer,Berlin

41

Efficient preconditioning techniques and ac-celerated primal-dual active set method forlarge-scale constrained optimization problems

A. El Maliki

Hassan II University, ISO Laboratory , ENCG Casablanca, Beau site, B.P 2725 Ain Sebaa, Casablanca, Morocco(abdelmaliki@gmail.com)

AbstractWe present efficient and robust preconditioning techniques and iterative solvers combined withprimal-dual set method for solving large-scale optimization problems with equality and inequalityconstraints. The solution of the primal and dual variables are solved simultaneously with well-integrated projection processus. Our tools are the primal-dual active set method (or equivalently,semismooth Newton method [1]) for the inequality constraints, an approximate block triangularpreconditioner factorization of the KKT matrix and an acceleration technique based on Krylovsubspace solver. Webelieve that these ingredients are essential for a successful primal-dual activeset strategy for large-scale optimization problems. We focus on the numerical solution of three-dimensional elasticity problems with unilateral contact constraints. Some numerical results arepresented to show the potential, efficiency and accuracy of the proposed approach.

Keywords. Krylov subspace solver, block triangular preconditioner, primal-dual active setmethod, semismooth Newton method, Schur complement approximation.

References

[1] Hintermller M, Ito K, Kunisch K, The primal-dual active set strategy as a semismoothnewton method. SIAM J. Optim. 13 (3) (2002) 865–888.

[2] El Maliki A, Fortin M, Deteix J, Fortin A, Preconditioned iteration for saddle-point sys-temswith boundconstraintsarising in contact problems. Computer Methodsin AppliedMechanicsand Engineering, 254, (2013), 114–125

[3] Saad Y, Iterative Methods for Sparse Linear Systems. PWS Publishing Company:Boston, MA, 1996.

[4] Elman HC, Silvester D, Wathen AJ. Performanceand analysisof saddlepoint precondi-tioners for the discrete steady-state Navier–Stokes equations. Numerische Mathematik2002; 90:641–664.

42

Polynomial chaos decomposition for sensi-tivity analysis of the OCP slurry rheology

Nabil El Mocayd1, Driss Ouazar1, Mohammed Seaid2,3

1International Water Research Institute, Mohammed VI Polytechnic University 43150 Benguerir, Morocco2Complex System Engineering and Human Systems, Mohammed VI Polytechnic University 43150 Benguerir, Morocco3School of Engineering and Computing Sciences, Durham University, Stockton Road DH1 3LE Durham, United Kingdom

AbstractIn April 2014, theJorf Lasfar phosphateslurry pipeline is inaugurated asoneof theworld’s largestslurry pipelines, see for example [1, 2]. Starting from the head station at Khouribga, the slurry ispumped 187 km in small batches of phosphate separated by water to the terminal station at JorfLasfar. Design and maintenance of the slurry pipeline involve complex techniques and representa challenge for the OCP group in Morocco. Hence, the implementation of a slurry pipeline forphosphate transport would help the OCP to move towards a great achievement for sustainability.However, as the demand is increasing as well, the industry needs to improve its productivity. Asa result, it is urgent to fully understand the dynamics that occurs during the transport of the slurrypipelines. The purpose of the current work is to define all the physico-chemical parameters withsignificant influence on the rheology of the slurry. We use polynomial chaos decomposition forsensitivity analysisbased on thegood performance that thismodel in theprobabilistic context. Thefocus is on recent applications of polynomial chaos for uncertainty representation and sensitivityin dataanalysis. Thefundamental concept on which polynomial chaos representations arebased isto regard uncertainty asgenerating anew set of dimensions, and thesolution asbeing dependent onthesenew dimensions. A spectral decomposition in termsof orthogonal basisfunctions isused, theevolution of the basis coefficients providing quantitative estimates of the effect of random modeldata. Included in the present study are an development of the mechanics of polynomial chaosdecompositions, an illustration of various means of implementing these representations, and aperspective on theapplicability of thecorresponding techniques to model and quantify uncertaintyin the OCPslurry rheology.

References

[1] J. Rusconi, A. Lakhouaja, and M. Kopuz. The Design and Engineering of the 187 kmKhouribga to Jorf Lasfar Phosphate Slurry Pipeline. Procedia Engineering, 138:142–150, 2016.

[2] A. Alouani. PhosphateBeneficiation Development for Customers’ Satisfaction in Sus-tainable Development Way OCP case Khouribga - Jorf Lasfar. Procedia Engineering,138:95–103, 2016.

43

A multiple reduced gradient algorithm for lin-early constrained multiobjective optimization

Mustapha El Moudden1, El Arbi Abdellaoui Alaoui2,3, Said Agoujil3

1Department of Mathematics and Computer Science, Faculty of Sciences, Moulay Ismail University, Meknes, Morocco2EIGSI, 282 Route of the Oasis, Maarif, 20140 Casablanca, Morocco3E3MI Research Team, Department of Computer Science, Faculty of Sciences and Techniques at Errachidia, Universityof Moulay Ismail, Meknes, Morocco

AbstractLinearly constrained multiobjectiveoptimization hasasignificant number of real-lifeapplications.Under thismotivation, anew algorithm isdeveloped for solving multiobjectiveoptimization prob-lems with both linear constraints and bound constraints on the variables. At each iteration ofthe algorithm, the search direction is obtained by solving an appropriate quadratic subprogram.Bisection technique is used to find stepsizes. We prove that the proposed algorithm convergesto points that satisfy the Kuhn-Tucker first-order necessary conditions for Pareto optimality (thePareto-critical points). Thismethod is implemented in MATLAB, and applied on somebenchmarkproblemsto test theperformanceof thealgorithm and theresultsarecompared with thoseobtainedusing existing methods.

References

[1] M. EL MOUDDEN, A. EL GHALI, Multiple reduced gradient method for multiobjectiveoptimization problems. Numerical Algorithms, 79 (2018), 1257-1282.

[2] M. EL MOUDDEN, A. EL GHALI, A new reduced gradient method for solving lin-early constrained multiobjective optimization problems. Computational Optimizationand Applications, 71 (2018), 719-741.

44

Extending the multivariate homogeneous qd-algorithm to two-point case for the estimationof parametrized eigenvalues

A. Elidrissi1, J. Abouir2, B. Benouahmane3

1,2,3University Hassan II of Casablanca, Faculty of Sciences and Technology, Laboratory LMCMAN, BP 146 Mohammedia20650 Moroccoeli.abdennasser@gmail.com, abouir@hotmail.com, benouah@hotmail.com

AbstractThe univariate qd-algorithm has been first introduced by H. Rutishauser in to calculate the eigen-values of a tridiagonal matrix. It is linked also to the classical problem of finding the singularities(poles) of a meromorphic function from the coefficients of its series. The scalar case has beenstudied for example in [3]. This algorithm was discovered also for the construction of two-pointPade approximant or for related problems. The multivariate qd-algorithm was discovered for theconstruction of homogeneous Pade approximants. However, A. Cuyt has been studied this algo-rithm in multivariate case to tackle some multivariate problems [4]. The main interest is to extendthe qd-algorithm to two-point in the scalar and multivariate cases. The multivariate homogeneoustwo-point Pade approximants have been introduced recently by J. Abouir and B. Benouahmanein [1]. It is well-known that there is a close relationship between the orthogonal polynomials [2],the eigenvalues and theqd-algorithm. The denominators of homogeneous two-point Padeapprox-imants can be considered as orthogonal polynomials with respect to a linear functional, while theconnection between these concepts will be established in the two-point case. This work has in-troduce the extended multivariate homogeneous qd-algorithm, and it has studied the convergenceof this algorithm. A direct application is to determine eigenvalues of multivariate matrix by usingthis new version of qd-algorithm.

References

[1] Abouir, J., Benouahmane, B.: Multivariatehomogeneoustwo-point Padeapproximants,Jaen J. Approx. Volume10, Number 1-2, 29-48 (2018).

[2] Benouahmane, B., Cuyt, A.: Propertiesof multivariate homogeneousorthogonal poly-nomials, J. Approx. Theory 113, 1-20, (2001).

[3] Brezinski, C.: Pade-type Approximation and General Orthogonal Polynomials, ISNM50, Birkhauser Verlag, Basel, 67-105, (1980).

[4] Cuyt, A.: Symbolic-Numeric qd-algorithms with Applications in Function Theoryand Linear Algebra, Symbolic Algebraic Methodsand Verification Methods, Springer-Verlag Wien GmbH, pp. 35-54, (2001).

45

Efficient numerical methods for singularly per-turbed differential equations

Fevzi Erdogan1, Mehmet Giyas Sakar1

1Yuzuncu Yil University, Faculty of Sciences, Department of Mathematics, Van, Turkey

AbstractThe purpose of this study is to present different uniform finite difference schemes for numericalsolution of a second order singularly perturbed differential equation. We have constructed somedifference schemes for given problem. The difference schemes are shown to be uniformly conver-gent to the continuous solution with respect to the perturbation parameter. Numerical results forthe test problem are given using the presented methods.

References

[1] P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E.O’Riordan and G.I.Shishkin, Robust Com-putational Techniquesfor Boundary Layers, Chapman-Hall/CRC, New York, 2000.

[2] H.G. Roos, M. StynesandL. Tobiska, Numerical Methodsfor Singularly PerturbedDif-ferential Equations, Convection-Diffusion and Flow Problems, Springer Verlag, Berlin,1996.

[3] P. Das, Comparisonof apriori andaposteriori meshesfor singularly perturbednonlinearparameterized problems, J. Comput. Appl. Math., 290 (2015) 16-25.

[4] G. M. Amiraliyev and H. Duru, A uniformly convergent finite difference method fora singularly perturbed initial value problem, Appl. Math. Mech.(English Edition) no.4,20(1999)379-387.

[5] F. Erdogan, G.M. Amiraliyev, Fitted finite difference method for singularly perturbeddelay differential equations, Numerical Algorithms59 (1) (2012) 131-145.

46

New algorithm for computing Lagrange multivariate-interpolation polynomials

M. Errachid, A. Essanhaji, A. Messaoudi1LabMIA-SI, Centre regional des metiers de l’education et de la formation Avenue Allal Al Fassi, 10000 Rabat,e-mail : errachid.m@gmail.com2LabMIA-SI, Centre regional des metiers de l’education et de la formation Avenue Allal Al Fassi, 10000 Rabat,e-mail : abdelhakessanhaji@gmail.com3LabMIA-SI, Ecole Normale Suprieure, University Mohammed V in Rabat,e-mail : abderrahim.messaoudi@gmail.com

AbstractRecently Errachid et al. [2] presented a new algorithm for computing the Hermite interpolationalgorithm, for a general case, called Generalized Recursive Polynomial Interpolation Algorithm(GRPIA). In this work we will give a new formulation of the Lagrange two-variate polynomialinterpolation [3], a result of the existence and the uniqueness of this polynomial will be given.We will show that this polynomial can be expressed as a Schur complement [1,2]. So using theSylvester’s identity [2,4], we will give a new algorithm for computing this polynomial. Someproperties of this algorithm will also be discussed and a generalization of this algorithm will beproposed for the Lagrange multivariate-interpolation polynomial.

Keywords : Schur complement- Sylvester’s identity - Multivariate interpolation polynomial -Lagrange polynomial interpolation - Generalized Recursive Polynomial Interpolation Algorithm.

References

[1] C. Brezinski, Other manifestations of the Schur complement, Linear Algebra Appl. 111(1988) 231-247.

[2] M. Errachid, K. Jbilou, A. Messaoudi et H. Sadok, GRPIA: a new algorithm for com-puting interpolation polynomials. To appear in Numer Algor (2018).

[3] M. Gasca et T. Sauer, On the history of multivariate polynomial interpolation, J. Com-put. Appl. Math. (2003) 23-35.

[4] D. V. Ouellette, Schur complements and statistics, Linear Algebra Appl. 29 (1980) 687-706.

47

Numerical method for solving the two-dimensionalSchrodinger equation with irregular singularities

Safaa El Gharbi1, Mouna Essaouini2, Hassan Safouhi3, Said El Hajji4

1Faculty of Sciences, Mohammed V University in Rabat, Rabat, Morocco2Faculty of Sciences, Chouaib Doukkali University, El Jadida, Morocco3Faculte Saint-Jean, University of Alberta Edmonton (AB), Canada4Faculty of Sciences, Mohammed V University in Rabat, Rabat, Morocco

AbstractIn thelast decades, Sinc numerical methodshavebeen widely used for theresolution of differentialequations. They are considered desirable since they excel for problems whose solutions may havesingularities, or infinitedomains, or boundary layers.In the present work, we aim to solve the two-dimensional Schrodinger equation with various typeof potentials, in particular the perturbed anharmonic coulombic potential with irregular singulari-ties, using the Sinc collocation method. By the mean of a suitable conformal transformations, weshow the single and the double exponential convergence rateof the proposed approach.Numerical experiments are reported to illustrate the behavior and the efficiency of the double ex-ponential Sinc methods comparing to thesingle exponential Sinc methods when applying to solvethe two-dimensional Schrodinger equation.

References

[1] M. DEHGHAN, F. EMAMI-NAEINI , The Sinc collocation and Sinc-Galerkin methodsfor solving thetwo-dimensional Schrodinger equation with nonhomogeneousboundaryconditions, Applied Mathematical Modelling 37 (2013).

[2] P. GAUDREAU, R. M. SLEVINSKY AND H. SAFOUHI , The double exponential Sinccollocation method for singular Sturm-Liouvilleproblems, (2014).

48

Mixed finite element methods for the Oseenproblem

Mohamed Farhloul

Departement de mathematiques et de statistique, Universite de Moncton, N.B., E1A 3E9, Canada

AbstractThe Oseen problem can be obtained as a linearization of the Navier–Stokes equations. The mostpopular formulation in computational incompressible Newtonian flows is the velocity–pressureformulation. However, for various reasons, one may need also information on the dual variablesas velocity gradient. To do so, one need to build appropriate mixed formulations. The purpose ofthiswork is to develop and analyze mixed finiteelement methods for theOseen problem using thetensor gradient of velocity as a new unknown. We prove that the new variational formulation andthe corresponding Galerkin scheme are well-posed. Wealso provide optimal order error estimatesfor the velocity, the pressure and the gradient of velocity when each row of the velocity gradientisapproximated by Raviart–Thomas or Brezzi–Douglas–Marini elementsand thevelocity and thepressure are approximated by discontinuous piecewise polynomials. Moreover, for the numeri-cal implementation, a hybrid form is presented. Finally, we include numerical experiments thatsupport the theoretical results.

49

Photometric Stereo under unknown lights position.

A. Concas1, R. Dessı2, C. Fenu1, G. Rodriguez1, M. Vanzi3

1Department of Mathematics and Computer Science, University of Cagliari2General Motors Global Propulsion Systems3Department of Electrical and Electronic Engineering, University of Cagliari

AbstractA classical problem in Computer Vision consists in reconstructing the 3D shape of an object,starting from a set of images. Photometric Stereo technique is used to extract shape and colorinformation from an object which isobserved from thesamefixed point of view but under differentlighting conditions. We will describe an algorithm to approximate the framed object, treating, inparticular, the case when the position of the light sources is unknown. Various attempts havebeen made to estimate the lights position directly from the data; see, e.g., [1, 2] where a linearcombination of special functions(spherical harmonics) isemployed. Obtaining such aresult wouldrelease the constraint on the precisepositioning of the light sources, making thesurveying processmuch simpler. Numerical experiments will be illustrated, showing that the lights position can beestimated directly from the data when at least 6 pictures of the observed object are available.

References

[1] R. Basri, D. Jacobsand I. Kemelmacher, Photometric stereo with general, unknown light-ing, Int. J. Comput. Vis., 72 (2007) pp. 239-257.

[2] C. P. Chen and C. S. Chen, The 4-source photometric stereo under general, unknownlighting, Computer Vision-ECCV 2006, Springer (2006) pp. 72-83.

[3] A. Concas, R. Dessı, C. Fenu, G. Rodriguez, and M. Vanzi, Identifying the lights positionin photometric stereo under unknown lighting, arXiv:1809.06933 [math.NA], 2018.

[4] R. Mecca, and M. Falcone, Uniqueness and approximation of a photometric shape-from-shading model, SIAM J. Imaging Sci., 6 (2013) pp. 616-659.

