Book of AbstractsThe International Conference

Mathematical Modeling of Complex Systems (M2CS)

Marrakech, 13--16 April 2020

Abdesslem Bentbib

Mustapha HachedKhalide Jbilou

Preface

The conference « Mathematical Modeling of Complex Systems » (M2CS), co-organized by 'UniversityMohammed VI Polytechnique, Benguérir, Morocco', Faculté des Sciences et Techniques UniversityCadi Ayyad, Marrakech, Morocco and University Littoral Côte d'Opale in Calais France, will be heldat Faculté des Sciences et Techniques, University Cadi Ayyad in Marrakech, Morocco, form April 13to April 16, 2020. The conference is celebrating the 70th birthday of Paul Van Dooren and the 60th birthday of HassaneSadok for their many contributions to numerical linear algebra. The aim of this conference is to bringtogether researchers working in different topics from mathematical modeling and applications. Therewill be more than 130 participants from Morocco and other countries around the world (23 differentnationalities). 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, computational statistics and applications in machine learning, datta mining andArticificial Intelligence . A special issue of the international journal “Electronic Transaction on Numerical Analysis” will beedited for the selected papers.

The Organizers (Chairs)

Abdesslem Bentbib Khalide Jbilou

Mat h ematical Modeling of Complex Systems (M2 CS)

Invited Plenary Speakers

Michele Benzi, ItalyClaude Brezinski, FranceFroilan Dopico, SpainMarilena Mitrouli, GreeceGérard Meurant, France

Marcos Raydan, PortugalMichela Redivo-Zaglia, ItalyHassane Sadok, FranceLothar Reichel, USAPaul Van Dooren

Organizing Committee

Noureddine Alaa, MoroccoFayssal Benkhaldoun, FranceAbdesslem Bentbib (Chair), MoroccoMustapha Hached, France

Khalide Jbilou (Chair), France Ahmed Kanber, MoroccoAbderrahim Messaoudi, MoroccoAhmed Ratnani, MoroccoRachid Sadaka, Morocco

Scientific Committee

Abdelkacem Abdous, MoroccoJilali Abouir , MoroccoSaid Agoujil, MoroccoNoure Eddine Alaa, MoroccoMichele Benzi, ItalyMohammed Benjelloun, FranceFayssal Benkhaldoun, FranceAbdesslem Bentbib, MoroccoAbderrahman Bouhamidi, FranceClaude Brezinski, FranceAbdellah Chkifa, MoroccoAlbert Cohen, FranceFroilan Dopico, SpainVladimir Druskin, NLSouad Elbernoussi MoroccoAbdellatif Ellabib, MoroccoJocelyne Erhel, FranceAbdelillah Hakim, MoroccoMohammed Heyouni, MoroccoKhalide Jbilou, France

Christos Koukouvinos, GreeceAbderrahim Messaoudi, MoroccoGérard Meurant, FranceMarilena Mitrouli, GreeceMarc Prévost, FranceSaid Raghay, MoroccoAhmed Ratnani, MoroccoMarcos Raydan, PortugalMichela Redivo Zaglia, ItalyLothar Reichel, USACarole Rosier, FranceYousef Saad, USARachid Sadaka, MoroccoHassane Sadok, FranceAhmed Salam, FranceMohammed Seaid, UKPaul Van Dooren, BelgiumMarc Van Barel, BelgiumRaf Vandebril, Belgiu

Sponsors of the Conference

• University Mohammed VI Polytechnique (UM6P), Benguérir

• University Cadi Ayyad (UCA), Marrakech

• Faculté des Sciences et Techniques (FST), Marrakech

• Université du Littoral Côte d'Opale (ULCO), France

• Ecole Normale Supérieure (ENS), Rabat

• Association Marocaine de Modélisation et Ingénierie Mathématiques (A2MIM)

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

• 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. Petridi, Athens, Greece.

Acim, M.A Comparison between the Financial and the

Islamic Market of MalaysiaMorocco

Adib, M.A generalized matrix Krylov subspace method for hybrid

variational model for image restorationMorroco

Ahusborde, E.Finite volume schemes for two-phase reactive flow in

porous mediaFrance

Aiane, N. Stochastic ill-posed problem with $\alpha$-mixing errors Algeria

Al Nazer, S.

Anderson acceleration and vector extrapolation

methods for solving thermodynamic chemical

equilibrium in porous media

France

Al Qahtani, H. Generalized block anti-Gauss quadrature rules Saudi Arabia

Alsenafi, A.

Parameter estimation for Gaussian mean-reverting

Ornstein–Uhlenbeck processes of the second kind: Non-

ergodic case (Poster)

Kuwait

Alsuwalih, F.Berry-Essen bounds for drift parameter estimation of

discretely observed fractional Vasicek-type boundsKuwait

Amir, L.Jacobian free methods for coupling transport with

chemistry in heterogenous porous mediaMorocco

Andjelic, M.A Sylvester-Kac matrix type and controllability

of half graphsKuwait

Anton, C. Stochastic Runge-Kutta pseudo-symplectic methods Canada

Arkoudis, IElectromagnetic scattering by a chiral obstacle in a chiral

environment using the method of fundamental solutionsGreece

Awais, AAnalysis of Common Fixed Point of F-contractions on

Closed BallPakistan

Bailoul, C.E.Study of a quasilinear parabolic equation with

superquadratic growth non-linearityMorocco

Balhag, A.Solution of the bi-level problem via first order differential

equation systemMorocco

Barkouki, H.A rational Krylov subspacemethod for the computation

of the matrix exponentialMorocco

Bellaihou, M. Spherical interpolatory geometric subdivision schemes Morocco

Ben Loghfyry, A. A fractional diffusion PDE in image denoising Morocco

Name Title of the abstract Country

Benzi, M.Block preconditioners for the coupled Darcy-Stokes

problem: eigenvalue and field-of-values analysisItaly

Bouchriti, A.Remarks on the asymptotic behavior of scalar auxiliary

variable (SAV) schemesMorocco

Bouhlal, L.Analysis of the dynamic response of a railway vehicle

modeled in 1D and 2DMorocco

Bourel, C.Modeling of shallow aquifers in interaction with overland

waterFrance

Brezinski, C. Richardson, Romberg, and others France

Cai, W.A phase shift deep neural network algorithm for high

frequency wave equationsUSA

Carrayrou, J.Equilirium chemistry modelling : from Newton to combined

algorithmsFrance

Chakir, Y.Multidimensional Laplace transform inversion using

multivariate rational approximantsMorocco

Charkaoui, A.

Existence and uniqueness of a renormalized periodic

solution of a parabolic equation with variable exponent

and L1 data

Morocco

Chehab, J.P.Times schemes with high-modes stabilization techniques

for nonlinear parabolic equationsFrance

Chhaibi, R. On the numerics of convolution flows in free probability France

Chkifa, M.A.On a fast hierarchical sparse grid quadrature and

applicationsMorocco

Cipolla, S. Extrapolation methods for Nonlinear Problems Italy

Cohen, A.Optimal hierarchical sampling and reconstruction using

weighed least squaresFrance

Dermoune, A. Interpolation, approximation and prediction France

Dmytryshyn, A. Canonical structure transitions of system pencils Sweden

DolbeaultAlgorithms for optimal sampling in weighted least-squares

methods on general domainsFrance

Name Title of the abstract Country

Dopico, F.Conditioning and backward errors of eigenvalues of matrix

polynomials under Möbius transformationsSpain

Dufrenois, FOnline learning in large-scale data sets for anomaly

detectionFrance

Eguillon, Y.Jacobian-free Newton-type implementation of IFOSMONDI

Co-simulation AlgorithmFrance

El Asnaoui, K.Slurry transport in pipelines usingmachine learning

techniquesMorocco

El Ghomari, M.Extended-rational global Arnoldi method for the matrix

function approximationMorocco

El Guide, M.Efficient numerical solvers for the RBF solution of Helmoltz

equations in three space dimensionsMorocco

El Halouy, S.Extended Krylov subspace methods for solving Sylvester

and Stein tensor equationsMorocco

El Idrissi, AExtended symbolic Gaussian cubature over concurrent

squaresMorocco

El Moussaoui, A.

2D meso modelling of crowd motion with the kinetic theory

approach : influence of model parameters on the

evacuation time

Morocco

El Yazidi, Y.Preconditioned RBF meshless method applied in free

boundary identification problemMorocco

Errachid, M.Bivariate polynomial interpolation of Lagrange and Hermite

: A general recursive resolutionMorocco

Ezzetouni, A.On the numerical solution of fractional stochastic

differential equations with jumpsMorocco

Fahim, H.An efficient identification of red blood cell equilibrium

shape in 2D using shape optimization and neural networksMorocco

Fakhouri, I.Systems of multidimensional BSDEs arising in the balance

sheet optimal switching problemMorocco

Gratien, JMBenchmark of various multi-level linear solvers with ALIEN,

an open generic and extensible linear algebra frameworkFrance

Hamad, D.Spectral clustering: incremental and evolutionary

algorithmsFrance

Name Title of the abstract Country

Hamadi, M.A.Stabilization of incompressible flow problems using Riccati

feedback approach based on Krylov subspace methodsMorocco

Handa, M.Modeling and optimization of an energy distribution

systemFrance

Hay, I.

Improvemments of some iterative methods for a special

common solution of split equilibrium problems and fixed

points problems

Morocco

Hourri, M. Credit risk management by artificial intelligence Morocco

Ibrahimoglu, B.A.A modification and extension of the fast algorithm for

computing the mock-Chebyshev nodesTurkey

K. Abou El HosseinMolecular dynamics modelling in ultra-high precision

manufacturingSouth Africa

Kaouane, Y.