50

Numerical methods for the computation of theabsorption inverse in graphs with absorptionwith applications

M. Benzi1, P. Fika2, M. Mitrouli3

1Scuola Normale Superiore, Pisa, Italy2University of Patras, Greece3National and Kapodistrian University of Athens, Greece

Abstract

Let L ∈ Rn×n be aLaplacian matrix that corresponds to aweighted, directed or undirected graph

G with anotion of absorption on itsnodes. In particular, let ussuppose that each nodeof thegraphis a transient state with transition ratedi > 0 to an absorbing state. Let d = [d1, . . . , dn]

T be thevector of the absorption rates. The absorption inverse of L with respect to d, denoted asLd, is aparticular 1-2 generalized inverseof graph LaplacianL, which providesawealth of informationon the structure of the underlying graph. In this work we discuss some interesting properties thatcharacterize the absorption inverse Ld and we develop and compare numerical methods for itscomputation and for the computation of other quantities associated with it, as well. Also, wediscuss and compare the centrality measures for ranking the nodes of graphs with absorption thatthe matrix Ld can provide.

References

[1] M. Benzi, A direct projection method for Markov chains, Linear Algebraand itsAppli-cations, 386 (2004), pp. 27–49.

[2] S. L. Campbell and C. D. Meyer, Generalized Inversesof Linear Transformations, Pit-man Publishing Ltd., London and San Francisco, 1979. Reprinted by Dover PublishingCo., New York, 1991.

[3] M. Benzi, P. Fika and M. Mitrouli, Graphswith absorption: numerical methods for theabsorption inverseand thecomputation of centrality measures, submitted.

[4] K. Jacobsen and J. Tien, A generalized inverse for graphs with absorption, Linear Al-gebraand itsApplications, 537 (2018), pp. 118–147.

51

Numerical approximation of an inverse prob-lem of image reconstruction

Anass Guitanou1, Chakir Tajani1

1Deaprtment of Mathematics, Polydisciplinary faculty of Larache, Morocco

AbstractImage reconstruction is the name given to the technique of completing deteriorated partsor fillingmissing parts of an image. This led us to present a model of the technique in partial differentialequations that we have solved by the finite element method. First, we will introduce the simpleharmonic reconstruction method. It is immediateand not very complicated to implement, howeverit fails to reconstruct images with sharp edges. Hence the insertion of a new model of reconstruc-tion using a non linear equation, in which we introduce the diffusion coefficient. In order to findthe latter, we solve an inverse problem using a numerical approximation based on the algorithmdeveloped by Kozlov, Maz’ya and Fomin called KMF. Several results and numerical simulationsare presented with FreeFem ++ software, showing the effectiveness of the algorithm.

References

[1] FREDRIK BERNTSSON, GEORGE BARAVDISH. Coefficient identification in PDEs ap-plied to image inpainting, 2014, Applied Mathematics and Computation, (242), 227-235.

[2] ANIS THELJANI. Partial differential equations methods and regularization techniquesfor image inpainting..Universit deHauteAlsace - Mulhouse, 2015.

[3] DIDIER AUROUX ,MOHAMED MASMOUDI. A one-shot inpainting algorithmbased on the topological asymptotic analysis,(25) 2016,251-267.

[4] LAMIA JAAFAR BELAID, AMEL BEN ABDA , NAWAL AL MALKI. The Cauchy Prob-lem for theLaplaceEquation and Application to ImageInpainting,ISRN MathematicalAnalysisVolume2011.

52

Approximate solutions to large nonsymmet-ric differential Riccati problems with applica-tions to transport theory

V. Angelova1, M. Hached2, K. Jbilou3

1Department of Intelligent Systems, Institute of Information and Communication Technologies, Akad. G. Bonchev, bl. 2,Sofia 1113, Bulgaria2Laboratoire P. Painleve UMR 8524, UFR de Mathematiques, Universite des Sciences et Technologies de Lille, IUT A,Rue de la Recherche, BP 179, 59653 Villeneuve d’Ascq Cedex, France3LMPA, 50 rue F. Buisson, ULCO Calais, France

AbstractIn our talk, we will be considering large scale nonsymmetric differential matrix Riccati equationswith low rank right hand sides. Thesematrix equationsappear in many applicationssuch ascontroltheory, transport theory, applied probability and others. We will show how to apply Krylov-typemethods such as the extended block Arnoldi algorithm to get low rank approximate solutions.The initial problem is projected onto small subspaces to get low dimensional nonsymmetric dif-ferential equations that are solved using the exponential approximation or via other integrationschemes such as Backward Differentiation Formula (BDF) or Rosenbrok method. We will alsoshow how these technique could be easily used to solve some problems from the well knowntransport equation. Some numerical experiments will be given to illustrate the application of theproposed methods to large-scale problems.

References

[1] E. J. DAVISON, M. C. MAKI ,, Thenumerical solution of thematrix Riccati differentialequation , IEEE Trans. Automat. Contr., Feb. 1973.

[2] Y. GULDOGAN, M. HACHED, K. JBILOU AND M. KURULAY, Low rank approximatesolutions to large-scaledifferential matrix Riccati equations, Appl. Math., to appear.

[3] K. JBILOU, Block Krylov subspacemethodsfor largecontinuous-timealgebraicRiccatiequations, Num. Alg., 34:339–353, 2003.

[4] L.-Z. LU, Solution form and simple iteration of a nonsymmetric algebraic Riccatiequation arising in transport theory, SIAM J. Matrix Anal. Appl., 26:679–685, 2005.

[5] P.H. PETKOV, N.D. CHRISTOV, AND M.M. KONSTANTINOV, Computational Meth-ods for Linear Control Systems, Prentice-Hall, Hemel Hempstead, 1991, ISBN 0-13-161803-2.

[6] H.H. ROSENBROCK Somegeneral implicit processesfor thenumerical solution of dif-ferential equations, J. Comput. , 5(1963), 329–330.

53

Stability and bifurcation analysis of non-linearepidemic models with vaccination and distributeddelays

Hajjami Riane1, Lahrouz Adil2, El Jarroudi Mustapha3

1Laboratory of Mathematics and Applications, Abdelmalek Essaadi University, FST Tangier, Morocco.2Laboratory of Mathematics and Applications, Abdelmalek Essaadi University, FST Tangier, Morocco.3Laboratory of Mathematics and Applications, Abdelmalek Essaadi University, FST Tangier, Morocco.

AbstractThe main purpose of this work is to qualitatively assess the impact of the vaccination and thevarying immunity period in the transmission of a disease governed by a general incidence rate.We first formulate an ODEs model with vaccination for susceptible individuals and temporaryimmunity where the sojourn time in the recovered compartment is assumed to be exponentiallydistributed. We show that the model has at most two steady states according to the value of thebasic reproduction number which plays theroleof a threshold between extinction and permanenceof the disease. We then introduce a DDEs model with distributed delay which takes into accountvarying immunity period. When the study is limited to the case of a varying immunity periodonly for vaccinated individuals, we show that the model dynamics depend on the probability ofimmunity lost. However, when the immunity acquired by vaccination is permanent while the oneacquired by infection is temporal and the probability density of losing immunity is modeled bythe weak delay kernel, the behavior of the obtained model is independent of the probability ofimmunity lost and the endemic level increases as the the mean delay decreases. Furthermore,for the strong delay kernel, by choosing time delay as bifurcation parameter the model is shownto undergo a Hopf-bifurcation at the endemic steady state under certain conditions. Numericalsimulations aregiven to support the theoretical analysis.

References

[1] A. Lahrouz, L. Omari, D. Kiouach, A. Belmaati , Completeglobal stability for an SIRSepidemic model with generalized non-linear incidenceand vaccination, Applied Math-ematicsand Computation 218 (2012) 6519.

[2] K.B. Blyuss, Y.N. Kyrychko, Stability and bifurcationsin an epidemicmodel with vary-ing immunity period, Bulletin of Mathematical Biology 72 (2010) 490.

54

The extended rational block Arnoldi methodfor model reduction in large-scale dynamicalsystems

O.Abidi1, M.A.Hamadi2, K.Jbilou3

1Universite du Littoral, Cote d’Opale, batiment H.Poincarre,50 rue F. Buisson, F-62280 Calais Cedex, France2Universite Mohammed VI Polytechnique, Ben Guerir, Morocco3K.JbilouUniversite du Littoral, Cote d’Opale, batiment H.Poincarre,50 rue F. Buisson, F-62280 Calais Cedex, France. E-mail: jbilou@lmpa.univ-littoral.fr

AbstractIn this talk, we propose a new block Krylov-type subspace method for model reduction in largescale dynamical systems. We project the initial problem onto a new subspace combined by theRational and the standard block Krylov subspaces. Then we get lower order dynamical systemsapproaching the original ones. We give some algebraic properties such as expressions of the errorbetween the original and reduced transfer functions. Numerical experiments are given for somebenchmark examples of large-scale linear dynamical systems.

References

[1] V. Druskin, V. Simoncini, Adaptiverational Krylov subspacesfor large-scaledynamicalsystems. Syst. Contr. Lett., 60(2011), 546–560.

[2] S. Gugercin, A.C. Antoulas and C. Beattie, A rational Krylov iteration for optimal H2

model reduction. J. Comp. Appl. Math., 53(2006), 1665–1667.

[3] OAbidi, M HachedandK. Jbilou, A global rational Arnoldi methodfor model reductionJournal of Computational and Applied Mathematics, 325, 175-187, 2017

[4] A. H. Bentbib, K. Jbilou and Y. Kaouane, A computational global tangential Krylovsubspace method for model reduction of large-scale MIMO dynamical systems. J. Sci.Comput., 75(2018), 1614–1632.

55

Estimation of the relative error in regressionanalysis under random left-truncation model

Hamrani, Farida1

1Faculty of Mathematics, USTHB,Algiers, Algeria

AbstractLet Y be a real random variable of interest and X an Rd-valued random vector of covariates. Wewant to estimate Y after observing X. There are several ways to estimate it and one of the mostpopular is the regression method which modeled by the following consideration: Y = m(X) + ǫ,wherem is the unknown regression function and ǫ is a random error variable.Classicaly, the regression function m is estimated by using the mean squared error as a loss func-tion. However, this loss function is very sensitive to outliers. One of the techniques we can use toovercome thisproblem is to use alternative loss function based on the squared relativeerror. Rela-tive error estimation has been recently used in regression analysis (see Park and Stefanski (1998),Jones et al.(2008)). This technique is useful in analyzing data with positive responses, such as lifetimes which we interest in this work. Another particularity of the life times is that are often notobserved completely. Censored and truncation are the most current forms of the incomplete data.In thiswork weare interested in the left truncated data, wheretheobservation (X,Y ) is interferedby another independent rv T such that all threerandom quantitiesY,X, andT areobservable onlyif Y ≥ T . Thismodel isoriginally appeared in astronomy (woodroofe (1985)), then extend to sev-eral domains as economics, epidemiology, demographics, actuarial. When we use the least squareerror as a loss function to determine the regression function m, Ould Saıd and Lemdani (2006)built a kernel type estimator of m(x) which take into account the truncation effect. Following thesame arguments, wedefine thekernel estimator of the truncated relativeerror regression of m andwe study itsasymptotic proprieties. Wegive also illustrationsof the results on simulated data.

References

[1] Jones, M. C., H. Park, K. L. Shin, S. K. Vines, and S. O. Jeong. 2008. Relative errorprediction viakernel regressionsmoothers. Journal of Statistical Planning and Inference138:288798.

[2] Ould Saıd, E. Lemdani, M. (2006). Asymptoticpropertiesof anonparametricregressionfunction estimator with randomly truncated data. Ann.Inst.Statist.Math. Vol 58:357-378.

[3] Park, H., Stefanski, L.A., 1998. Relative-error prediction. Statist. Probab. Lett. 40,227236.

[4] Woodroofe, M. (1985). Estimating a distribution function with truncated data. Ann.Statist. Vol 13:163-177.

56

The BCMRH method for multiple linear sys-tems

I. Abdaoui1, L. Elbouyahyaoui2, M. Heyouni1

1Laboratoire LM2N. Equipe MSN, ENSA, Universite Mohammed Premier, Oujda, Maroc2Centre Regional des Metiers de l’Education et de la Formation, Fes, Maroc

AbstractIn thiswork, weare interested in methodsbased on theblock Hessenberg process and weconsiderthe block CMRH (BCMRH) method introduced in [1] for solving linear systems of the form

AX = B, (1)

whereA isanonsingular matrix of order n andB, X are rectangular matricesof dimension n× r

with r largely smaller than n, i.e., r ≪ n. Considering the fact that block CMRH shares muchsimilarity with the block GMRES method [3], with less arithmetic and storage requirements, andinspired by the numerical performance of the simpler GMRES (SGMRES) method proposed fora single linear system (cas r = 1) by Walker and Zhou in [4] and generalized to the block case(r 6= 1) by Liu and Zhong in [2], wepropose anew implementation of theblock CMRH algorithmthat will be called the simpler block CMRH (SBCMRH). The proposed method is less expensivein terms of computational needs since that the block Hessenberg process is reformulated to startwithAR0 instead of R0 and this leads to a triangular linear system easier to solve. Moreover, thesimpler version of the block CMRH method allows to check the convergence within each cycleof the block Hessenberg process by using a recursive relation that updates the residual at eachiteration. Similarly, this new implementation avoids the use of Givens rotations to solve the leastsquares problem produced by the quasi-minimization condition.

References

[1] M. ADDAM , M. HEYOUNI , H. SADOK. The block Hessenberg process for matrix equa-tions. Electronic Transactionson Numerical Analysis, 46 (2017), pp. 460–473.

[2] H. L IU, B. ZHONG, Simpler block GMRES for nonsymmetric systems with multipleright-hand sides. Electronic Transactionson Numerical Analysis, 30 (2008), pp. 1–9.

[3] B. V ITAL, Etudedequelquesmethodesde resolution deproblemes lineairesde grandetaille sur multiprocesseur, Phd thesis, UniversitedeRennes I.

[4] H. F. WALKER AND L. ZHOU, A simpler GMRES. Numerical Linear Algebra withApplications, 1(6) (1994), pp. 571–581.

57

Bayesian inference for a continuous time hy-brid switching diffusion process

El Houcine Hibbah1, Hamid El Maroufy1, Christiane Fuchs2

1Department of Mathematics, Faculty of Sciences and Technics, Sultan Mouly Slimane University, Morocco.2Biostatistics Group, Institute of Computational Biology, Helmholtz Zentrum Munchen, Germany

AbstractWe adopt a Bayesian approach to estimate a hybrid switching diffusion process (HSD); where thediffusion component is dependent on the switching regime and the transition rates of the regimeswitching are dependent on the diffusion observations. Since in reality phenomena are only ob-served in discrete times, Data imputation is called for to create more observations so as to havegood approximations for the density of the diffusion process. Consequently, we will alternatebetween the estimation of the latent imputed observations, the regime switching states and the pa-rameters of the models in a Bayesian context. The latent imputed data is updated in block using aMetropolis hastings algorithm. The switching states are computed by an adaptation of a forwardfiltering backward smoothing algorithm to aHSD model. Theparameters areestimated after priorspecifications and conditional posterior densities formulation.

References

[1] Adesi, G.B., Rasmussen, H., and Ravanelli, C. AN OPTION PRICING FORMULA FOR

THE GARCH DIFFUSION MODEL, Computational Statistics & Data Analysis. 49(2)287-310 (2005).

[2] GOLIGHTLY, A. AND WILKINSON, D. Bayesian Inference for Nonlinear MultivariateDiffusion ModelsObserved with Error. Computational Statistics and Data Analysis 521674-1693 (2008).

[3] GOUTTE, S., ISMAIL , A. AND PHAM , H.. Regime-switching stochastic volatilitymodel: estimation and calibration to VIX options. Applied Mathematical Finance 24(1)38-75 (2017).

[4] SORWAR, G. Estimating single factor jump diffusion interest rate models. Applied Fi-nancial Economics. 21:22 1679-1689, DOI: 10.1080/09603107.2011.591729(2011).

[5] Y IN, G. AND ZHU, C. Propertiesof solutionsof stochastic differential equationswithcontinuousstate-dependent switching. J. Differential Eqs. 249 24092439 (2010) .

58

Properties of the correlation matrix impliedby a recursive path model

Zouhair El Hadri1, Mbarek Iaousse2, Mohamed Hanafi3, Pasqual Dolce4,Yousfi El Kettani2

1Department of Mathematics, Faculty of Sciences, Mohamed V University, Rabat, Morocco2Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, Morocco3Oniris, StatSC, Nantes, France4Department of Public Health, University of Naples Federico II, Naples, Italy

AbstractPath analysis isaset of statistical techniques used to examine the relationships of cause and effectbetween a set of variables using the correlations between variables [1-2-3]. The implied correla-tion matrix can be computed in two different methods: Joreskogs method [4] and Finite Iterativemethod [3]. In this communication, we present the properties of the last method which allowus to do explicit computations. We have announced and demonstrate these properties and givesome examples of illustration. In addition, we have implemented an R program to do simulatedbenchmarking between thetow methods: weused lavaan packagefor Joreskogsmethod, whilethemethod based on these properties is programmed. The results show that both methods are practi-cally identical in termsof estimations, with theadvantage of last method due to itsproperties usedin the estimation in the sens that the estimated correlation matrices conserve theire characteristicsusing theFinite IterativeMethod (elements of thediagonal areequal to 1 and elements outside thediagonal are between -1 and 1 unlike Joreskog’s method).

References

[1] Bollen, K. A. (1989). Structural equationswith LV. New York: Wiley.

[2] Duncan, O. D. (1966). Path analysis: Sociological examples. American Journal of So-ciology, 72, 1-16.

[3] El hadri, Z and Hanafi, M. (2015). TheFiniteIterativeMethod for calculating thecorre-lationmatrix impliedby arecursivepathmodel. ElectronicJournal of AppliedStatisticalAnalysisVol. 08, Issue01, pp 84-99.

[4] Joreskog, K. G. (1970). A general method for the analysis of covariance structures.Biometrica, 57, 239251.

59

Numerical approximation of an inverse bound-ary coefficient problem

Bouchta Jouilik1, Chakir Tajini2, Jaafar Abouchabaka1

1Faculty of sciences of Kenitra, Morocco2Polydisciplinary faculty of Larache, Morocco

AbstractCorrosion is a complex natural phenomenon, it has always been a major industrial problem. De-spite the scientific breakthroughs and technological advances accumulated over the last decades,corrosion still causes a great deal of damage and affects many areas. This industrial evil can takevarious forms ranging from simple uniform corrosion to more complex aspects, encountered insevere industrial environments ....In this work, we are interested in the problem of detection of corrosion which can be modeledus an inverse problem which consists to identifying a parameter called ”Robin coefficient” repre-senting the damage corrosion in the inaccessible part of the boundary. We proposed an iterativealgorithm to resolve thisproblem and westudied thestability of thesolution of the inverse bound-ary coefficient problem which is a crucial issue for numerical resolution which is investigated byseveral authors. Numerical results are presented showing the effectiveness and the stability of theproposed algirithm.

References

[1] P. KAUP AND F. SANTOSA, Nondestructiveevaluationof corrosion damageusing elec-trostatic measurements. J. Nondestruct. Eval., 14 (1995), 127-136.

[2] S. CHAABANE, C. ELHECHMI , M. JAOUA, A stable recovery method for the Robininverse problem. Math. Comput. Simulation, 66 (2004), 367-383.

[3] C. TAJANI , B. JOUIL IK , J. ABOUCHABAKA, Numerical identification of robin coeffi-cient by iterativemethod. Palestine journal of Mathemtics, 7(1) (2018), 64-72.

60

A block tangential Lanczos method for modelreduction of large-scale first and second or-der dynamical systems

K. Jbilou1, Y. Kaouane2

1Universite du Littoral, Cote d’Opale, Laboratoire LMPA, Calais , France, email: Khalide.Jbilou@univ-littoral.fr2Universite du Littoral, Cote d’Opale, Laboratoire LMPA, Calais , France, email: yassine.kaouane@etu.univ-littoral.fr

AbstractIn this work, we present a new approach for model reduction in large scale first and second orderdynamical systems with multiple inputs and multiple outputs (MIMO). This approach is basedon the projection of the initial problem onto tangential Krylov subspaces to produce a simplerreduced-order model that approximates well the behaviour of the original model. We present analgorithm named: Adaptive Block Tangential Lanczos-type (ABTL) algorithm. We give somealgebraic properties and present some numerical experiences to show the effectiveness of the pro-posed algorithms.

References

[1] V. Druskin and V. Simoncini. Adaptive rational Krylov subspaces for large-scale dy-namical systems. SystemsControl Lett., 60(8), 546–560, 2011.

[2] [4] V. Druskin, V. Simoncini, and M. Zaslavsky. Adaptive tangential interpolation inrational Krylov subspaces for MIMO dynamical systems. SIAM J. Matrix Anal. Appl.,35(2), 476–498, 2014.

61

A semi-Lagrangian finite element conjugategradient method for three-dimensional incom-pressible flows

Bassou Khouya1, Mofdi El-Amrani2, Mohammed Seaid3,1

1Mohammed VI Polytechnic University, Lot 990 Hay Moulay Rachid, Benguerir 43150, Morocco2Faculty of sciences and technics of Tangier, Abdelmalek Essady University, Morocco3School of Engineering and Computing Sciences, Durham University, South Road, Durham DH1 3LE, UK

AbstractAnalysis and implementation of a class of semi-Lagrangian finite element methods for solvingtwo-dimensional incompressible flows have been investigated in [1, 2] among others. In the cur-rent study we present an extension of these techniques to three-dimensional flow problems us-ing unstructured grids formed with tetrahedral elements. The method solves the incompressibleNavier-Stokes equations in the primitive-variable form and it consists of coupling a second-orderbackward time discretization based on the characteristics method with a spatial discretization offinite element type. The method combines advantages of the semi-Lagrangian method to accu-rately solve the convection-dominated flow problems with a finite element method for space dis-cretization of the governing equations. It can be interpreted as a fractional-step technique wherethe transport part and the Stokes part are treated separately. The implementation of the proposedGalerkin-characteristic method differs from its Eulerian counterpart in the fact that it is appliedduring each time step, along the characteristic curves rather than in the time direction. Therefore,due to the Lagrangian treatment of convection, the standard Courant-Friedrichs-Levy conditionis relaxed and the time truncation errors are reduced in the Stokes part. To solve the generalizedStokesproblem weimplement aconjugategradient algorithm. Thismethod avoidsprojection tech-niques and does not require any special correction for the pressure. The focus is on constructingefficient algorithms with a large stability region to solve three-dimensional incompressible flowproblems. We verify the method for well-established benchmark problems including lid-drivencubic cavity and flow around a cylinder. We also present numerical results for a problem of slurryflow in pipelines. The proposed semi-Lagrangian finite element conjugate gradient method hasbeen found to be feasible and satisfactory.

References

[1] M. El-Amrani, and M. Seaid. A conjugategradient algorithm for solving theGalerkin-characteristic approximation of interfacial flows. Applied Numerical Mathematics,62:1197–1214, 2012.

[2] M. El-Amrani, and M. Seaid. Convergence and stability of finite element modifiedmethod of characteristics for the incompressible Navier-Stokes equations. Journal ofNumerical Mathematics, 15:101–135, 2007.

62

A multi-level adaptive mesh refinement methodfor the simulation of flow and transport in het-erogeneous porous media

Hanen Amor1, Fayssal Benkhaldoun2, Imad Kissami1

1IRSN / PSE-ENV / SEDRE / UEMISBP 17 - 92262 Fontenay-aux-Roses Cedex, Francehanen.amor,imad.kissami@irsn.fr2LAGA, Universite Paris 1399 Av. J.B. Clement 93100 Villetaneuse, Francefayssal@math.univ-paris13.fr

AbstractThis paper presents a multi-level adaptive mesh refinement strategy based on an a posteriori errorestimator implemented within the MELODIE software developed by IRSN to assess the safetyof deep geological disposal for high level and intermediate long-lived radioactive waste. Indeed,numerical simulations reproducing the transfer of radionuclides throughout the disposal facilitiesand geological formations must take into account thediscrepancies within thiscomplex system, intermsof geological layersand heterogeneities (faults) or of thevariousgeometrical scales to tackle(from cm to km). The solute transport ismodeled by asystem combining the Darcy flow equationand the diffusion-convection concentration equation. This work consist in developing a multi-level local refinement (built on vertex bisection method) and coarsening of the mesh elements intwo- and three-dimensional cases. This method is based on a posteriori error estimator, derivedthanks to an H(div)-conforming flux reconstruction. This method is then illustrated with 2D and3D semi-realistic heterogeneous cases.

References

[1] Mathieu, G., Dymitrowska, M., Bourgeois, M.: Modeling of radionuclide transportthrough repository components using finite volume finite element and multidomainmethods. Physicsand Chemistry of theEarth, PartsA/B/C 33, S216–S224 (2008)

[2] Vohralık, M.: Guaranteed and fully robust a posteriori error estimates for conformingdiscretizationsof diffusion problemswith discontinuouscoefficients. Journal of Scien-tific Computing 46(3), 397–438 (2011)

[3] Ern, A., Vohralık, M.: A posteriori error estimation based on potential and flux recon-struction for the heat equation. SIAM Journal on Numerical Analysis 48(1), 198–223(2010)

[4] AmazianeB., BourgeoisM., El Fatini M.: AdaptiveMesh Refinement for a Finite Vol-umeMethod for Flow and Transport of Radionuclidesin HeterogeneousPorousMedia.Oil GasSci. Technol. 69(4), 4, 687–699 (2013)

63

A variational method for solving two-dimensionalBratu’s problem

R. Akhrif1, A. Kouibia2, M. Pasadas3

1Faculte des Sciences Juridiques et Economiques et Sociales de Tetouan, Maroc.2Dept. Applied Mathematics, Faculty of Sciences, University of Granada, Spain.3Dept. Applied Mathematics, Faculty of Sciences, University of Granada, Spain.

AbstractThe non-linear boundary problems occur in engineering and science, including of diffusion pro-cesses of the chemical reactions and heat transfers. Particularly, Bratu Problem appears in a largevariety of applications areas such as the fuel ignition model of thermal combustion, radiative heattransfer, the Chandrasekhar model of the expansion of the universe, chemical reactor theory andnanotechnology [3, 4, 1, 2]. In [1] a summary of the history such kind of problem isgiven.

The two-dimensional Bratu Problem is an elliptic partial differential equation with homoge-neous Dirichlet boundary conditions. Thisproblem isgiven by

∆u+ λeu = 0, in Ω = (0, 1)× (0, 1),u = 0 on ∂Ω,

(1)

whereλ > 0 andΩ isabounded domain withboundary ∂Ω. Theproblem isanonlinear eigenvalueproblem that iscommonly used as a test problem for many numerical methods.

In this work we present a variational approximation method for solving Bratu’s problem. Weformulate the problem and we prove some characterization of the approximate solution of theproblem. Moreover we construct a sequence of approximate solutions of the problem from theknots of a domain partition. We finish this paper showing some graphic and numerical examplesin order to prove the effectiveness and the useful of our method.

References

[1] J. Jacobsen, K. Schmitt, The Liouville.Bratu.Gelfang problem for radial operators, J.Differential Equations184 (2002) 283–298.

[2] R. Jalilian, Non polynomial splinemethod for solving Bratu’s problem, Comput. Phys.Comm. 181 (2010) 1868-1872.

[3] A. Mohsen, A simple solution of the Bratu problem, Computersand Mathematicswithapplications67 (2014) 16–33.

[4] Y.Q. Wan, Q. Gou, N. Pan, Thermo.electro.hydrodynamic model for electrospinningprocess, Int. J. Nonlinear Sci. Numer. Simul. 5 (2004) 5–8.

64

Design of numerical method with adaptivelytransformed Chebyshev nodes for solving 1Dproblem of non-isothermal polymeric fluid flow∗

Boris Semisalov1,2, Ekaterina Kruglova1,2, Alexander Blokhin1,3

1Novosibirsk State University, Novosibirsk, Russia2Institute of Computational Technologies SB RAS, Novosibirsk, Russia3Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia

AbstractThe activedevelopment of 3D printing using polymer materialscauses aneed in development andanalysis of new mathematical models describing with high reliability the flow of solutions andmelts of polymers in the channels of printing devices. The work [2] is devoted to solving thisproblem on the basis of mesoscopic approach (see [1]). It includes the development of modifiedrheological Pokrovskii–Vinogradov model that takesinto account thermo-mechanical propertiesofpolymeric liquids. In [2] thecaseof steady-statenonisothermal flowsof apolymeric fluid betweentwo coaxial cylinders was studied. Using pseudo-spectral Chebyshev method the solution wasobtained for extremely small values of the relative radius of inner cylinder, wherein the methodshowed geometric rate of convergence. The authors found that the singularity of the solution islocated in the complex plane, and its position depends on the radius r0 of the inner cylinder.

According to ideasof [3], in thiswork numerical algorithm from [2] wasmodified to take intoaccount the exact location of the singularity. The modified algorithm uses the rational barycentricinterpolants to approximate analytic functions. The locations of the grid points are adapted to sin-gularities of the desired solution, which areapproximated by the locations of poles of Chebyshev-Padeapproximants. Thisapproach helps to increasecardinally therateof convergence in problemswith small r0. Thedependence of thesingularity position on thevalueof theproblem’sparametersis investigated. The graphs of the solution obtained in [2] and by the new algorithm are presented.

References

[1] Pokrovskii V. N. The Mesoscopic Theory of Polymer Dynamics, 2nd ed., Springer, Berlin, 2010.

[2] Blokhin A.M., Kruglova E.A., Semisalov B.V. Estimation of Two Error Components in the Nu-merical Solution to the Problem of Nonisothermal Flow of Polymer Fluid between Two Coax-ial Cylinders // Computational Mathematics and Mathematical Physics, 2018, vol. 58. No. 7, P.1099–1115.

[3] T. W. Tee, Lloyd N. Trefethen. A Rational Spectral Collocation Method with Adaptively Trans-formed Chebyshev Grid Points // SIAM J. Sci. Comput., 28(5), P. 1798-1811.

∗Two of the authors of the paper, Boris Semisalov and Ekaterina Kruglova, would like to acknowledge RussianScience Foundation (project No. 17-71-10135) for financial support.

65

Algorithms for the Stokes flow with stick-slipboundary conditions

T. Brzobohaty1, J. Haslinger2, M. Jarosova1, R. Kucera1, V. Satek1

1IT4Innovations, VSB-TU Ostrava, Ostrava, CZ2Department of Numerical Mathematics, FMP, Charles University, Prague, CZ

AbstractThe contribution deals with the Stokes problem when the slip of a fluid along the wall may oc-cur if the shear stress attains certain a-priori given bound [6]. The mathematical model of thevelocity-pressure formulation leads to the variational inequality of the second kind. Our finite el-ement approximation that uses the P1-bubble/P1 finite elements [2] is combined with the TFETIdomain decomposition method [1]. Thedual algebraic problem arising after theelimination of thevelocity component leads to the minimization of the quadratic, strictly convex function in termsof four Lagrange multipliers representing the gluing condition, the pressure solution component,the impermeability condition on the slip part of the boundary, and the stick-slip condition. Thefourth Lagrange multiplier is subject to box constraints and, due to the use of the TFETI method,all Lagrange multipliers have to satisfy linear equality constraints. The solution is computed bya path-following variant of the interior-point method [3] adapted for box and linear equality con-straints[4, 5]. Themain ideaof thepath-following algorithm based on thedamped Newton methodwill be presented. The inner linear systems are solved by the preconditioned projected conjugategradient method for which we will show the spectral analysis of preconditioning.

References

[1] Dostal Z, et al.: Total FETI - an easier implementable variant of the FETI method fornumerical solution of elliptic PDE, Comm Numer Meth Eng, 22:1155–1162, 2006.

[2] Koko J: Vectorized Matlab codes for the Stokes problem with P1-bubble/P1 finite ele-ment, 2012. Onlineat: http://www.isima.fr/∼jkoko/Codes/StokesP1BubbleP1.pdf.

[3] Kucera R, et al.: An interior point algorithm for the minimization arising from 3Dcontact problemswith friction, Optim Method Softw, 28:1195–1217, 2013.

[4] Kucera R, Haslinger J, Satek V, Jarosova M: Efficient methods for solving the Stokesproblem with slip boundary conditions, Math Comput Simul, 145:114–124, 2018.

[5] Brzobohaty T, Jarosova M, Kucera R, Satek V: Path-following interior point method:theory and applications for the Stokes flow with a stick-slip boundary condition, AdvEng Softw, 2018. Onlineat: https://doi.org/10.1016/j.advengsoft.2018.06.010.

[6] Rao I J, Rajagopal K R: The effect of the slip boundary condition on the flow of fluidsin achannel, Acta Mechanica, 135:113–126, 1999.

66

From space frequency representation to theanalysis of 2D-images

Lamzouri Fatima Zahra1, Bouali bouchta2

1University Mohamemd the first Oujda2University Mohamemd the first Oujda

AbstractIn this paper, we extend the work of Gonzalo Galiano and Julien Velasco [1], which focuses onthe analysis of one-dimensional signals by exploiting the theory of image processing of time-frequency representations. Therefore, we generalize the work of G. Galiano and J. Velasco forstudying multi-dimensional signals through their space frequency representations, which is thewigner ville distribution. Indeed, we show that the wigner ville distribution of 2D images is a4-D space frequency representation which can be considered as a color image. In this way, wecould analyze two-dimentional images by exploiting the theory of color images processing ofspace-frequency representation, in order to find applications, which can serve medicine domain(synthetic and biomedical images).