Model reduction for large scale first and second order

dynamical systems via an adaptive tangential Lanczos-type

method

Morocco

Karow, MApproximation of pseudospectra of block \triangular

matricesGermany

Kouibia, A. Numerical solution of second kind Volterra integral systems Spain

Laayouni, L.Algebraic optimized Schwarz methods for Black-Scholes

modelsMorocco

Lamghari, A.An efficient algorithm to compute pedestrians desired

directions in a dense crowdMorocco

Mas, J.Preconditioners to compute the least squares solution of

overdetermined linear systems with dense rowsSpain

Meurant, G.On the Gauss-Radau upper bounds for the $A$-norm of the

error in the Conjugate Gradient algorithmFrance

Mitrouli, M.Variable selection in statistical modelling via a numerical

linear algebra approachGreece

Moore, E.S.Space-time multipatch discontinuous galerkin isogeometric

analysis for parabolic evolution problemsGhana

Mula, O. State estimation with reduced models France

Nichols, J.Nonlinear reduced modeling and state estimation of

parametrics PDEsAustralia

Title of the abstract Country

Name Title of the abstract Country

Name

Ousaadane, A.On the convergence of Robin-Robin method for solving an

inverse problemMorocco

Pranic, M.Generalized Gauss-Kronrod quadrature rules for the

approximation of matrix functionalsBosnia

Psarrakos, P. Householder sets for matrix polynomials Greece

Quenjel, E.H.A free diminishing finite volume scheme for nonlinear

diffusion-convection problemsFrance

Raibi, O. FETI method in topology optimization Morocco

Raydan, M. Gauss-Newton approach for large-scale Riccati equations Portugal

Redivo Zaglia, M.Treatment of breakdown and near-breakdown in Lanczos-

type algorithms (the MRZ gang!)Italy

Reichel, L.Numerical methods for ill-posed problems with a sparse

and nonnegative solutionUSA

Rosier, C.A nonlinear optimization method applied to the hydraulic

conductivity identification in unconfined aquifersFrance

Rump, S. Estimates of the determinant of a perturbed identity matrix Germany

Ryckelynck, P. A model for the behaviour of an innovative entrepreneur France

Sadek, M.Extended Krylov methods for approximate of the matrix

exponentialMorocco

Sadok, H.Convergence analysis of some preconditioned Block Krylov

subspace methodsFrance

Salam, A. On the kernel of the vector Epsilon-algorithm France

Smail, L. Variable elimination algorithm: an updated version UAE

Spiteri, P.Asynchronous parallel subdomain methods coupled with

Krylov methodsFrance

Talali, K.Finite volume simulation of a sharp-diffuse model for

seawater intrusion in coastal aquifersMorocco

Title of the abstract CountryName

Taourirte, L.Weak solutions of some quasilinear singular elliptic

equations with data measuresMorocco

Tebbens, J.Efficient matrix computations for the minimum covariance

determinant estimatorCzech Republic

Tromeur-Dervout, D.Acceleration of the convergence of the Asynchronous RAS

methodFrance

Van Dooren, P. Linearizations of polynomial and rational matrices Belgium

Zoi, SScattering of thermoelastic waves by a multi-layered

obstacleGreece

Name Title of the abstract Country

A Comparison between the Financial and theIslamic Market of Malaysia

M.Acim1, B. Roukiane1, M.Zahid1, K.Akdim1

1Laboratory LAMSAFA, University Cadi Ayyad, Marrakesh, Morocco

AbstractIn this paper we study sukuk(Islamic bonds) of the Islamic financial market of Malaysia, since theypresent favorable conditions for investment which are compliant with Shariah principles, especiallypresent an attractive environment for both government and Islamic financial institutions. We usegovernment indexes of several maturities while comparing them with their counterparts (conven-tional bonds)in each maturity, over the periods from 2007 to 2017, while basing on the rate ofreturn, risk measures that we calculate using Garch and EGarch Models; we have confirmed thatmore the maturity is higher more the volatility is higher, for the conventional bonds and converselyfor the Islamic bonds. We searched also the causal links existing(we performed the Granger causal-ity tests that allow us to establish the causal relationships between bonds and Sukuk, where we haveconfirmed the existence of a single unidirectional relationship between these indices.

References[1] Haque, M. M., Chowdhury, M. A. F., Buriev, A. A., Bacha, O. I., and Masih, M., ”Who drives whom - sukuk

or bond? A new evidence from granger causality and wavelet approach” Review of Financial Economics 36(2018): 117132.

[2] Sew Lai Ng, Wen Cheong Chin, Lee Lee Chong,”Multivariate market risk evaluation between MalaysianIslamic stock index and sectoral indices” Borsa Istanbul Review xx (2016): 1-13.

[3] Mohamed Abulgasem .A. Elhaj, Nurul Aini Muhamed, Nathasa Mazna Ramli,”The Influence of CorporateGovernance, Financial Ratios, and Sukuk Structure on Sukuk Rating” Procedia Economics and Finance 31(2015) 62-74.

1

A generalized matrix Krylov subspace methodfor Hybrid variational model for image restora-tion

M. Adib1, J. Abouir1, K. Jbilou2

1University Hassan II- Casablanca, Faculty of Science and Technology Mohammedia, Laboratory LMCMAN, B.P. 146 Mohamme-dia, Morocco2L.M.P.A., Universite du Littoral, 50 rue F. Buisson BP699, F-62228 Calais-Cedex, France

AbstractRestoration is a fundamental research field in image processing. The purpose of image restorationis to recover an original image that has been digitized and has been degraded by blur and additivenoise. Recently Zhu et al [1] proposed a new hybrid total variation model by combining the secondorder total variation regularization and the total variation regularization. In this work we presenta fast algorithm to solve this new hybrid variational model.An augmented Lagrangian method isdeveloped to handle the constraints when the model is given with matrix variables, and an alternatingdirection method (ADM) is used to iteratively find solutionsof the subproblems. The numericalresults show the performance of our proposed algorithm.

References[1] J.ZHU, K.L I , B.HAO, (2019), Adv.Differ.Equ; Hybrid variational model based on alternating direction

method for image restoration.

[2] A.H.BENTBIB, M.EL GUIDE, K.JBILOU , (2019),Journal of Computational and Applied Mathematics; Ageneralized matrix Krylov subspace method for TV regularization, 112405.

[3] Y.H UANG, M.K.NG, Y.W.WEN, (2008),Multiscale Model.Simul.7(2); A fast Total variation minimizationmethod for image restoration, 774-795.

[4] J.LIU , T.Z.HUANG, X.G.LV, S.WANG, (2013),Abstr.Appl.Anal; An Efficient Variational Method for Im-age Restoration, 213536.

2

Finite volume schemes for two-phase reactiveflow in porous media

Etienne Ahusborde1, Brahim Amaziane1, Mustapha El Ossmani2, Mo-hamed Id Moulay1

1Universite de Pau et des Pays de l’Adour, E2S UPPA, CNRS, LMAP, Pau, France2University Moulay Ismaıl, EMMACS-ENSAM, Meknes, Maroc

AbstractReactive multiphase multicomponent flows in porous media are involved in many applications re-lated to subsurface environment and energy issues. Such flows are governed by a set of highly non-linear system of degenerate partial differential equations (describing a multiphase compositionalflow) coupled with algebraic and/or ordinary differential equations (related to geochemical model)requiring special numerical treatment. The numerical strategies for solving this system are dividedinto two categories: the global implicit and the sequentialapproaches. The global implicit approachsolves one nonlinear system gathering all equations at eachtime step while for the sequential ap-proach, flow and reactive transport are solved sequentiallyat each time step.In the framework of the parallel platform DuMuX [1], we have developed and implemented severalfinite volume schemes to tackle such problems. Firstly, we developed a sequential approach solvinga two-phase compositional flow problem and then a reactive transport problem using a direct substi-tution approach (DSA) [2]. Both subsystems are discretizedin a fully implicit manner. Nonetheless,sequential approaches can introduce operator splitting errors. By consequence, we developed andimplemented a fully-coupled, fully implicit method to solve reactive two-phase flows to achieveimproved stability [3]. Both strategies were validated by numerous test cases including High Per-formance Computing. An advanced comparison between both strategies for a three dimensionalscenario of geological storage of CO2 will be presented.