References

[1] G.Galianoand J. Velasco, On anonlocal spectrogramfor denoisingone-dimensionalsignals, Appl. Math. Comput. vol. 244, pp. 859869, 2014

67

Entropy stable discontinuous Galerkin finiteelement methods with multidimentional slopelimitation for hyperbolic systems of conser-vation laws

Aziz Madrane1

1McGill University 845 Sherbrooke St W, Montreal, QC H3A 0G4, Canada

AbstractWe present an entropy stable discontinuous Galerkin (DG) finite element method to approximatesystems of 2-dimensional symmetrizable conservation laws on unstructured grids [2],[3] . Thescheme is constructed using a combination of entropy conservative fluxes and entropy-stable nu-merical dissipation operators. Themethod isdesigned to work on structured aswell ason unstruc-tured meshes.The variables are approximated by discontinuous piecewise linear or bilinear.Thenumerical scheme has two steps: thefirst step isafiniteelement calculation which includes calcu-lations of fluxes across the edges of theelements using 1-D entropy stablesolver. The second stepis a procedure of stabilization through a truly multidimensional slope limiter. We compared themethod versusRoe’ssolversassociated with entropy corrections [1] and Osher’ssolver.Special at-tention isgiven to piecewise linear approximation which isshown to betotal variation diminishingand convergent. The method is illustrated by computing solution of the two stationary problems:a regular shock reflection problem and a2-D flow around a double ellipse at high Mach number.

References

[1] A. MADRANE, E.TADMOR, Entropy stability of Roe-typeupwind finite volumemeth-ods on unstructured grids, in Hyperbolic Problems: Theory, Numerics, Applications,Proceedingsof the 12th International Conference in 2008, Vol. 67(2) (E. Tadmor, J.-G.Liu and A. Tzavaras, eds), AMS Proc. Symp. Applied Math., University of Maryland(2009) 775-784

[2] A. MADRANE, U. S. FJORDHOLM , S. M ISHRA , E. TADMOR , Entropy conservativeand entropy stable finite volume schemes for multi-dimensional conservation laws onunstructured meshes, ECCOMAS 2012, Proc. J. Eberhardsteiner et al. (eds.), Vienna,Austria, September 10-14, 2012.

[3] A. MADRANE, S. M ISHRA , E. TADMOR , Entropy conservative and entropy stablefinite volume/finite element schemes for the Navier-Stokes equations on unstructuredmeshes, ECCOMAS2014, Proc. July 2025, 2014, Barcelona, Spain.

68

The generalized finite volume SUSHI schemefor the discretization of the peaceman model

Ouafa Soualhi1, Mohamed Rhoudaf2, Mohamed Mandari3

O. SOUALHI,Department of Mathematics,University Moolay ismail, Faculty of sciences Meknes,Morroco.

M. Rhoudaf,Department of Mathematics,University Moolay ismail, Faculty of sciences Meknes,Morroco.

M. Mandari,Department of Mathematics,University Moolay ismail, Faculty of sciences Meknes,Morroco.

AbstractIn thispaper, weprovetheconvergenceof aschemausing stabilisation and hybrid interfacesof par-tial differential equations describing miscible displacement in porous media. This system is madeof two coupled equations: an anisotropic diffusion equation on the pressure and a convection-diffusion-dispersion equation on theconcentration of invading fluid. Theanisotropic diffusion op-erators in both equations requirespecial carewhilediscretizing by afinitevolumemethod SUSHI.Later, we present some numerical experiments.

References

[1] C. CHAINAIS-HILLAIRET, J. DRONIOU, Convergenceanalysisof a mixedfinitevolumescheme for an elliptic-parabolic system modeling miscible fluid flows in porous media.SIAM J. Numer. Anal., 45(5):2228-2258(electronic), 2007.

[2] C. CHAINAIS-HILLAIRET, S. KRELL , A. MOUTON, ” Convergence analysis of aDDFV scheme for a system describing miscible fluid flows in porous media. Numeri-cal Methodsfor Partial Differential Equations, Wiley, 2014, pp.38.

[3] C. CHAINAIS-HILLAIRET, S. KRELL , A. MOUTON, ” study of discrete duality finitevolume schemes for the peaceman model. SIAM J. Sci. Comput., 35(6):A2928-A2952,2013.

69

On prescribing CG convergence

Gerard Meurant1

1Paris, France

AbstractIn this talk we show how to construct real symmetric positive definite square matricesA of ordern and real right-hand sides b for which the Conjugate Gradient (CG) algorithm of M. Hestenesand E. Stiefel [3] applied to Ax = b has a prescribed residual norm convergence curve. For adescription of the CG algorithm and a study of its properties, we refer to [4]. We also considerprescribing as well theA-norms of the error.

The Full Orthogonalization Method (FOM) of Y. Saad [5] for solving nonsymmetric linearsystems is equivalent to CG when the matrix is symmetric. Prescribing the residual norms forFOM was studied some years ago. One can construct linear systems for which the matrix hasprescribed eigenvalues and such that FOM delivers prescribed residual norms and also prescribedRitz values at all iterations; see [1].

We are interested in discussing what can be done for CG. We will construct symmetric tridi-agonal matricesT such that CG yields prescribed relative residual norms with the right-hand sidee1, the first column of the identity matrix. Then, the linear system Ax = b is obtained fromA = V TV

T and b = V e1 whereV is any orthonormal matrix. Moreover, it turns out that in ourconstruction there are some free parameters that can be chosen to prescribe as well theA-normsof the error.

We consider two ways of constructing T . The first one uses results from A. Greenbaum andZ. Strakos [2] to construct T−1. The second one is a more direct construction of the non-zeroentries of T . Then, we illustrate the results with some numerical examples.

References

[1] J. DUINTJER TEBBENS AND G. MEURANT, Any Ritz value behavior is possible forArnoldi and for GMRES, SIAM J. Matrix Anal. Appl., v 33, n 3 (2012), pp. 958–978.

[2] A. GREENBAUM AND Z. STRAKOS, Matrices that generate the same Krylov resid-ual spaces, in Recent advances in iterative methods, G.H. Golub, A. Greenbaum andM. Luskin Eds., Springer (1994), pp. 95–118.

[3] M.R. HESTENES AND E. STIEFEL, Methodsof conjugategradients for solving linearsystems, J. Nat. Bur. Standards, v 49 n 6 (1952), pp. 409–436.

[4] G. MEURANT, The Lanczos and conjugate gradient algorithms, from theory to finiteprecision computations, SIAM, Philadelphia(2006).

[5] Y. SAAD, Krylov subspace methods for solving large nonsymmetric linear systems,Math. Comp., v 37 (1981), pp. 105–126.

70

Fast and stable estimates for matrix functionals

M. Mitrouli1, P. Fika2

1National and Kapodistrian University of Athens, Greece2University of Patras, Greece

AbstractThe approximation of matrix functionals appears in many applications arising from the fields ofmathematics, statistics, mechanics, networks, machine learning and physics. In this paper, weestimate matrix functionals of the form XT f(A)Y , where A ∈ R

p×p is a given diagonalizablematrix, X, Y ∈ R

p×k areskinny “block vectors” with k ≪ p columnsand f isasmooth functiondefined on the spectrum of the matrix A. We apply a direct backward stable approach based onthe extrapolation of the moments of the given matrix, for estimating this kind of matrix function-als. This approach avoids the application of the polarization identity, relies only on inner productsand matrix vector multiplications, and therefore, the computational speed for large p is quite re-markable. Specific applications from Statisticsand Network Analysis arepresented, for which theextrapolation estimates are highly recommended.

References

[1] E. Estrada and D. J. Higham. Network properties revealed through matrix functions.SIAM review, 52(4):696–714, 2010.

[2] C. Fenu, L. Reichel, and G. Rodriguez. GCV for Tikhonov regularization via globalGolub-Kahan decomposition. Numerical Linear Algebra with Applications, 23(3):467–484, 2016.

[3] P. Fikaand M. Mitrouli. Estimation of thebilinear form yT f(A)x for Hermitian matri-ces. Linear Algebra and its Applications, 502:140–158, 2016.

71

A new trisection method to solve lipschitzianand holderian bi-objectives optimizationproblems

Naffeti Bechir1, Hamadi Ammar2

1bechir.naffeti@gmail.com2hamadi.ammar63@gmail.com

AbstractA multi-objective optimization problem consists in minimizing simultaneously several objectivefunctions that arecompetitors:which means that improving an optimum of one of theseobjectives,deteriorates thoseof theother functions. For amulti-objectivesoptimization problem there isnot asingle optimal solution of all the objectives, but a compromise of solutions named Pareto optimalsolutions and the set of these solutions is denoted by PO (F ) . A multi-objectives optimizationproblem is defined by:

minx∈A

F (x) = (f1 (x) , f2 (x) , ..., fm (x)) , A ⊂ Rd,m ≥ 2, d ≥ 1

wherex iscalled thedecision variableand fi the i−th objective. In thisframework, weconsider abi-objective unidimensional optimization problem (m = 2, d = 1). In the first part, the objectivefunctions fi are supposed to be lipschitzian..Several approaches, suited to lipschitzian objectives,havebeen proposed to find themajority of pointswho belong to theset PO (F ). In thisframework,the algorithm we elaborate, belongs to the family of branch and bound methods. The algorithmis based on the trisection division of the intervals and is easy to implement and apply. It is ableto find the majority of the points that are minimum in the sense of Pareto. Numerical tests provethe efficiency of our method when compared to known ones. In the second part, we extend theelaborated algorithm to the case of holderian objective functions never considered before. Somenumerical experiments allowsus to appreciate the performance of our method.

References

1. Zilinskas, A. and J. Zilinskas, Adaptation of aone-step worst-caseoptimal univariatealgorithm of bi-objective Lipschitz optimization to multidimensional problems. Com-munications in Nonlinear Scienceand Numerical Simulation, 2015. 21(1): p. 89-98.2. Zilinskas, A. and G. Gimbutien, On one-step worst-case optimal trisection in uni-variatebi-objectiveLipschitz optimization. Communicationsin Nonlinear ScienceandNumerical Simulation, 2016. 35: p. 123-136.

72

A new gmres for Schur complement domaindecomposition method

N. Nagid1, H. Belhadj2, M.R. Amattouch3, S.E. Maaqoul4

1,2,3,4Department of Mathematics, Faculty of Sciences and Techniques, University of Abdelmalek Essaadi, BP. 416 Tang-ier, Morocco

AbstractThispaper concernstheuseof an accelerated iterativemethod for thesolution of schur complementsystems. We show that the Generalized Minimum Residual Method GMRES in conjunction witha truncated primal schur complement decomposition can be accelerated using an extrapolationmethode and used to solve large linear systems of equations. Since in most cases of interest, thegmressequencesmay convergevery slowly, extrapolation strategy can often beapplied to enhanceits convergence.Theoretical resultsand numerical experiences haveillustrated that thevector epsilon algorithm arein general more economical and yields faster convergence than the polynomial type extrapolationmethods, with respect to thecomputing and storage requirements.

References

[1] Meza, J. C. (1995). A modification to the GMRES method for ill-conditioned linearsystems. Techni-cal Report SAND95-8220, Scientific Computing Deptartment, SandiaNational Laboratories, Livermore, CA.

[2] Pu, B. Y., Huang, T. Z., and Wen, C. (2014). A preconditioned and extrapolation-accelerated GMRESmethod for PageRank. Applied MathematicsLetters, 37, 95-100.

[3] Sidi, A. (2008). Vector extrapolation methods with applications to solution of largesystemsof equationsand to PageRank computations. Computersand MathematicswithApplications, 56(1), 1-24.

[4] Pu, B. Y., Huang, T. Z., and Wen, C. (2014). A preconditioned and extrapolation-accelerated GMRESmethod for PageRank. Applied MathematicsLetters, 37, 95-100.

[5] Roberts, D. E. (2002). Thevector epsilon algorithmaresidual approach. Numerical Al-gorithms, 29(1-3), 209-227.

[6] Jbilou, K., and Sadok, H. (2000). Vector extrapolation methods. Applications and nu-merical comparison. Journal of Computational and Applied Mathematics, 122(1-2),149-165.

[7] Paige, C. C., Rozloznk, M., and Strakos, Z. (2006). Modified Gram-Schmidt (MGS),least squares, and backward stability of MGS-GMRES. SIAM Journal on Matrix Anal-ysisand Applications, 28(1), 264-284.

73

Stein’s unbaised risk estimate : image pro-cessing

Nihad Nouri1, Fatiha Mezoued2

1Ecole Nationale de Statistique et d’Economie Applique, LAMOPS, 42003 Tipaza, Algeria, Email: nouri.enssea@gmail.com2Ecole Nationale de Statistique et d’Economie Applique, LAMOPS, 42003 Tipaza, Algeria, Email: famezoued@yahoo.fr

AbstractThis paper concerned with additive model of the form Y = M + ǫ, where Y is an observedn × m matrix (noisy image) with m < n, M is an unknown n × m matrix of interest with lowrank and ǫ isarandom noisewhosedistribution iselliptically symetric suggested by D.Fourdrinierand S. Canu in 2017 ([CF17]), which generalise the gaussien case proposed by E.J.Candes in2012 ([CST13]). These authors developed Steins unbiased risk estimate (SURE) in this ellipticalcontext. In thepresent work weproposean application to imageprocessing whereweuseSURE toevaluate thequality of thedenoising method, which isvery important since the true mean-squarederror (MSE) of an estimator isafunction of theunkown parameter to beestimated, and thuscan notbe determined exactly. SURE behaves well in both gaussien and elliptical distribution comparingwith the true mean-squared error.

References

[ANW12] A. Agarwal, S. Negahban, M.J. Wainwright, Noisy matrix decompositionviacon-vex relaxation: Optimal rates in high dimensions, Ann. Statist. 40 (2012) 1171–1197.

[BCFS14] A. Boisbunon, S. Canu, D. Fourdrinier, W. Strawderman, M.T. Wells, Akaike’sinformation criterion,Cp and estimatorsof loss for elliptically symmetric distributions.International Statistical Review, 82 (2014) 422–439.

[CST13] E.J. Candes, C.A. Sing-Long, J.D. Trzasko, Unbiased risk estimates for singularvalue thresholding and spectral estimators, IEEE Trans. Signal Proc. 61 (2013) 4643–4657.

[CF17] S. Canu, D. Fourdrinier,Unbiasedrisk estimatesfor matrixestimation in theellipticalcasesJournal of MultivariateAnalysis, YJMVA 4236 (2017).

[FAN04] K.T. Fang, Elliptically contoured distributions, In: Encyclopediaof Statistical Sci-ences, Wiley, 2004.

74

High-order implicit time integration for the en-riched finite element solution of transient dif-fusion problems

A. Ouardghi1, M. El Amrani2, M.S. Mohamed3, M. Seaid1,4

1Laboratory of CSEHS, University Mohammed VI Polytechnic, Benguerir, Morocco.2Abdelmalek Essaadi University, Faculty of Sciences and Techniques, Tangier, Morocco.3School of Energy, Geoscience, Infrastructure and Society, Heriot-Watt University, Edinburgh EH14 4AS, UK.4School of Engineering, University of Durham, South Road, Durham DH1 3LE, UK.

AbstractHigher order implicit diagonal Runge-Kutta schemes are investigated for their applicability to theenriched finite element solution of transient diffusion problems. A second-order, two new third-order accurateschemeswith different stageorder accuracy, and afourth-order accurateschemearetested and compared to thecommonly used implicit first-order backward differencemethod. In thepresent work, an enriched partition of unity finiteelement method isdeveloped to solvetimedepen-dent diffusion problems. In this formulation, multiple exponential functions describing the spatialand temporal diffusion decay are embedded in the finiteelement approximation space. Theresult-ing enrichment is in the form of a local asymptotic expansion. Unlike previous works in this areawhere the enrichment must be updated at each time step, here, the temporal decay in the solutionis embedded in the asymptotic expansion. Thus, the system matrix that is evaluated at the firsttime step may be decomposed and retained for every next time step by just updating therighthandside of the linear system of equations. The advantage is a significant saving in the computationaleffort where, previously, the linear system must be reevaluated and resolved at every time step. Incomparison with the traditional finite element analysis with pversion refinements, the present ap-proach ismuch simpler, moreefficient, and yieldsmoreaccuratesolutionsfor aprescribed numberof DoFs. Numerical results are presented for a transient diffusion equation with known analyticalsolution. The performance of the method isanalyzed on two applications: the transient heat equa-tion with a single source and the transient heat equation with multiple sources. The aim of sucha method compared with the classical finite element method is to solve timedependent diffusionapplications efficiently and with an appropriate level of accuracy.

References

[1] M.S. MOHAMED, M. SEAID, J. TREVELYAN, O. LAGHROUCHE. A partition of unityFEM for time-dependent diffusion problemsusing multipleenrichment functions. Inter-national Journal for Numerical Methods in Engineering. 93:245-265 (2013)

[2] M.S. MOHAMED, M. SEAID, J. TREVELYAN, O. LAGHROUCHE. Time-independenthybrid enrichment for finiteelement solution of transient conduction-radiation in diffu-sive grey media. Journal of Computational Physics. 251:81-101 (2013)

75

Robust regression analysis for a censored re-sponse and functional regressors

Aıt Hennani, Larbi1, Lemdani Mohamed2, Ould Saıd Elias3

1 Lab. Biomaths., Univ. Lille 2., 59006 Lille, France2 Lab. Biomaths., Univ. Lille 2., 59006 Lille, France3IUT de Calais, ULCO, LMPA, Calais France

AbstractLet (Tn)n≥1 be an independent and identically distributed sequence of interest random variables(rv) distributed as T. In censorship models, T is subject to random censoring by another rv C.

Based on the so-called synthetic data, we define an M-estimator for the regression function of Tgiven a functional covariate χ. In this paper we built a new kernel estimator based on the so-called synthetic data of the mean squared relative error prediction for the regression function. Weestablish the uniform almost sure convergence with rate over a compact set and its asymptoticnormality. The asymptotic variance is explicitly given and as product we give a confidence bands.A simulation study has been conducted to comfort our theoretical resullts.

References

Attouch, M., Laksaci, A., and Ould Saıd, E. (2012). ’Robust regression for functionaltime seriesdata’ , Journal of the Japan Statistical Society, 42, 125–143.Carbonez, A., Gyorfi, L., and van der Meulin, E.C. (1995). ’Partition-estimate of a re-gression function under random censoring’ , Statistics & Decisions, 13, 21–37.Guessoum, Z., and Ould Saıd, E. (2008). ’On nonparametric estimation of the regres-sion function under random censorship model’ , Statistics & Decisions, 26, 159–177.Kaplan, E.M., and Meier, P. (1958),.’Nonparametricestimation from incompleteobser-vations’ , Journal of the American Statistical Association, 53, 457–481.Lemdani, M., and Ould Saıd, E. (2017). ’Nonparametric robust regression estimationfor censored data’ , Statistical Papers, 58, 505–525.

76

Robin-Robin domain decomposition algorithmcoupled with boundary elements method forsolving an inverse problem in elasticity equation

Abdellatif Ellabib1, Abdeljalil Nachaoui2, Abdessamad Ousaadane1

1Departement de Mathematiques, Laboratoire des Mathematiques Appliquees et Informatique, Faculte des Sciences etTechniques, Universite Cadi Ayyad Marrakech.2Laboratoire de Mathematiques Jean Leray, Universite de Nantes, 2 rue de la Houssiniere, BP92208 44322 Nantes,France.

AbstractThiswork dealswith domain decomposition likemethodsfor solving an inverseproblem in elastic-ity equation. Wesearch to reconstruct theunknown boundary conditions on apart of theboundarybased on the other boundaries. We construct a Steklov-Poincare kind’s operator. This allows usto reformulate our inverse problem into a fixed point one involving a Steklov kind’s operator. Theproposed approach offers the opportunity to exploit domain decomposition methods for solvingthis inverse problem. Finally, we present a numerical study of this problem, using Robin-Robinalgorithm and the boundary elements method.

Key words: Cauchy Inverseproblem, Elasticity equation, Steklov-Poincareoperator, Degreeof Leray-Schauder, Boundary elements method.

References

[1] V. BADEVA and V. MOROZOV, Problemes incorrectement posees, theorie et Appli-cations, Masson, Paris, 1997.

[2] A. BREBBIA and J. DOMINGUEZ, Boundary elements: An introductory course,Second Edition, WIT Press, Computational Mechanics Publications, Boston, South-hampton.

[3] K. DEIMLING, Nonlinear functional analysis Springer-Verlag, 1985.

[4] G. DUVAUT and J.L. LIONS, Inequalities in Mechanics and Physics, DUNOD,PARIS, 1972.

[5] A. ELLABIB and A. NACHAOUI, A domain decomposition method for boundaryelement approximationsof theelasticity equations, Annalsof theUniversity of Craiova,Mathematicsand Computer ScienceSeriesVolume42(1), Pages211-225, 2015.

[6] A. ELLABIB and A. NACHAOUI, An iterative approach to the solution of an inverseproblem in linear elasticity, Mathematics and Computers in Simulation 77, 189-201,2008.

77

The tensor Golub–Kahan bidiagonalization forill-posed tensor equations with applicationsto color image restoration

Fatemeh Panjeh Ali Beik1, Khalide Jbilou2, Mehdi Najafi1

1Department of Mathematics, Vali-e-Asr University of Rafsanjan,P.O. Box 518, Rafsanjan, Iran2Laboratory LMPA, 50 Rue F. Buisson, ULCO calais cedex, France

AbstractThe Tikhonov regularization technique in conjunction with the Golub–Kahan bidiagonalizationbased on tensor format (GKB

−BTF) is proposed to solve ill-posed linear tensor equations (LTE)

with a noise–contaminated RHS. In practice, an ill-posed LTE may arise from discretization of afully three-dimensional microscale dual phase lag problem by a mixed–collocation finite differ-ence method [4] being mentioned as a test problem to numerically examine the performance ofGKB

−BTF in conjunction with Tikhonov regularization. Wecomment that developing the idea in

[3], the connection between GGK−

BTF and Gauss-type quadrature rules is exploited to cheaplycomputeboundsbeing useful for determining theregularization parameter by thediscrepancy prin-ciple. We further disclose some experimental results corresponding to color image restoration astest problems.

References

[1] Bader BW and Kolda TG. MATLAB Tensor Toolbox Version 2.5.http://www.sandia.gov/ tgkolda/TensorToolbox.

[2] Beik FPA, Movahed FS and Ahmadi-Asl S. On the Krylov subspacemethodsbased ontensor format for positive definite Sylvester tensor equations. Numer. Linear AlgebraAppl. 2016; 23: 444–466.

[3] Bentbib AH, Guide EM, Jbilou K and Reichel L. Global Golub–Kahan bidiagonaliza-tion applied to large discrete ill-posed problems. J Comput. Appl. Math. 2017; 322:46–56.

[4] Malek A, Bojdi ZK, Golbarg PNN. Solving fully three-dimensional microscale dualphase lag problem using mixed-collocation finite difference discretization. Journal ofHeat Transfer. 2012; 134(9), 094504.

78

Mathematical Structures defined by Identities

Constantin M. Petridi1

1cpetridi@math.uoa.gr , cpetridi@hotmail.com

AbstractWe propound the thesis that there is a “ limitation” to the number of possible structures which areaxiomatically endowed with identities involving operations. In the case of algebras with a binaryoperation satisfying a formally irreducible (to be explained) identity between two n-iterates of theoperation, it isestablished that the frequency of such algebrasgoes to zero likee−n/16 asn → ∞.This isproved by a suitable ordering and labeling of the expressions (words) of the correspondingfree algebra and the formation of a series of tableaux whose entries are the labels. The tableaux

reveal surprising symmetry properties, stated in terms of the Catalan numbers1

n+ 1

(

2n

n

)

and their partitions. Asa result of the defining identity and the tableaux all iteratesof order higherthan n fall into equivalence classes of semantically equal ones. Classnumbers depending on thetableaux arecalculated for all algebrasof order n = 3 (and partially n = 4). Certain classnumbersdepending on homomorphismsof thestructureisproposed and corresponding generating functionsset up. Asan exampleof this, akind of skein polynomialsareconstructed characterizing theformalbuild-up of iterates. As no distinction is made between the various algebras, isomorphic or not,which are models of the identity the results can also be formulated in terms of varieties withsignature the identity in question.

References

[1] Petridi, C. M., Krikelis, P. B., Mathematical Structures defined by Identities,arXiv:math/0110333v1 [math.RA] 31 Oct 2001.

79

Mathematical Structures defined by IdentitiesIII

Constantin M. Petridi1

1cpetridi@math.uoa.gr , cpetridi@hotmail.com

AbstractIn this work we extend the theory (formal part only) of algebras with one binary operation [1],to algebras with several operations of any arity. We briefly outlined our ideas of generalizing themethod of tableaux to algebras with several operations

V1(x1, x2, · · · , xa1), V2(x1, x2, · · · , xa2), · · · , Vk(x1, x2, · · · , xak)

satisfying axiomatically defined identities and indicated the way of how to proceed. The projectis now carried out. The technique applied is the same as in Formal Part of [1]. The crucialfact that the number IV1V2...Vk

n of formally reducible identities can be calculated by exactly thesame method used for IV1

n (= In) seems to hold true. Algebras with only binary operations arediscussed. For algebras with two binary operations V (x, y) and W (x, y) the proof is given indetail. Algebras with operations of any arity can be treated by reduction to a well defined set ofalgebras with binary operations. Research and exposition of the general theory are impeded byproblems of construction and inspection of the tableaux Tn whenever n is greater than 3. This is

due to the fast growth of the Catalan numbers (Sn ∼

4n

π1

2n3

2

) and their generalizations, let alone

problems of printing and publication. Programs designed to seek the structures resulting froma given identity failed after a few steps (blow-ups). Exposition therefore is limited to illustratethe theory on the worked example of tableau T3. Still, the concrete new findings reached in thiscase corroborate further our fundamental thesis that there is a scarcity of existing mathematicalstructures in the sense that the frequency of irreducible identities goes to zero with increasing n.Seen historically, thisalso explainswhy mathematics, in thecourseof time, hasdeveloped thewayit did with associativity V (V (x, y), z) = V (x, V (y, z)), the simplest structure, reigning supremeover the mathematical landscape. All other essential mathematical structures, found or created byresearch such as e.g. Groups, Fields, Vector Spaces, Lie Algebras, etc, ... include in their axiomsystem (signature) at least onebinary operation obeying the law of associativity. Weconcludewitha note on the connection with Formal Languages.

——————————————————-

References

[1] Petridi, C. M., Krikelis, P. B., Mathematical Structures defined by Identities,arXiv:math/0110333v1 [math.RA] 31 Oct 2001.

[2] Petridi, C. M., Mathematical Structures defined by Identities II, arXiv:1009.1006v1[math.RA] 6 Sep 2010.

80

A data-driven modeling approach to computekilling dynamics and clearance of a pathogenusing backward birth-death-immigration clus-tering processes of killer cells

Harshana Rajakaruna1, Vitaly V. Ganusov2

1Dept. of Microbiology, University of Tennesse, Knoxville TN 37996, USA

AbstractIdeal design of sampling, required to answer a research question at hand, is often limited by var-ious unavoidable sampling constraints. And also, applied mathematical and statistical modelingquestions, often asked, are constrained by the data already collected for some other research pur-poses. Hence, themethod of modeling and computation of system properties, to answer a researchquestion, may depend on thetypeof thedataalready available. Here, wemodel asystem of aclus-tering population of cells (antigen-specific killer T cells), involved in killing pathogens (malariaparasites) in a mouse liver, to answer (i) how long (mean-time) it takes T cells to clear malariaparasites from the liver, and (ii) the probability of the number of T cells involved in this processover time. The data available are the numbers of T cells that cluster around the parasites and thevitality index of the parasites (i.e., parasites health) over time. A limitation of the data is that theyarenot available from thebeginning of time that T cellswere injected into infected mice, but sam-pled scattered over the time line. This limits the possibility of estimating the states of the systembefore the data were collected, resulting an immeasurability at which time points the lethal shotswere given to parasites by T cells, and the number of T cells involved in it, along the course. Thisis because the death of parasites can be slow. That is, the available data limits the modeling andestimating the underlying processes of evolution and decay of the system fully. Here, we show amethod of backward computation of birth-death-immigration process of T cell clustering, coupledwith stochastic killing processes, to compute the mean time to, and the probability of the numberof T cells involved in the process of pathogen clearance. We show the estimations of the quan-tities of interest using different modeling methods and simulations, and compare their degree ofconsistencies.

81

Matrix Shanks transformations

C. Brezinski1, M. Redivo-Zaglia2

1Laboratoire Paul Painleve, UMR CNRS 8524, UFR de Mathematiques, Universite de Lille, France2Universita degli Studi di Padova, Dipartimento di Matematica “Tullio Levi-Civita”, Italy

AbstractShanks’ transformation is a well know sequence transformation for accelerating the convergenceof scalar sequences. It can berecursively implemented by thescalar ε–algorithm of Wynn who alsoextended it to sequences of vectors and of square matrices satisfying a linear difference equationwith scalar coefficients [7]. Another extension of the transformation to sequences of elements ofa general vector space was proposed and studied in [1] and can be implemented by the simplifiedtopological ε–algorithm [2, 3].

In this talk, we propose a more general extension to the matrix case where the matrices canbe rectangular and satisfy a difference equation with matrix coefficients. Our approach is related,but not similar, to those presented in [4], [5] and [6]. In the particular case of square matrices, thenew transformation can be recursively implemented by the matrix ε–algorithm of Wynn. We alsorelate it to Pade-type and Pade approximants for power series with matrix coefficients.

Numerical experiments showing the interest of this new transformation end the talk.

References[1] C. Brezinski, Generalisation de la transformation de Shanks, de la table de Pade et de l’ε–

algorithme, Calcolo, 12 (1975) 317-360.

[2] C. Brezinski, M. Redivo-Zaglia, The simplified topological ε–algorithms for accelerating se-quences in a vector space, SIAM J. Sci. Comput., 36 (2014) A2227-A2247.

[3] C. Brezinski, M. Redivo-Zaglia, The simplified topological ε–algorithms: software and applica-tions, Numer. Algorithms, 74 (2017) 1237-1260.

[4] K. Jbilou, A. Messaoudi, Block extrapolation methods with applications, Appl. Numer. Math.,106 (2016) 154-164.

[5] K. Jbilou, A. Messaoudi, K. Tabaa, Some Schur complement identitiesand applications to matrixextrapolation methods, Linear Algebra Appl., 392 (2004) 195-210.

[6] K. Jbilou, H. Sadok, Matrix polynomial and epsilon–typeextrapolation methodswith applications,Numer. Algorithms, 68 (2015) 107-119.

[7] P. Wynn, Acceleration techniquesfor iterated vector and matrix problems, Math. Comp., 16 (1962)301-322.

82

Stochastic diffusion process based on Goel-Okumoto curve

O. Rida1, A. Nafici1, B. Achchab1

1Univ.Hassan 1, LAMSAD, Ecole Nationale des Sciences Appliquees de Berrechid, Avenue de l’universit, BP 280,Berrechid, Morocco

AbstractThe principal aim of the present work is to study a stochastic diffusion process based on the Goel-Okumoto curve ([1], [2]). Such a process can be considered as an extension of the nonhomoge-neous log-normal diffusion process [3]. From the corresponding Itos stochastic differential equa-tion (SDE), firstly we establish the probabilistic characteristics of the studied process, such as thesolution to theSDE, theprobability transition density function and their distribution, themomentsfunction in particular the conditional and unconditional trend functions. Secondly, we treat theparameters estimation problem by using the maximum likelihood method in basis of the discretesampling, thus we obtain non-linear equations that can be solved by numerical methods.

Keyword: Diffusion process; Goel-Okumoto curve ; Stochastic differential equation ; maximumlikelihood estimation.

References

[1] Goel, A. L., & Okumoto, K. (1979). Time-dependent error-detection rate model forsoftware reliability and other performancemeasures. IEEE transactions on Reliability,28(3), 206-211.

[2] Goel, A. L. (1985). Softwarereliability models: Assumptions, limitations, and applica-bility. IEEE Transactionson softwareengineering, (12), 1411-1423.

[3] R. Gutierrez, P. Roman and F. Torres. Inference and first-passage time for the lognor-mal diffusion process with exogenous factors: application to modelling in economics.Applied Stochastic Models in Business and Industry, 15:325332, 1999.

83

Numerical Linear Algebra for data-relatedapplications

Yousef Saad1

1Department of Computer Science and Engineering University of Minnesota Minneapolis, USA.