References[1] B. Flemisch, M. Darcis, K. Erbertseder, B. Faigle, A. Lauser, K. Mosthaf, S. Muthing S., P. Nuske, A.

Tatomir, M. Wolf, and R. Helmig. DuMu X : DUNE for multi-Phase, Component, Scale, Physics, ... flowand transport in porous media. Advances in Water Resources,Vol 3, 1102-1112, 2011.

[2] E. Ahusborde, B. Amaziane, M. El Ossmani, Improvement ofnumerical approximation of coupled two-phase multicomponent flow with reactive geochemical transport in porous media, Oil & Gas Science andTechnology - Rev. IFP Energies nouvelles, Vol 73, 73, 2018.

[3] E. Ahusborde, B. Amaziane, M. El Ossmani, M. Id Moulay, Numerical modeling and simulation of fullycoupled processes of reactive multiphase flow in porous media, Journal of Mathematical Study, Vol 52, 359-377, 2019.

3

Stochastic ill-posed problem withα-mixing errors

Nabila Aiane1, Abdelnasser Dahmani21Universite de Bejaia, Laboratoire de Mathematiques Appliquees, Bejaia, Algerie;2Centre universitaire de Tamanrasset, Laboratoire de Mathematiques Appliquees, Tamanrasset, Algerie

AbstractIn this work we consider the linear ill posed problem described by the operator equation Ax = u, inHilbert space in which the second member is measured with α-mixing errors. To solve this problemwe propose a stochastic procedure of Robbins-Monro type which converges almost completely tothe exact solution. To check the validity of our results, we consider some numerical examples.

References[1] Azadivar, F. (1999). Simulation Optimization Methodologies. In proceedings of the 1999 Winter Simulation

Conference. 93-100.

[2] Dahmani, A. Zerouati, H. Bouhmila, F. (2012). The L-pseudo-solution using stochastic algorithm of Landwe-ber. Meccanica. 47(8):1935-1943.

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

4

Anderson acceleration and vector extrapola-tion methods for solving thermodynamic chem-ical equilibrium in porous media

Safaa Al-Nazer1, Carole Rosier2, Mustapha Jazar3

1Lebanese University-Tripoli-Lebanon / ULCO-Calais-France2LMPA, Centre Universitaire de la Mi-Voix,50 rue F. Buisson, CS80699, 62228 Calais3Lebanese University-LaMA Liban-Laboratoire de Mathematiques et Applications.

AbstractModeling thermodynamic equilibrium of complex nonlinear chemical systems with the most usedNewton-Raphson (NR) method can lead to non convergence or excessive numbers of iterations.The efficiency of this method is improved by preconditioningtechniques and/or by coupling theNR method with a fixed point method on a particular formulation of the equilibrium system calledthe Positive Continuous Fraction (PCF) method. In this work, we solve the fixed point problemresulting from PCF method by using two iterative methods, other than NR, which are AndersonAcceleration method and Vector extrapolation methods. TheAnderson Acceleration method [2] isan algorithm for accelerating the convergence of fixed-point iterations, including the Picard method.Compared with a NR method, an advantage of Anderson acceleration is that there is no need toform the Jacobian matrix. Thus the method is easy to implement. A strategy is used to improvethe robustness of Anderson acceleration method which is to reduce the matrix condition number ofthe least squares problem in the Anderson-acceleration implementation so that numerical stabilitycan be guaranteed. Among the Vector Extrapolations methods, we consider the two polynomialmethods: reduced rank extrapolation (RRE) and the minimal polynomial extrapolation (MPE). Theaim of this methods is to transform a sequence of vectors generated by some process to a newone with the goal to converge faster then the initial sequence. We present our numerical results incomparison with those of J.Carrayrou [1] who used the NR method to solve the thermodynamicequilibrium. This comparison can show the high efficiency ofthe methods that we used.

References[1] J.Carrayrou. Modelisation du transport de solutes ractifs en milieu poreux sature. These de doctorat, Univer-

site Louis Pasteur, Strasbourg, 2001.

[2] H.F. Walker, P. Ni, Anderson ecceleration for fixed-point iterations, SIAM J. Numer. Anal. 49 (4) (2011)17151735.

5

Berry-Essen bounds for drift parameter esti-mation of discretely observed fractional Vasicek-type bounds

Fares Alsuwalih1, Abdulaziz Alsenafi2, Khalifa Es-Sebaiy3

1,2,3Department of Mathematics, Kuwait University, Kuwait, fares.alazemi@gmail.com, alsenafi@sci.kuniv.edu.kw,khalifa.essebaiy@ku.edu.kw

AbstractWe consider the least square-type estimation of the drift parameters for the fractional Vasicek-typemodel of the second kind Xt, t ≥ 0 defined as dXt = θ(µ+Xt)dt+dY

(1)t , t ≥ 0 with unknown

parameters θ > 0 and µ ∈ R, where Y (1)t :=

∫ t0 e

−sdBas with at = HetH , and Bt, t ≥ 0 is a

fractional Brownian motion of Hurst indexH ∈ (0, 1). We study the consistency and the asymptoticdistribution of our estimators of θ and µ based on the continuous-time observations Xt, t ∈ [0, T ]as T → ∞. Assume µ = 0, the parameter estimation of θ has been studied for H ∈ (0, 12). Herewe present a study valid for all H ∈ (0, 1). Our method is based on pathwise properties of X andY (1) proved in the sequel.