AbstractThis tutorial will focus on linear algebra techniques for applications related to machine learningand other applications that rely of large data sets. A common tool that is exploited in solvingdatamining and machine learning problems isthat of ‘dimension reduction’ . Dimension reductionis based on the precept that the observed data often lies in a noisy version of a low-dimensionalsubspace and so it is critical to work in this subspace not only to reduce computational cost butalso to improve accuracy. The presentation will start with an overview of the key concepts andthen illustrate dimension reduction methods with applications such as information retrieval, facerecognition and matrix completion for recommender systems. Oneof themain difficulties in manyof themethodsbased on dimension reduction isto find the inherent approximate rank of thedataathand. Wewill show how afew simple random sampling methods for computing spectral densitiesand counting eigenvalues can be used for this purpose. Among other topics, the tutorial willalso cover applications of graph Laplaceans, such as clustering and image segmentation, as wellmethods for analyzing networks.

84

Minimal residual method for solving symmet-ric positive Sylvester tensor equations.

El M. Sadek1, A.H. Bentbib2, S. El-Halouy2

1Laboratory labSIPE, ENSA, d’EL Jadida, University Chouaib Doukkali, Morocco2Laboratory LAMAI, University Cadi Ayyad, Marrakesh, Morocco

AbstractOur purpose in thiswork isanew iterativemethod in tensor format for solving theSylvester tensorminimization problem

X ∗ = arg minX∈R

I1×I2×···×IN

‖D − (X ×1 A1 + X ×2 A2 + · · ·+ X ×N AN ) ‖ (1)

where An ∈ RIn×In , n = 1, 2, · · · , N, are symmetric, positive matrices, and the tensor D ∈

RI1×I2×···×IN isknown and X ∗ ∈ R

I1×I2×···×IN is the unknown tensor to be determined.Thewell known minimal residual method isproposed here on tensor format (MINRES-BTF).

TheLanczosprocesson thetensor format isalso used. Theoretical resultsaregiven and numericalexperiments are presented comparing MINRES-BTF with the conjugate gradient CG on tensorformat.

References

[1] A. Beik, F. Panjeh, F. S. Movahed, On the Krylov subspace methods based on tensorformat for positivedefiniteSylvester tensor equations, Numer. Linear AlgebraAppl, 23,pp. 444–466. (2016).

[2] Saeed Karimi and MaryamDehghan, Global least squaresmethod based on tensor formto solve linear systems in Kronecker format, Transactions of the Institute of Measure-ment and Control, 2017.

[3] Kolda, T. G., Bader, B.W, Tensor Decompositionsand Applications. SIAM Rev. 51, pp.455–500 (2009).

[4] Chen Z, Lu LZ. A gradient based iterative solutions for Sylvester tensor equations.Mathematical Problemsin Engineering 2013.

[5] C. C Paige, M. A. Saunders, Solution of sparse indefinite systems of linear equations.SIAM journal on numerical analysis, (1975)

85

Block extended Krylov method for solving large-scale generalized differential Sylvester equations

Lakhlifa Sadek1, Abdeslem Hafid Bentbib2, Hamad Talibi Alaoui1

1 lakhlifasadek@gmail.com, FS, d’EL Jadida, University Chouaib Doukkali, Morocco2Laboratory LAMAI, University Cadi Ayyad, Marrakesh, Morocco

AbstractWe propose in this work a new method for solving large-scale generalized differential Sylvesterequations.

X(t) = AX(t) +X(t)BT +k

∑

i=1

NiX(t)MTi − EF

T, t ∈ [t0, Tf ]

X(t0) = X0,

(1)

whereA,Ni ∈ Rn×n, B,Mi ∈ R

p×p, and E ∈ Rn×s, F ∈ R

p×s, with s ≪ n, p. The matricesA andB are assumed to be large, sparse, and nonsingular.

Our approach is to project the initial problem onto the block extended Krylov subspace. Thisprojection allows us to obtain a low-dimensional generalized differential Sylvester equation. Wethen solve the obtained equation by BDF or Rosenbrock method. Theoretical results are reportedand numerical experiments are given to show the effectiveness of the proposed methods.

References

[1] A.H. Bentbib, K. Jbilou, and El. M. Sadek, On some Krylov subspace based methodsfor large-scale nonsymmetric algebraic Riccati problems. Comput. Math. Appl. 2015,pp. 2555–2565.

[2] V. Druskin, L. Knizhnerman, Extended Krylov subspaces approximation of the matrixsquare root and related functions, SIAM J. Matrix Anal. Appl., 19(3)(1998), pp. 755–771.

[3] M. Hached K. Jbilou. Numerical solutions to large-scale differential Lyapunov matrixequations. Numer. Algor, 2017.

[4] M. Hached K. Jbilou. Computational Krylov-based methodsfor large-scaledifferentialSylvester matrix problems. Numer Linear AlgebraAppl. 2018.

[5] Jarlebring E, Mele G, Palitta D, Ringh E. Krylov methods for low-rank commutinggeneralized Sylvester equations. Numer Linear AlgebraAppl. 2018

86

Review of the convergence of some Krylovsubspace methods for linear systems of equa-tions with one or several right hand sides

Hassane Sadok1

1 LMPA, Universite du Littoral,50, rue F. Buisson, BP 699,62228 CALAIS-Cedex, FRANCE sadok@lmpa.univ-littoral.fr

AbstractKrylov subspace methods are widely used for the iterative solution of a large variety of linearsystems of equations with one or several right hand sides or for solving nonsymmetric eigenvalueproblems.It is the purpose of the present talk to compare three variants of GMRES for multiple right hand-sides including a block GMRES [1, 3, 4, 5]. These schemes are based on block Arnoldi-typesmethods and differ in thechoice of inner product. Weprovide aunified description of themethodsdiscussed and derive new expressions and bounds for the residual errors.We will also discuss the block version of the BiCGSTAB algorithm[2], for solving multiple linearsystems using some algebraic orthogonalities.

References

[1] L. ELBOUYAHYAOUI , A. MESSAOUDI AND H. SADOK, Algebraic Properties of BlockArnoldi algorithm and Block GMRES method, Elect. Trans. Num. Anal., 33 (2009) pp.207–220.

[2] M. EL GUENNOUNI , K. JBILOU AND H. SADOK, A Block version of BICGSTAB forlinear systems with multiples right-hand sides, Elect. Trans. Num. Anal., 16 (2003) pp.129–146.

[3] K. JBILOU, A. MESSAOUDI AND H.SADOK, Global FOM and GMRES algorithms formatrix equations, App. Num. Math., 31 (1999) pp. 49–43.

[4] M.H. GUTKNECHT, Block Krylov Space Methods for Linear Systems With MultipleRight-hand Sides: an Introduction, [Aug. 2005; to appear 2006 in: Modern Mathemat-ical Models, Methodsand Algorithmsfor Real World Systems(A.H. Siddiqi, I.S. Duff,and O. Christensen, eds.), AnamayaPublishers, New Delhi, India.]

[5] V. SIMONCINI AND E. GALLOPOULOS, Convergence properties of block GMRES andmatrix polynomials, Linear AlgebraAppl., 247(1996), pp. 97-119.

87

An extended conjugate duality for generalizedsemi-infinite programming problems via a con-vex decomposition

Abdelmalek Aboussoror1, Samir Adly2, Said Salim1

1Universite Cadi Ayyad, Faculte Polydisciplinaire de Safi, Laboratoire LMC, B.P. 4162, Sidi Bouzid, Safi, Morocco.E-mails: aboussororabdel@hotmail.com, salimsaidabdou24@gmail.com2Universite de Limoges, Laboratoire XLIM, 123 avenue Albert Thomas, 87060 Limoges, France.E-mail: samir.adly@unilim.fr

AbstractWe are concerned with the following generalized semi-infinite programming problem

(P) minx∈X

G(x,y)≤0,∀y∈Y (x)

F (x)

whereF : Rn −→ R, G : Rn × Rm −→ R, are respectively convex and concave functions, X is

a nonempty convex compact subset of Rn, and Y (x) is an infinite index subset of Rm dependingon x, defined by

Y (x) = y ∈ Rm/g(x, y) ≤ 0,

where g = (g1, . . . , gp)T : Rn × R

m −→ Rp is a convex function. We assume that for every

x ∈ X, the set Y (x) isnonempty.

Generalized semi-infiniteprogramming (GSIP) have several applications as engineering, eco-nomics, optimal control and transportation problems .

In thiswork, wepresent an extended conjugateduality for problem (P). Thisduality isdefinedvia adecomposition of the global optimization problem (P) into a family of convex subproblems.Under appropriate assumptions, we first establish strong duality. Then, via this duality we pro-vide necessary and sufficient global optimality conditions for problem (P). These conditions areexpressed in terms of subdifferentials and normal cones in the sense of convex analysis.

References

[1] A. Aboussoror and S. Adly, Generalized semi-infinite programming: optimality condi-tions involving reverseconvex problems, Numerical Functional Analysisand Optimiza-tion, Vol. 35, 816-836, 2014.

[2] G. Still, Generalized semi-infinite programming: Theory and methods, European Jour-nal of Operational Research, Vol. 119, 301-303, 1999.

88

Efficient preconditioning of linear systems aris-ing from the enriched finite element discretiza-tion of conduction-radiation equations

Mohammed Seaid1,2

1School of Engineering, University of Durham, South Road, Durham DH1 3LE, UK.2Laboratory of CSEHS, University Mohammed VI Polytechnic, Benguerir, Morocco.

AbstractWeinvestigate theeffectivenessof preconditioning techniques for theenriched finiteelement solu-tion for transient conduction-radiation problems in diffusive grey media. Thegoverning equationsconsist of a semi-linear transient heat equation for the temperature field and a stationary diffusionapproximation to theradiation in grey media. Thecoupled equations are integrated in timeusing asemi-implicit method in thefiniteelement framework. Weshow that for theconsidered problems, acombination of hyperbolic and exponential enrichment functionsbased on an approximation of theboundary layer leads to improved accuracy compared to the conventional finite element method.Most of enriched finite element methods use dense direct solvers for the resulting linear systems.This ismainly thecasedueto theillconditioned linear systemsthat areassociated with thesemeth-ods. In thisstudy, to solvetheresulting linear system weformulatediffusion synthetic accelerationand Krylov subspace methods. We also introduce a fast multilevel algorithm. All these methodscan be viewed as preconditioned iterative methods with different preconditioners. Numerical re-sults along with comparisons of effectiveness and efficiency of these solvers are carried out onseveral test examples for conduction-radiation in diffusive grey and non-grey media.

References

[1] M.S. MOHAMED, M. SEAID, A. BOUHAMIDI. Iterative solvers for generalized finiteelement solution of boundaryvalueproblems. Numerical Linear Algebra with Applica-tions. 25:1-18 (2018)

[2] M.S. MOHAMED, M. SEAID, J. TREVELYAN, O. LAGHROUCHE. An enriched finiteelement model with q-refinement for radiativeboundary layers in glasscooling. Journalof Computational Physics. 258:718-737 (2014)

[3] M.S. MOHAMED, M. SEAID, J. TREVELYAN, O. LAGHROUCHE. Time-independenthybrid enrichment for finiteelement solution of transient conduction-radiation in diffu-sive grey media. Journal of Computational Physics. 251:81-101 (2013)

89

Markov decision processes with discountedcost : new action elimination procedure

Abdellatif Semmouri1, Mostafa Jourhmane2

1dep. of Mathematics, Laboratory of TIAD, Faculty of Siences and Techniques, Sultan Moulay Slimane University, BeniMellal, Morocco2dep. of Mathematics, Laboratory TIAD, Faculty of Siences and Techniques, Sultan Moulay Slimane University, BeniMellal, Morocco

AbstractMarkov Decision Processes (MDPs) are a popular decision model for stochastic systems. As thecomputational complexity is an open problem, all researchers are interesting to find methods andtechnical tools in order to solve the proposed problem.

In this paper, we provide an efficient algorithms based on the backward induction to find op-timal policies and value functions for solving finite-horizon discrete-timeMDPs in the discountedcase. In the first time, we present a backward induction algorithm for solving Markov decisionproblem employing the total discounted expected cost cretirion over the finite horizon planning.Second, we propose a new test to identify nonoptimal decisions in the same context. Combiningthepreceding presented results, wegive an useful algorithm by linking therecursiveapproach andthe action elimination procedures.

It isapowerful analytical tool used for sequential decision making under uncertainty, that havebeen widely used in many industrial, manufacturing and financial fields.

References

[1] C. Derman, FinteState Markovian Decision Processes, Academic Press, 1970.

[2] M.L. Puterman, Markov decision processes, Wiley, New York, 1994.

[3] J. Filar and K. Vrieze, Competitive Markov Decision Processes, Springer, New York,1997.

[4] J. MacQueen, A modified dynamic programming method for Markovian decision prob-lems, J. Math. Anal. Appl., vol. 14, pp. 3843, 1966.

[5] J. MacQueen, A test for suboptimal actionsin Markoviandecisionproblems, Oper. Res.,vol. 15, pp. 559561, 1967.

[6] E. L. Porteus, Some bounds for discounted sequential decision processes, Mgmt Sci.,vol. 18, pp. 711, 1971.

[7] R. Grinold, Eliminationof suboptimal actionsin Markov decisionproblems, OpnsRes.,vol. 21, pp. 848851, 1973.

[8] N. A. J. Hastingsand J. M. C. Mello, Testsfor suboptimal actionsin discountedMarkovprogramming, Mgmt Sci., vol. 19, pp. 10191022, 1973.

90

Recursive orthogonal polynomials algorithm

Sidki Sara1, Sadaka Rachid1, Benazzouz Abderrahim1

1Ecole Normale Superieure, Mohammed V University in Rabat, Av. Mohammed Belhassan El Ouazzani, Takaddoum,B.P. 5118, Rabat, Morocco

AbstractThis work presents the expression of orthogonal polynomials in an other form using Schur com-plement. For a linear functional c defined inR[X], the family of orthogonal polynomial isdefinedas follows : c(xiPk(x)) = 0, for i = 0, . . . , k − 1 and Pk(0) = 1, for k ≥ 0. The propertiesof Schur complement and the Sylvester identity allows us to express the orthogonal polynomialasa Schur complement and giveahybrid three-termsrecurrence relationship( Orthogonal polyno-mials, adjacent polynomials). A similar relation was obtained in other work without a recursivealgorithm to compute the terms. In our case, we get directly the terms of our relation as a ratio ofHankel determinants. In order to compute theseterms, weused thewell known recurrencerelationof the Hankel determinants, which weobtained easily using the Schur complement. Wepropose arecursive algorithm to compute the family of orthogonal polynomials and the families of adjacentpolynomials. Finally we give some examples.

References

[1] C.Brezinski: Other manifestationsof theSchur complement. Linear AlgebraAppl.1988

[2] C.Brezinski: Recursive interpolation, extrapolation and projection. J. Comput. Appl.Math. 9, 369376 (1983)

[3] Brezinski, C.: Some determinantal identities in a vector space, with applications. In:Werner, H.Bunger, H.J. (eds.) Pad Approximation and its Applications, Bad-Honnef,1983, LectureNotes in Math., vol. 1071, pp. 111. Springer, Berlin (1984)

[4] C. Brezinski, M. Redivo Zaglia: A Schur complement approach to a general extrapola-tion algorithm. 2002

[5] A.Messaoudi: Recursive interpolation algorithm: a formalism for linear equations-I:Direct methods. J. Comp. Appl. Math. 76, 1330 (1996)

[6] Messaoudi, A.: Matrix recursive interpolation algorithm: a formalism for linearequations-II: iterativemethods. J. Comp. Appl. Math. 76, 3153 (1996)

91

Identification of the source function for a sea-water intrusion problem in unconfined aquifer

S. Slimani1, K. Najib1,2, I. Medarhri1,2, A. Zine3

1LERMA, Mohammadia Engineering School, Science and Applied MathematicsMohammed V University, Agdal-Rabat, Morocco.2Superior National School of Mines, 1000 Agdal-Rabat, Morocco.3 Institute Camille Jordan, Central Lyon High School , 69134 cully, France.

AbstractIn thiswork, weareinterested in an inverseproblem of seawater intrusion in an unconfined aquifer(with sharp interface approach), this problem consists to identify the unknown space-dependentsource of the problem [1]:

φ∂th1 − div(KTf (h− h1) + Ts(h1))h1)− div(KTs(h)h) = −(Qs +Qf )

φ∂th− div(KTs(h)h)− div(KTs(h)h1) = −Qs

(1)

from boundary datah1 = h1,D on Γ× [0, T ]. We applied the technique of variable separationused by M.Kulbay and B.Mukanova in [2]. Such as the model associated to the direct problem isnon-linear, a method of linearization is adopted. The inverse source problem minimization of acost function :

Jβ(Fn+1) =

1

2‖ V

n+1(l, t)− gε(t) ‖2L2(0,T ) +

β

2‖ Fn+1 ‖22

whereV n+1(t, x) = hn+11 − h1,D(0, t), gε(t) = hn+1

1 (L, t)− h1,D(0, t), β is the regular-ization parameter, which can be determined by the L-curve criterion.Our aim is to prove the convergence of the solution theoretically and validate the results numeri-cally.

References

[1] P. Marion, K.Najib, C. Rosier, Numerical simulations for a seawaterintrusion problem in a free aquifer, Applied Numerical Mathematics, 2014.

[2] M.Kulbay, B.Mukanova, C.Sebu, Identification of separable sources foradvection-diffusion equations with variable diffusion coefficient from boundarymeasured data, Inverse problems in science and engineering, March 2016.

92

New implementation of the Block GMRES method

L. Elbouyahyaoui1, A. Messaoudi2, H. Sadok3, L. Smoch4

1Departement de Mathematiques, Faculte des Sciences et Techniques, Universite Hassan II, Mohammadia, Maroc.2Departement de Mathematiques, Ecole Normale Superieure, Rabat, Maroc.3Universite du Littoral, zone universitaire de la Mi-voix, batiment H. Poincarre, 50 rue F. Buisson, BP 699, F-62280 CalaisCedex, France.4Universite du Littoral, zone universitaire de la Mi-voix, batiment H. Poincarre, 50 rue F. Buisson, BP 699, F-62280 CalaisCedex, France.

AbstractThe Block GMRES is a block Krylov solver for solving nonsymmetric systems of linear equa-tions with multiple right-hand sides. This method is classically implemented by first applying theArnoldi iteration as a block orthogonalization process to create a basis of the block Krylov spacegenerated by thematrix of thesystem from the initial residual. Next, themethod issolving ablockleast-squares problem, which is equivalent to solving several least squares problems implying thesame Hessenberg matrix. These latter are usually solved by using a block updating procedure fortheQR decomposition of theHessenberg matrix based on Givens rotations. A moreeffectivealter-native was given in [?] which uses theHouseholder reflectors. In thispaper wepropose anew andsimple implementation of the block GMRES algorithm, based on a generalization of Ayachour’smethod [2] given for theGMRESwith asingleright-hand side. Several numerical experimentsareprovided to illustrate the performance of the new implementation.

References

[1] W.E. Arnoldi. The principeof minimized iterations in the solution of the matrix eigen-valueproblem, Quart. Appl. Math., vol. 9 (1951), pp. 17–29.

[2] E.H. Ayachour. A fast implementation for GMRES method, J. Comput. Appl. Math.,vol. 159 (2003), pp. 269–283.

[3] J. Baglama. AugmentedBlock Householder Arnoldi method, Linear AlgebraAppl., vol.429 (2008), pp. 2315–2334.

[4] H. Calandra, S. Gratton, J. Langou, X. Pinel, X. Vasseur. Flexible variants of blockrestarted GMRES methods with application to geophysics, SIAM J. Sci. Comput. vol.34, no. 2 (2012), pp. 714–736.

[5] I.S. Duff, R.G. Grimes, J.G. Lewis. Users Guide for the Harwell-Boeing sparse matrixcollection (Release I), Technical Report TR/PA/92/86, CERFACS, Toulouse, France,1992. Matricesavailableat <http://math.nist.bov/MatrixMarket/>.

93

Finite volume method in a general mesh for acoupled system of parabolic-parabolic equations

Ouafa Soualhi1, Mohamed Rhoudaf2

O. SOUALHI,Department of Mathematics,University Moolay ismail, Faculty of sciences Meknes,Morroco.

M. Rhoudaf,Department of Mathematics,University Moolay ismail, Faculty of sciences Meknes,Morroco.

AbstractIn this paper, we present the finite volume method DDFV applied to a model of (Patlak) Keller-Segel, this model consists of a coupled system of parabolic-parabolic equations. The existence ofa discrete solution isproved. Numerical simulations are performed to verify accuracy.

References

[1] G. CHAMOUN, M. SAAD AND R. TALHOUK, Monotone combined numerical schemefor anisotropic chemotaxis-fluid model, Numer. MethodsPartial Differential Equations30 (2014), no. 3, 10301065. 65M08 (65M12-65M60).

[2] G. CHAMOUN, M. SAAD AND R. TALHOUK, Finite volume scheme for isotropicKeller-Segel model with general scalar diffusivefunctions. CongrsSMAI 2013, 128137,ESAIM Proc. Surveys, 45, EDP Sci., LesUlis, 2014. (Reviewer: Jochen Gerhard Schtz)65M08 (65M12).

[3] G. CHAMOUN, M. SAAD AND R. TALHOUK, Mathematical and numerical analysisof a modified Keller-Segel model with general diffusive tensors. J. Biomath 2, 1312273,http://dx.doi.org/10.11145/j.biomath.2013.12.07, 2013.

[4] G. CHAMOUN, M. SAAD AND R. TALHOUK, A coupled anisotropic chemotaxis-fluidmodel: the case of two-sidedly degenerate diffusion. Computer and Mathematics withApplications, http://dx.doi.org/10.1016/j.camwa.2014.04.010, 2013.

94

Numerical analysis of a sliding frictional con-tact problem with unilateral contact and nor-mal compliance

Yahyeh Souleiman1

1Avenue Djanaleh, BP:1904, Universite de Djibouti, Djibouti

AbstractWe consider a mathematical model which describes the equilibrium of a viscoelastic body in fric-tional contact with a moving foundation. The contact is modeled with a multivalued normal com-pliance condition with unilateral constraint and memory term, associated to a sliding version ofCoulomb’s law of dry friction. We present a description of the model, list the assumption on thedata and derive a variational formulation of the problem, which is in a form of a system couplinga nonlinear equation for the stress field with a variational inequality for the displacement field.Then, we introduce a fully discrete scheme for the numerical approximation of the sliding contactproblem. Under certain solution regularity assumptions, wederive an optimal order error estimateand weprovidenumerical validation of thisresult. Finally, weprovidesomenumerical simulationsin the study of a two-dimensional problem.

References

[1] M. Barboteu & Y. Souleiman, Numerical Analysis of a Sliding frictional contact prob-lem with Unilateral Contact and Normal Compliance in preparation.

[2] M. Sofonea, F. Patrulescu& Y. Souleiman, A Contact Problemwith Wear andUnilateralConstraint, Applicable Analysis. 95 (2016), 2590–2607.

[3] M. Sofonea & Y. Souleiman, Analysis of a Sliding Frictional Contact Problem withUnilateral Constraint, Mathematics and Mechanics of Solids. 22 (2015), 324–342.

[4] M. Sofonea & Y. Souleiman, A Viscoelastic Sliding Contact Problem with NormalCompliance, Unilateral Constraint and Memory Term, Mediterranean Journal of Math-ematics. 13 (2016), 2863–2886.

95

New perturbation bounds for matrix decom-positions

Weiwei Sun1

1City University of Hong Kong, Hong Kong, China

AbstractThis talk focuses on recent results of perturbation bounds in matrix decompositions, such as (gen-eralized) polar decompositions, eigen-decomposition and QR decomposition. Classical resultscan be found in [1, 2, 3, 4, 5]. First, we present several new perturbation bounds for (generalized)polar decompositions under the Frobenius norm for both complex and real matrices. Our boundsare simpler and sharper than those in previous studies. Secondly, we present some new perturba-tion bounds in a combined form for all these three matrix decompositions. Both factors in eachmatrix decomposition are involved in the combined bounds and these bounds areoptimal for eachfactor.

References

[1] N.J. Higham, Computing thepolar decomposition-withapplications, . SIAM J Sci StatistComput, 7)1986) pp.1160–1174.

[2] W. Li and W. Sun, Combined perturbation bounds: I eigensystem and Singular valuesystem, SIAM Matrix Analysis, 29(2007), pp.643–655.

[3] W. Li and W. Sun, New perturbation bounds for unitary polar factors. SIAM J MatrixAnal. Appl., 25(2003), pp. 362–372.

[4] W. Li and W. Sun, Perturbation boundsof unitary and subunitary polar factors, SIAM J.Matrix Anal. Appl., 23(2002), pp.1183–1193.

[5] G.W. Stewart and J.G. Sun, Matrix PerturbationTheory, Boston: AcademicPress, 1990.

96

Numerical solution of a new fractional epidemicmodel

Mostafa Tahiri1, Moulay Rchid Sidi Ammi1

1Department of Mathematics, AMNEA Group, Faculty of Sciences and Techniques, Moulay Ismail University, B. P. 509,Errachidia, Morocco

AbstractThere are several definitions for fractional derivatives; in this paper we choose to work with theAtangana-Baleanu-Caputo fractional derivative. One of the advantages of such derivative is al-lowing us to consider classical initial conditions to be included in the formulation of the problem.Also, the Caputo fractional derivative of a constant is zero, which is not true for other fractionalderivatives, for example, Riemann-Liouville.

In order to solve the initial value problems, Atangana and Baleanu suggested a new operatorwith fractional order based upon the Mittag-Leffler function. Their operators have all the benefitsof that of Caputo and Fabrizio in addition the kernel used is nonlocal. The operators have allthe benefits of those of Riemann–Liouville and Caputo; in addition the kernel is non-singular.Interesting observation is that their fractional integral is the fractional average of the Riemann–Liouville fractional integral of the given function and the function itself.

Several research papers have been published using a new concept of fractional differentiationwith Mittag-Leffler. More importantly the results revealed that, the new concept of more adequatefor modeling real world problems to take into account the non-locality and also to have a memoryeffect.

In this work, we study an epidemic model with Atangana-Baleanu-Caputo fractional deriva-tives. We have obtained a special solution using an iterative scheme via Laplace and the Sumudutransformation. Uniqueness and the existence of a solution using the Banach fixed point theo-rem are studied. A detailed analysis of the stability and convergence of the special solution waspresented. Finally, our generalized model in the derivative sense was solved numerically by theAdams-Bashforth-Moulton method.

——————————————————-

References

[1] Abdeljawad, T. and Baleanu, D. (2016). Integration by parts and its applications ofa new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel. arXivpreprint arXiv, 1607.00262.

[2] Alkahtani, B.S.T. (2016). Chua’scircuit model with Atangana–Baleanu derivativewithfractional order. ChaosSolitonsand Fractals. 89, 547–551.

[3] Atangana, A. and Koca, I. (2016). Chaos in a simple nonlinear system with Atangana–Baleanu derivativeswith fractional order. Chaos, Solitonsand Fractals. 89, 447-454.

97

A fully implicit finite volume scheme for sea-water intrusion models in coastal aquifers

B. Amaziane1, M. El Ossmani2, K. Talali2

1 CNRS / Univ Pau & Pays Adour /E2S UPPA, LMAP, Federation IPRA, UMR5142, 64000, Pau, France.2 University Moulay Ismail, EMMACS-ENSAM, Marjane 2, 50000 Meknes, Morocco.

AbstractThis work deals with numerical simulations of the sharp interface seawater intrusion model [1] incoastal aquifers, performed by meansof afinitevolumemethod. Modeling such problems leads toastrongly nonlinear coupled system of two parabolic partial differential equations with freshwaterand saltwater heads are the unknowns.

In this talk, we propose a fully implicit cell-centered finite volume scheme to discretize thesharp interface seawater intrusion model. An implicit, Euler time discretization is employed, thatallowstheuseof much longer timestepsthan explicit schemes. To approximatethediffusiveterms,a conforming finite element scheme with piecewise linear elements is performed. The nonlinearsystem is solved by Newton’s method and a preconditioned BiConjugate Gradient STABilized(BiCGSTAB) is used to solve the linear systems. Numerical differentiations techniques are usedto approximate the derivatives in the calculation of the Jacobian matrix. The control of the timestep is based on the number of iterations required by the Newton method to achieve convergencefor the last time iteration.

Wehavedeveloped and implemented thisschemein anew modulein thecontext of theparallelopen source platform DuMux[3], based on the Distributed and Unified Numerics EnvironmentDUNE. The accuracy and effectiveness of this new module is demonstrated through numericalinvestigation. Numerical simulationsarecompared to an analytical solution proposed by Keuleganand thetestsproposed in [4]. Thepreliminary resultsachieved arepromising and numerical resultsare in good agreement with those in [4]. Future works will focus on the numerical analysis of thescheme and the extension of the methodology to more advanced models, like the mixte sharp-diffuse interface model recently introduced in [2].

References

[1] J. Bear. Dynamicsof fluids in porousmedia. American Elsevier. New York, 1972.

[2] C. Choquet, M. Diedhiou, C. Rosier. Derivation of a sharp-diffuse interfaces modelfor seawater intrusion in a free aquifer. Numerical simulations. SIAM J. Appl. Math.76:138–158, 2016.

[3] DuMux: Dune for multi- Phases, Component, Scale, Physics, ... flow and transportin porousmedia. http://www.dumux.org/, last accessed January 15, 2019.

[4] P. Marion, K. Najib, C. Rosier. Numerical simulations for aseawater intrusion problemin a freeaquifer. Appl. Numer. Math. 75:48–60, 2014.

98

A seventh order bi-quintic B-spline colloca-tion method applied to biharmonic Dirichletproblems

H. Allouche1, A. Tazdayte1

1Moulay Ismail University, Faculty of Sciences Department of Mathematics and Computer ScienceMAN-TA Research Team B.P. 11 201, Zitoune, Meknes, Morocco

AbstractIn this paper, we investigate the application of the bi-quintic B-spline collocation method to thenumerical solution of biharmonic Dirichlet problem on the rectangular domain. The standardbi-quintic B-spline collocation provides only second order convergence. To overcome this limi-tation, we formulate two seventh order collocation schemes, namely, one-step left and right-handsidesperturbation (OSLRP) collocation method and tow-step left and right-hand sidesperturbation(TSLRP) collocation method. Themain ideaof theproposed techniqueconsists in annihilating thederivatives in the residual of the differential operator by applying a high order perturbation to theresidual of the differential operator which in consequence increase therateof convergence. A ma-trix decomposition algorithm is developed for the fast solution of the linear system in the case oftwo-step approach. Convergence analysis isdiscussed and thenumerical illustration arepresentedto confirm the efficiency of the proposed method.

99

Approximating the extreme Ritz values in CG

Gerard Meurant1, Petr Tichy2

130 rue du sergent Bauchat, 75012 Paris, France2Charles University, Sokolovska 83, 18675 Prague, Czech Republic

AbstractIn practical conjugate gradient (CG) computations it is important to monitor the quality of theapproximate solution to Ax = b so that the CG algorithm can be stopped when the requiredaccuracy is reached. The relevant convergence characteristics, like theA-norm of the error or thenormwise backward error, cannot be easily computed. However, they can be estimated. Suchestimates often depend on approximations of the smallest or largest eigenvalue of A.

The smallest and largest eigenvalues of A can be approximated during the CG computationsusing the smallest and largest Ritz values. We will describe a very cheap algorithm, based onthe incremental norm estimation technique, which allows to estimate the smallest and largest Ritzvalues. An improvement of the accuracy of these estimates is possible, at the cost of storing theCG coefficients and solving a linear system with a tridiagonal matrix at each CG iteration. Thecheap approximations to the smallest and largest Ritz values can be used not only to approximateconvergence characteristics, but also the condition number of A, or the ultimate level of accuracy.

Wefurther introduceanew upper bound for theA-norm of theerror, which isclosely related totheGauss-Radau upper bound, and discuss theproblem of choosing theparameter µ which shouldrepresent a lower bound for the smallest eigenvalue of A. The new bound has several practicaladvantages, themost important one is that it can beused asan approximation to theA-norm of theerror even if µ is not exactly a lower bound for the smallest eigenvalue of A. Therefore, µ can bechosen, e.g., as a cheap approximation to the smallest Ritz value mentioned above.

The numerical experiments demonstrate the efficiency of the estimates of the extreme Ritzvalues, and show their practical use in error estimation in CG.

References

[1] G. Meurant and P. Tichy. On computing quadrature-based bounds for the A-norm ofthe error in conjugategradients. Numer. Algorithms, 62(2):163–191, 2013.

[2] G. Meurant and P. Tichy. Approximating theextremeRitz valuesand upper boundsforthe A-norm of theerror in CG. Numer. Algorithms, published onlinefirst, 2018.