6

Generalized block anti-Gauss quadrature rules

Hessah Alqahtani1, Lothar Reichel2

1Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University, Rabigh 21911, Saudi Arabia. E-mail:hfalqahtani@kau.edu.sa2Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA. E-mail: reichel@math.kent.edu

AbstractGolub and Meurant describe how pairs of Gauss and GaussRadauquadrature rules can be appliedto determine inexpensively computable upper and lower bounds for certain real-valued matrix func-tionals defined by a symmetric matrix. However, there are many matrix functionals for which theirtechnique is not guaranteed to furnish upper and lower bounds. In this situation, it may be possibleto determine upper and lower bounds by evaluating pairs of Gauss and anti-Gauss rules. Unfor-tunately, it is difficult to ascertain whether the values determined by Gauss and anti-Gauss rulesbracket the value of the given real-valued matrix functional. Therefore, generalizations of anti-Gauss rules have recently been described, such that pairs ofGauss and generalized anti-Gauss rulesmay determine upper and lower bounds for real-valued matrixfunctionals also when pairs of Gaussand (standard) anti-Gauss rules do not. The available generalization requires the matrix that definesthe functional to be real and symmetric. The present paper reviews available anti-Gauss and general-ized anti-Gauss rules and extends them in several ways that allow applications in new situations. Inparticular, the genarlized anti-Gauss rules for a real-valued non-negative measure described in [5],are extended to allow the estimation of the error in matrix functionals defined by a non-symmetricmatrix, as well as to matrix-valued matrix functions. Modifications that give simpler formulas andthereby make the application of the rules both easier and applicable to a larger class of problemsalso are described.

References[1] C. Fenu, D. Martin, L. Reichel, and G. Rodriguez, Block Gauss and Anti-Gauss Quadrature with Application

to Networks, SIAM J. Sci. Comput., 35 (2013), pp. A2046–A2068.

[2] G. H. Golub and G. Meurant, Matrices, moments and quadrature, in Numerical Analysis 1993, eds. D. F.Griffiths and G. A. Watson, Longman, Essex, England, 1994, pp. 105–156.

[3] G. H. Golub and G. Meurant, Matrices, Moments and Quadrature with Applications, Princeton UniversityPress, Princeton, 2010.

[4] D. P. Laurie, Anti-Gauss quadrature formulas, Math. Comp., 65 (1996), pp. 739–747.

[5] M. Pranic and L. Reichel, Generalized anti-Gauss quadrature rules, J. Comput. Appl. Math., 284 (2015), pp.235–243.

7

Parameter estimation for fractional Vasicek-typemodel of the second kind: non-ergodic case

Fares Alazemi1, Abdulaziz Alsenafi2, Khalifa Es-Sebaiy3

1,2,3Department of Mathematics, Kuwait University, Kuwait, fares.alazemi@gmail.com, alsenafi@sci.kuniv.edu.kw,khalifa.essebaiy@ku.edu.kw

AbstractWe consider the least square-type estimation of the drift parameters for the fractional Vasicek-typemodel of the second kind Xt, t ≥ 0 defined as dXt = θ(µ+Xt)dt+dY

(1)t , t ≥ 0 with unknown

parameters θ > 0 and µ ∈ R, where Y (1)t :=

∫ t0 e

−sdBas with at = HetH , and Bt, t ≥ 0 is a

fractional Brownian motion of Hurst indexH ∈ (0, 1). We study the consistency and the asymptoticdistribution of our estimators of θ and µ based on the continuous-time observations Xt, t ∈ [0, T ]as T → ∞. Assume µ = 0, the parameter estimation of θ has been studied for H ∈ (0, 12). Herewe present a study valid for all H ∈ (0, 1). Our method is based on pathwise properties of X andY (1) proved in the sequel.

8

Jacobian free methods for coupling transportwith chemistry in heterogenous porous media

Laila Amir1, Michel Kern2

1FSTG, Univ. Cadi Ayyad, Marrakech, Morocco.l.amir@uca.ma2INRIA, Paris, France & ENPC, Univ. Paris–Est, France.michel.kern@inria.fr

AbstractReactive transport problems involve the coupling between the chemical interactions of differentspecies and their transport by advection and diffusion. Under the hypothesis of local equilibrium,the model couples transport PDEs with local algebraic equations.

In this talk, we present a model that allows a separation of transport and chemistry at the softwarelevel, while keeping a tight numerical coupling between both subsystems [1] . We give a formulationthat eliminates the local chemical concentrations and keeps the total concentrations as unknowns.

The coupled system is solved by a NewtonKrylov method, wherethe Jacobian is not stored,one just needs to be able to compute the product of the Jacobian with a vector, leading to Jacobianfree methods. The Jacobian matrix vector product is a directional derivative, so this product can beapproximated by finite differences. For this problem, it canalso be computed exactly.

The block structure of the model is exploited both at the nonlinear level, by eliminating someunknowns, and at the linear level by using block Gauss-Seidel or block Jacobi preconditioning [2].The methods are illustrated on various test cases.

References[1] L. Amir and M. Kern,A global method for coupling transport with chemistry in heterogeneous porous media,

Computational Geosciences, 14, pp. 465–481, 2010.

[2] L. Amir and M. Kern,Preconditioning a coupled model for reactive transport in porous media, InternationalJournal of Numerical Analysis and Modeling., 16, pp. 18–48,2019.

9

A Sylvester-Kac matrix type and controllabilityof half graphs

Milica Andjelic1

1 Department of Mathematics, Kuwait University, Kuwait

AbstractWe provide a family of tridiagonal matrices whose eigenvalues are the first positive squares. Inparticular, we compute the spectrum of the Hankel bidiagonal matrix. As an application we considerthe Laplacian controllability of half graphs (a particular subclass of chain graphs).