100

The erosion of a sand bed by the flow effect

H. Torbi1, F. Boushaba2, A. Yachoutti3, S. Daoudi4

1University Mohammed 1st, laboratory of mechanical and energetic, Faculty of sciences2University Mohammed 1st, laboratory of Mechanics and Numerical Modeling, ENSAO3University Mohammed 1st, laboratory of Mechanics and Numerical Modeling, ENSAO4University Mohammed 1st, laboratory of mechanical and energetic, Faculty of sciences

AbstractIn this work we are concerned with the construction and application of finite volume methodfor solving the morphodynamics models in contracting channel flows. The governing equationsdescribe thehydrodynamics for the free-surfaceflow and morphodynamics for thesediment trans-port. Here, we consider thewell-established shallow water equations to model thehydrodynamicsand Exner equation to model the morphodynamics. We use a new version of Roe scheme namedSRNH written in finite volumes method. This is to model the spread of a flood wave along thecanal equipped with regularly spaced identical grooves which plays the role of macro roughness.Finally the comparison is illustrated with a simple test case of contracting channel flow and weobserve the influence of macro roughness on the transport and deposition of sediments.

References

[1] F. BENKHALDOUN , I . ELMAHI , M. SEAID, Well-balanced finite volume schemesfor pollutant transport by shallow water equationson unstructured meshes;J. Comput.Phys. 226 180203, 2007.

[2] J. HUDSON, Numerical techniquesfor morphodynamicmodelling;Dissertation, Univer-sity of Reading , 2001.

[3] F. BENKHALDOUN , S. SAHMIM , M. SEAID, Solution of the sediment transportequations using a finite volume method based on sign matrix;SIAM J. Sci. Comp. 3128662889, 2009.

[4] F. BENKHALDOUN , S. DAOUDI , I . ELMAHI , M. SEAID, Comparison of unstructuredfinite-volume morphodynamic models in contracting channel flows; Mathematics andComputers in Simulation 81 20872097, 2011.

101

Biorthogonal rational Krylov subspacemethods

Niel Van Buggenhout1, Marc Van Barel1, Raf Vandebril1

1KU Leuven, Department of Computer Science, Celestijnenlaan 200A box 2402, 3001 Leuven, Belgium

AbstractRational Krylov subspace methods use rational functions of a given matrix instead of polynomi-als to construct a Krylov subspace. The poles of these rational functions can be chosen and allowusto manipulatetheconvergenceof themethodsto an areaof interest in thespectrum of thematrix.

Here we show that biorthogonal projection onto rational Krylov subspaces can be represented bya matrix pencil consisting of two tridiagonal matrices. This is a very elegant generalization of thestandard biorthogonal Lanczosalgorithm and therational Arnoldi algorithm. Theformer results inatridiagonal matrix and the latter in amatrix pencil of upper-Hessenberg matrices. Thetridiagonalmatrix pencil naturally leads to a six-term recursion and a Lanczos-type algorithm that is valid ina very general setting, i.e., no special structure of the given matrix is required and the poles of theKrylov subspaces do not have to be known a priori.

A novel nonsymmetric rational Lanczos-type algorithm isproposed and itsvalidity is tested usingsomenumerical experiments. Thisalgorithm generalizes, besidesLanczosand biorthogonal Lanc-zos, several known algorithms, e.g., AGR/CMV-factorization and more recent results concerningextended Krylov subspaces.

References

[1] Watkins, D. S. (1993). Some perspectives on the eigenvalue problem. SIAM review,35(3), 430-471.

[2] Ruhe, A. (1998). Rational Krylov: A practical algorithm for largesparsenonsymmetricmatrix pencils. SIAM Journal on Scientific Computing, 19(5), 1535-1551.

[3] Jagels, C., & Reichel, L. (2011). Recursion relations for the extended Krylov subspacemethod. Linear AlgebraAppl, 434(7), 1716-1732.

[4] Van Buggenhout, N., Van Barel, M., & Vandebril, R. (2018). Biorthogonal ExtendedKrylov SubspaceMethods. arXiv preprint arXiv:1809.07660.

102

Dimensionality reduction and class predictionalgorithm with application to microarray BigData

Abdelhak Zoglat1, Fadoua Badaoui2, Amine Amar3, Laila Ait Hassou4,Cyrille Guei Okou5

1Laboratory of Mathematics, Statistics and Applications, Department de Mathematics, Faculty of Sciences, MohammedV University in Rabat. Rabat, Morocco.2Department of Statistics, Demography et Actuarial Science, National Institute of Statistics and Applied Economics.Rabat, Morocco.3Moroccan Agency for Sustainable Energy, Rabat, Morocco.4Laboratory of Mathematics, Statistics and Applications, Department de Mathematics, Faculty of Sciences, MohammedV University in Rabat. Rabat, Morocco.5Unité de Formation et Recherche (UFR) Environnement, Université Jean Lorougnon Guédé de Daloa, Daloa, Côted’Ivoire

RésuméThe recent technology development in the concern of microarray experiments has provided

many new potentialities in terms of simultaneous measurement. But new challenges have arisenfrom these massive quantities of information qualified as Big Data. The challenge consists to ex-tract themain information containing thesense from thedata. To thisend researchersareusing va-rioustechniquesashierarchical clustering, mutual information and self-organizing mapsto nameafew. However, themanagement and analysisof themillionsresulting dataset haven?t yet reached asatisfactory level, and thereisno clear consensusabout thebest method/methods revealing patternsof gene expression. Thus, many efforts are required to strengthen the methodologies for optimalanalysis of Big Data. In this paper, we propose a new processing approach which is structured onfeature extraction and selection. The feature extraction, is based on correlation and rank analysisand leads to a reduction of the number of variables. The feature selection, consists in eliminatingredundant or irrelevant variables, using some adapted techniques of discriminant analysis. Our ap-proach is tested on threetypeof cancer geneexpression microarray and compared with concurrentother approaches. It performs well, in terms of prediction results, com- putation and processingtime.

Keywords : Linear discriminant analysis, Tumor classification, Basis vectors, Kendall rankcorrelation, Cancer

103

Imaging of an elastic ellipsoidusing near-field data

E. S. Athanasiadou1, S. Zoi2

1Department of Mathematics, National and Kapodistrian University of Athens.2Department of Mathematics, National and Kapodistrian University of Athens.

AbstractThe scattering problem of time-harmonic elastic waves from a triaxial ellipsoidal obstacle is con-sidered. A low-frequency formulation of the direct scattering problem is described. A methodusing near-field data for solving the corresponding inverse scattering problem in order to specifythe size and the orientation of the ellipsoid is presented. A matrix with elements given in terms ofafinitenumber of measurements of thescattered field isconstructed. Theeigenvalues of thismea-surement matrix provide information for the semi-axes of the ellipsoid whereas the eigenvectorstogether with a rotation matrix whose elements are given in terms of the Euler angles are used tospecify the orientation of the ellipsoid. Corresponding results for geometrically degenerate casesof the ellipsoid such as spheroids, spheres, needles and discs are obtained for appropriate valuesof the geometrical parameters.

References

[1] C. E. Athanasiadis, E. Athanasiadou, S. Zoi and I. Arkoudis, An inversescattering problem for thedielectric ellipsoid, AIPConference Proceedings 1978, 470055 (2018), doi: 10.1063/1.5044125.

[2] T. Apostolopoulos, K. Kiriaki and D. Polyzos, Theinversescattering problem for arigid ellipsoidin linear elasticity, Inverse Problems, 6, pp. 1-9, (1990).

[3] G. Dassios and K. Kiriaki, The rigid ellipsoid in the presence of a low-frequency elastic wave,Quarterly of Applied Mathematics, 43, 4, pp. 435-457, (1986).

[4] G. Dassios and R. Kleinman, Low Frequency Scattering, Oxford University Press, (2000).

[5] V. D. Kupradze, T. G. Gegelia, M. O. Basheleishvili and T. V. Burchuladze, Three DimensionalProblemsof theMathematical Theory of Elasticity and Thermoelasticity, North Holland Publish-ing Company, (1979).

104

First Name Last Name City, country

Ouaddah Abdelhamid Algeria

Mohammed Abdellaoui Fez, Morocco

Nawel Abdesselam Laghouat, Algeria

Belkacem Abdous Rabat, Morocco

Jilali Abouir Mohammedia, Morocco

Marwan Abukhaled Sharjah, United Arab Emirates

Karima Adel-Aissanou Bejaia, Algeria

Mehdi Adib Mohammedia, Morocco

Imane Agmour Casablanca, Morocco

Said Agoujil Errachidia, Morocco

Ahmed Ahmed Mahdi Dhahran, Saudi Arabia

Nabila Aiane Bejaia, Algeria

Fatima Zahra Ait Bella Morocco

Mohamed Aitichou Morocco

Rim Akhrif Tetouan, Morocco

Nader Al Rachidi United Kingdom

Mohammad W. Alomari Irbid, Jordan

Sultan Alyobi Newcastle upon Tyne, United Kingdom

Mohamed Ridouan Amattouch Tangiers, Morocco

Siham Aouissi Morocco

Abderrahim Aslimani Morocco

Ilham Asmouh Benguerir, Morocco

Evagelia Athanasiadou Athens, Greece

Bassem S. Attili Sharjah, United Arab Emirates

Adnane Azzouzi Taroudant, Morrocco

Nossaiba Baba Morocco

Achraf Badahmane Marrakesh, Morocco

Leila Baiche Bejaia, Algeria

Domingo Barrera Granada, Spain

Khouya Bassou Benguerir, Morocco

Mohammed Bassoudi Laghouat, Algeria

Benaoumeur Bayour Mascara, Algeria

Farah Belaadich Fez, Morocco

Hassan Belhadj Tangiers, Morocco

Walid Ben Aribi Tunis, Tunisia

Lahbib Benahmadi Casablanca, Morocco

Abderrahim Benazzouz Rabat, Morocco

Abdessalem Bennamar Cheraga, Algeria

Brahim Benouahamne Mohammedia, Morocco

Abdeslem Hafid Bentbib Marrakesh, Morocco

El Houcine Bergou Jouy en Josas, France

Mohammed Berkhouch Agadir, Morocco

Francisco Bernal Huelva, Spain

Abdelouahab Bibi Constantine, Algeria

Abdelkader Bouajaja Settat, Morocco

LIST OF PARTICIPANTS

Naziha Bordj Algiers, Algeria

Hssaine Boualam Settat, Morocco

Redouane Bouchouirbat Benguerir, Morocco

Karima Boufi Settat, Morocco

Feriel Bouhadjera Annaba, Algeria

Ouafae Boulerhcha Oujda,Morocco

Khalifa Boumzough Agadir,Morocco

Hamza Boutayeb Morocco

Habib Bouzir Mascara, Algeria

Claude Brezinski Lille, France

Mikhail Bulatov Irkutsk, Russia

Hassan Cheriha Nice, France

Moulay Abdallah Chkifa Benguerir, Maroc

Anna Concas Cagliari, Italy

Mohamed Dalah Constantine, Algeria

Hanane Dalimi Casablanca, Maroc

Jaouad Danane Casablanca, Morocco

Achiya Dax Jerusalem, Israel

Dries DeSamblanx Leuven(Heverlee), Belgium

Patricia Díaz de Alba Cagliari, Italy

Mohamed A. Djehaf Sidi bel Abbes, Algeria

Youcef Islam Djilani Kobibi Mascara, Algeria

Froilán Dopico Madrid, Spain

Irada Dzhalladova Ukraine

Abdelwahab Elaassri Morocco

Abdessamad El Ansari Temara, Morocco

Safaa El Gharbi Rabat, Morocco

Mohamed El Ghomari Marrakesh, Morocco

Mounia El Hafyani Temara, Morocco

Badr El Haji Fez, Morocco

Smahane El Halouy Marrakesh, Morocco

Raghia El Hanine Marrakesh, Morocco

Atmane El Houch Casablanca, Morocco

Abdennasser Elidrissi Mohammedia, Morocco

Amine El Koufi Casablanca, Morocco

Abdessamad El Madkoury Marrakesh, Morocco

Abderrahman El Maliki Casablanca, Morocco

Nabil El Moçayd Benguerir, Morocco

Mustapha El Moudden Meknes, Morocco

Youness El Yazidi Marrakesh, Morocco

Abdenasser Elidrissi Mohammedia, Morocco

Fevzi Erdogan Van, Turkey

Mohamed Errachid Rabat, Morocco

Youssef Errai Morocco

Abdelhak Essanhaji Rabat, Morocco

Mohammed Es Sebaiy Marrakesh, Morocco

Mouna Essaouini El Jadida, Morocco

Mohamed Farhloul Moncton, Canada

Abderrahmane Ettahiri Ouarzazate, Morocco

Mohamed Amin Farid Casablanca, Morocco

Caterina Fenu Cagliari, Italy

Najib Guessous Fez, Morocco

Paraskevi Fika Patras, Greece

Anass Guitanou Larache, Morocco

Mustapha Hached Lille, France

An Haichao Montreal, Canada

Riane Hajjami Tangiers, Morocco

Mohamed Amine Hamadi Benguerir, Morocco

Farida Hamrani Algiers, Algeria

Mohamed Heyouni Oujda, Morocco

El Houcine Hibba Beni Mellal, Morocco

Najima Houari Algeria

M'barek Iaousse Kenitra, Morocco

Abdennaceur Jarray Gabes,Tunisia

Khalide Jbilou Calais, France

Bouchta Jouilik Kenitra, Morocco

Ahmed Kanber Marrakesh, Morocco

Yassine Kaouane Calais, France

Kamal Khalil Marrakesh, Morocco

Imad Kissami Fontenay aux Roses, France

Abdelouahed Kouibia Granada, Spain

Karima Kreit Marrakesh, Morocco

Ekaterina Kruglova Novosibirsk, Russia

Radek Kucera Ostrava, Czech Republic

Ahlam Labdaoui Constantine, Algeria

Kamal Lachhab Morocco

Meriem Lamraoui Annaba, Algeria

Fatima Zahra Lamzouri Oujda, Morocco

Hamid Lefraich Settat, Morocco

Leila Loudiki Safi, Morocco

Salaheddine Maaqoul Tangiers, Morocco

Aziz Madrane Montreal, Canada

Mohamed Mandari Meknes, Morocco

Abdelouahab Mani Bordj Bou Areridj, Algeria

Meryem Meddahi Chlef, Algeria

Abderrahim Messaoudi Rabat, Morocco

Ismail Merabet Ouargla, Algeria

Ahcene Merad Oum el Bouaghi, Algeria

Marilena Mitrouli Athens, Greece

Jabrane Moustaaid Marrakesh, Morocco

Bechir Naffeti Tunis, Tunisia

Nabila Nagid Tangiers, Morocco

Nihad Nouri Tipaza, Algeria

Abdelouahed Ouardghi Benguerir, Morocco

Elias Ould Said Lille, France

Abdessamad Ousaadane Marrakesh, Morocco

Miguel Pasadas Granada, Spain

Constantin Petridi Athens, Greece

Harshana Rajakaruna Knoxville TN, USA

Michela Redivo-Zaglia Padova, Italy

Imad Rezzoug Oum el Bouaghi, Algeria

Mohamed Rhoudaf Meknes, Morocco

Oussama Rida Berrechid, Morocco

Mohamed Rossafi Morocco

Yousef Saad Minneapolis, USA

Rachid Sadaka Rabat, Morocco

El Mostafa Sadek El Jadida, Morocco

Lakhlifa Sadek El Jadida, Morocco

Hassane Sadok Calais, France

Abdelali Saidi Kenitra, Morocco

Nourddin Saidou Fez, Morocco

Fatima Ezzahra Saissi Safi, Morocco

Said Salim Safi, Morocco

Abdellatif Semmouri Beni Mellal, Morocco

Fatima Sennouni Morocco

Sarah Sidki Rabat, Morocco

Soumaia Slimani Rabat, Morocco

Mohammed Smihi Fez, Morocco

Ouafa Soualhi Meknes, Morocco

Yahyeh Souleiman Djibouti, Djibouti

Anton Sudirman Bandung, Indonesia

Weiwei Sun Hong Kong, China

Mostafa Tahiri Errachidia, Morocco

Kamel Tahri Tlemcen, Algeria

Khadija Talali Meknes, Morocco

Imran Talib Lahore, Pakistan

Kenza Tamiti Algeria

Abderrahim Tazdayte Meknes, Morocco

Jurjen Duintjer Tebbens Prague, Czech Republic

Abdelmoutia Telli Biskra, Algeria

Petr Tichy Prague, Czech Republic

Halima Torbi Oujda, Morocco

Azzeddine Tsouli Casablanca, Morocco

Niel Van Buggenhout Leuven, Belgium

Anis Younes Tunis, Tunisia

Faiza Zaamoune Biskra, Algeria

Az-eddine Zakrad Marrakesh, Morocco

Benaissa Zerroudi Kenitra, Morocco

Lhassan Zmour Agadir, Morocco

Stefania Maria Zoi Athens, Greece