10

Stochastic Runge-Kutta pseudo-symplectic meth-ods

Cristina Anton1

1Department of Mathematics and Statistics, Grant MacEwan University, Edmonton, AB T5J 4S2, Canada

AbstractConsider the stochastic Hamiltonian system (SHS) in the sense of Stratonovich

dP i

t = −

∂H0

∂Qit

dt−

d∑

r=1

∂Hr

∂Qit

dwr

t , dQi

t =∂H0

∂P it

dt+

d∑

r=1

∂Hr

∂P it

dwr

t , (1)

whereP0 = p, Q0 = q, P , Q, p, q aren-dimensional column vectors, andwrt , r = 1, . . . , n are

independent standard Wiener processes. The flowφt(p, q) = (Pt(p, q), Qt(p, q))T of (1) preserves

the symplectic structure [2].Symplectic schemes [2] show better accuracy in numerical simulations over a long time interval,

but unless we consider a special type of SHS, they are implicit methods, so the computing time islarge. Pseudo-symplectic methods are explicit methods that preserves the symplectic structure upto a certain order of accuracy, so they offer a compromise between the long-term accuracy and thecomputing time.

In [2] we use an approach based on generating functions to construct pseudo-symplectic schemesfor SHS (1), but these schemes require many derivatives of high order of the functionsHr, r =1, . . . , d. To deal with this issue, here we construct explicit Runge-Kutta pseudo-symplectic schemes.Our approach is based on a rooted tree analysis and stochastic B-series for order condition [1]. Wealso illustrate numerically the order of convergence of theproposed schemes.

References[1] Sverre Anmarkrud, Kristian Debrabant, and Anne Kværnø.General order conditions for stochastic parti-

tioned Runge–Kutta methods.BIT Numerical Mathematics, 58(2):257–280, 2018.

[2] Cristina Anton. Explicit pseudo-symplectic methods based on generating functions forstochastic Hamiltonian systems. Journal of Computational and Applied Mathematics, 2019.https://doi.org/10.1016/j.cam.2019.112433.

11

Electromagnetic scattering by a chiralobstacle in a chiral environment usingthe method of fundamental solutions

E. S. Athanasiadou1, I. Arkoudis1

1National and Kapodistrian University of Athens, Department of Mathematics

AbstractThe scattering of a time-harmonic plane chiral electromagnetic wave by a penetrable chiral body isconsidered. An extension of the standard method of fundamental solutions is presented and appliedto our problem in order to obtain numerically the solution. The electric fields are expressed as linearcombinations of the dyadic fundamental solutions of the corresponding chiral Maxwell equations.Based on the elements of the dyadic fundamental solutions for infinite singularity points, appropriatesystems of functions are constructed. The linear independence and the completeness on the surfaceof the scatterer of these systems are proved using the chiralvector potentials. These propertieslead to the convergence of the constructed approximate solution to the exact one. Applying thetransmission conditions, the scattering problem is transformed into a linear algebraic system withcoefficients in block matrix form. Moreover, introducing the Beltrami fields, the block matrix isseparated into a left circularly polarized part and a right circularly polarized part. When the measureof chirality in the exterior or the interior medium vanishes, our results cover the case of a chiralscatterer in an achiral environment and the case of a dielectric scatterer in a chiral environmentrespectively.

12

Analysis of Common Fixed Point of F-contractionson Closed Ball

Awais AsifDepartment of Math & Stats, International Islamic University Islamabad, Pakistan.

AbstractThis article advances the approach of finding fixed point of a single mapping and two mappingsin F-metric spaces. Existence of fixed point and common fixed point of F-contraction is investi-gated while the contractive condition is imposed only on a subset closed ball of the F-metric spaceinstead of imposing it on the whole space. The paper explains fixed point without using the thirdaxiom of the famous Wardowskis F-contraction. The results are supported by providing an exam-ple. Moreover, few important corollaries are also developed from the main results. The paper isan improvement to both F-contraction and the newly defined F-metric. Keywords: F-metric, Reichtype Contraction, Kannan type Contraction.

13

Study of a quasilinear parabolic equation withsuperquadratic growth non-linearity

Charaf Eddine Bailoul1, Fatima Aqel2, Nour Eddine Alaa1

1Laboratory LAMAI, University Cadi Ayyad, Marrakesh, Morocco2Laboratory IR2M, University Hassan Ier, Settat, Morocco

AbstractIn this work, we consider the following quasilinear parabolic equation :

(Pλ)

ut − uxx = |ux|p in QT

u(t, 0) = u(t, 1) = 0 for t ∈]0, T [

u(0, x) = λu0(x) for x ∈]0, 1[

Where we setQT =]0, T [×]0, 1[, λ ∈ R+, the initial datau0 is nonnegative belonging toC([0, 1])

and the exponentp > 2.We prove the existence of a critical valueλ∗ satisfying0 < λ∗ < ∞, such that ifλ > λ∗ the problem(Pλ) has no solution and ifλ ≤ λ∗, (Pλ) has exactly one solution inC1+

α

2,2+α(QT )∩C0(QT ). We

also present a robust numerical method to compute this critical value.

References[1] N. Alaa, Etude d’equations elliptiques non lineaires a dependance convexe en le gradient eta donnees

mesures, Thesis, University of Nancy I 1989.

[2] N. Alaa, Contribution a l’ etude d’equations elliptiques et paraboliques avec donnees mesures, These dedoctorat d’Etat, University Cadi Ayyad, Faculty of Sciences Semlalia 1996.

[3] N. Alaa, and M. Pierre,Weak solutions of some quasilinear elliptic equations withdata measures, J. SIMA,Vol. 24, No.1, 1993, p. 23-35.

[4] N. Alaa, and F. Aqel,Characterization of the critical value for a quasilinear elliptic equation with arbitrarygrowth with respect to the gradient, Mediterranean Journal of Mathematics,16 (6) 2019.

14

Solution of the bi-level problem via first orderdifferential equation system

Aicha Balhag1, Zaki Chbani2, Hassan Riahi3

1Laboratory LIBMA Mathematics, Faculty of Sciences Semlalia Cadi Ayyad University2 Mathematics and Population Dynamics Laborator, Faculty of Sciences Semlalia Cadi Ayyad University3Mathematics and Population Dynamics Laborator, Faculty of Sciences Semlalia Cadi Ayyad UniversityC

AbstractIn a Hilbert framework, we introduce continuous and discrete dynamical systems which aim atsolving constrained optimization problem

infx∈SF

ϕ(x),

where the functionϕ : H → is supposed to be strongly convex, differentiable, with∇ϕ Lips-chitz continuous andα-strongly monotone. andSF is the solution set of equilibrium problem;find x ∈ K, such thatF (x, y) ≥ 0, ∀y ∈ K. We first consider a first-order dynamical systemformulated in terms of the resolventJF of the maximal monotone bifunctionF, we show the con-vergence properties of the orbits of these systems. The timediscretization of these dynamics givesvarious forward-backward splitting methods for solving constrained optimization problem. Theconvergence of these algorithms is obtained under mild and standard conditions imposed on thecost bifunction and control parameters strong convergenceof the algorithms is established. Wepresent several numerical examples to illustrate the behavior of our schemes and emphasize theirconvergence advantages compared with some related methods.

References[1] B.Abbas and H. Attouch, Dynamical systems and forward-backward algorithms associated with the sum

of a convex subdifferential and a monotone cocoercive operator,Optimization, 64:10, 2223-2252, DOI:10.1080/02331934.2014.971412.

[2] D.V. Hieu and A. Gibali, Strong convergence of inertial algorithms for solving equilibrium prob-lems,Optimization Letters, https://doi.org/10.1007/s11590-019-01479-w

15

A rational Krylov subspace method for the com-putation of the matrix exponential

H. Barkouki1, A. H. Bentbib2, K. Jbilou3

1houda.barkouki@edu.uca.ac.ma2a.bentbib@uca.ac.ma3Khalide.Jbilou@univ-littoral.fr

AbstractThe approximation ofetAB where A is a large sparse matrix and B a rectangular matrix is the keyingredient in many scientific and engineering computations. A powerful way to consider this prob-lem is to use Krylov subspace methods. The purpose of this work is to approximate the exponentialon a block vectorB of Rn×p using the rational block Lanczos algorithm. We also derive some errorestimates and error bounds for the convergence of the rational approximation and finally numericalresults attest to the computational efficiency of the proposed method.

References[1] O. ABIDI , M. HACHED, AND K. JBILOU, Adaptive rational block Arnoldi methods for model reductions in

large-scale MIMO dynamical systems, New Trends in Mathematical Sciences., 4 (2), p. 227-239 (2016).

[2] Z. BAI , D. DAY, AND Q. YE, ABLE: An adaptive block Lanczos method for non-Hermitian eigenvalueproblems, SIAM J. Matrix Anal. Appl., 20, 1060-1082 (1999).

[3] V. D RUSKIN AND L. K NIZHNERMAN, Krylov subspace approximation of eigenpairs and matrix functionsin exact and computer arithmetic, Numer. Linear Algebra Appl., 2 (1995), pp. 205-217.

[4] V. D RUSKIN AND L. K NIZHNERMAN, Extended Krylov subspaces: Approximation of the matrix squareroot and related functions, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 755-771.

[5] S. GUTTEL, Rational Krylov Methods for Operator Functions, PhD thesis, Institut fur Numerische Mathe-matik und Optimierung der Technischen Universitat Bergakademie Freiberg, 2010.

[6] L. K NIZHNERMAN , V. DRUSKIN AND M. ZASLAVSKY , On optimal convergence rate of the rational Krylovsubspace reduction for electromagnetic problems in unbounded domains. SIAM J. Numer. Anal., 47(2)(2009), 953–971.

[7] L. L OPEZ AND V. SIMONCINI , Analysis of projection methods for rational function approximation to thematrix exponential, SIAM J. Numer. Anal., 44 (2006), pp. 613-635.

16

Spherical interpolatory geometric subdivisionschemes

Mohamed Bellaihou1, Aziz Ikemakhen2

1Cadi-Ayyad University, FSTG BP 549 Marrakesh, Morocco.2Cadi-Ayyad University, FSTG BP 549 Marrakesh, Morocco.

AbstractSubdivision is the process of generating curves and surfaces by iteratively refining a given initialpolygon according to certain refinement rules. Subdivisionschemes are said to be interpolatoryif the limit curve interpolates the vertices of the initial control polygon. Interpolatory geometricschemes for curves in the plane are well studied ([1]). Such schemes take the geometry of the con-trol polygons into account by using non-linear refinement rules and generate limit curves. Thesecurves are in generalG1-continuous which means they are good 1-sub-manifolds on the plane. Sub-division schemes on manifolds ([2], [3]) are method to refining geodesic polygon to approach someembedded curves on the manifold . The analysis of these schemes is based on their proximity to thelinear schemes which they are derived from. Such schemes cangenerateC1-limit curves.We define general geometric subdivision schemes generatingcurves on the 2-dimensional unitsphere by using geodesic polygons and spherical geometry. We show that a spherical interpolatorygeometric subdivision scheme is convergent if the sequenceof maximum edge lengths is summableand the limit curve isG1-continuous if in addition the sequence of maximum angular defects issummable. Some experimental examples are given to demonstrate the excellent properties of theseschemes.

References[1] N. DYN AND K. HORMANN, Geometric conditions for tangent continuity of interpolatory planar subdivision

curves, Computer Aided Geometric Design, 29(2012): 332–347.

[2] J. WALLNER, Smoothness analysis of subdivision schemes by proximity, Constructive Approximation ,24(2006): 289–318.

[3] J. WALLNER AND N. DYN, Convergence and C1 analysis of subdivision schemes on manifolds by proximity,Computer Aided Geometric Design , 22(2005): 593–622.

17

A fractional diffusion PDE in image denoising

Anouar Ben-Loghfyry1, Abdelilah Hakim2

1FSTG Marrakech, Morroco.

AbstractIn this paper, we will present a partial differential equation based on the fractional diffusion. Inorder to defend our model, a theoretical analysis is provided, and also some numerical results. Wewill compare our approach with some known models to see the efficacy of our proposed model.

18

Block preconditioners for the coupled Darcy-Stokes problem: eigenvalue and field-of-valuesanalysis

Michele BenziScuola Normale Superiore, Pisa, Italy (michele.benzi@sns.it)

AbstractThe coupled Darcy-Stokes problem is an important problem in fluid flow modeling. Upon finiteelement discretization, the resulting linear system is of double saddle point type. We investigatedifferent preconditioners including block triangular and augmented Lagrangian-based ones. Wepresent a spectral analysis and fieldof-value analysis of the exact versions of these preconditioners,and present the results of numerical experiments illustrating the performance of inexact variants ofthese methods on 3D problems with large jumps in the permeability coefficients.

This is joint work with Fatemeh Beik (Vali-e-Asr University of Rafsanjan, Iran).

19

Remarks on the asymptotic behavior of scalarauxiliary variable (SAV) schemes

Anass Bouchriti1, Morgan Pierre2, Nour Eddine Alaa1

1LAMAI Laboratory, Faculty of Science and Technology, Cadi Ayyad University, Marrakesh, Morocco.2Laboratoire de Mathematiques et Applications, Universite de Poitiers, CNRS, F-86962 Chasseneuil, France

AbstractWe introduce a time semi-discretization of a modified Allen-Cahn equation (Damped Wave) by aSAV scheme with second order accuracy. The energy dissipation law is shown to hold withoutany restriction on the time step, referring to unconditional stability. We confirm a strong featureof SAV, stating the possibility of reforming the semi-discrete equation into a nonlinear powerfulsolver. We finally prove that any sequence generated by the scheme converges to a steady state (upto a subsequence). However, we notice that the steady state equation associated to the SAV schemeis a modified version of the steady state equation associatedto the damped wave equation. We showthat a similar result holds for a SAV fully discrete version of the Cahn-Hilliard equation and wecompare numerically the two steady state equations.

References[1] N. E. ALAA AND M. PIERRE, Convergence to equilibrium for discretized gradient-likesystems with analytic

features; IMA J. Numer. Anal., 33, 4,1291–1321. (2013)

[2] S. ALLEN AND J. W. CAHN , A microscopic theory for antiphase boundary motion and its application toantiphase domain coarsing; Acta. Metall., 1084–1095, 27. (1979)

[3] P.F. ANTONIETTI, B. MERLET, M. PIERRE AND M. V ERANI, Convergence to equilibrium for a second-order time semi-discretization of the Cahn-Hilliard equation; AIMS Mathematics, 178–194, 1, 3. (2016)

[4] D. GILBARG AND N. S. TRUDINGER, Elliptic partial differential equations of second order; Springer-Verlag,Berlin, Series Classics in Mathematics, xiv+517. (2001)

[5] O. GOUBET, Remarks on some dissipative sine-Gordon equations; Taylor And Francis, Complex Var. EllipticEqu.,1–7. (2019)

[6] O. KAVIAN , Introduction a la theorie des points critiques et applications aux problemes elliptiques; Springer-Verlag, Paris, Series Mathematiques & Applications (Berlin), 13, viii+325. (1993)

[7] J. SHEN, J. XU AND J. YANG, A new class of efficient and robust energy stable schemes for gradient flows;arXiv:1710.01331. (2017)

20

Analysis of the dynamic response of a railwayvehicle modeled in 1D and 2D

Laila Bouhlal1, Nouzha Lamdouar2

1lailabouhlal@research.emi.ac.ma2lamdouar@emi.ac.ma

AbstractThe objective of this study is to analyze the dynamic response of a railway vehicle while travellingon a track with irregularities considered as a source of excitement. Two types of vehicle models havebeen tested, one modelled by three unidimensional rigid elements representing: the body, boogieand wheel linked by spring damping couples, with three degrees of freedom, the second model, thevehicle was modelled by seven bi-dimensional rigid elements representing: the body, tow boogiesand four wheels, this system based on ten degrees of freedom.The vehicle motion equations of the two models respectively with 1D and 2D were obtained directlyby applying fundamental principles of dynamics. After assembling the stiffness, the damping andthe mass matrices and the vectors of nodal loads, global equations were solved step-by-step inte-gration using the Newmark β and Wilson θ methods to simultaneously obtain dynamic responsesof vehicle.

References[1] S.A. Mosayebi, M. Esmaeili and J.A. Zakeri. Numerical investigation of the effects of unsupported railway

sleepers on train-induced environmental vibrations.Journal of Low Frequency Noise, Vibration and ActiveControl 2017, Vol. 36(2) 160176.

[2] I.Sebean, M. Dumitriu. Validation of the theoretical model for the study of dynamic behavior on verticaldirection for railway vehicles. ANNALS of Faculty Engineering Hunedoara International Journal of Engi-neering, 2014, Tome XII 153-160.

[3] P. Lou. A vehicle-track-bridge interaction element considering vehiclespitching effect.Finite Elements inAnalysis and Design 41 (2005) 397427

21

Modeling of shallow aquifers in interactionwith overland water

Christophe Bourel1, Catherine Choquet2, Carole Rosier1, MunkhgerelTsegmid3

1Univ. Littoral Cote d’Opale, LMPA, F- 62228 Calais, France2La Rochelle Universite, MIA, F-17031 La Rochelle, France3Mongolian University of Science and Technology, Mongolia

AbstractWe present a new class of efficient models for water flow in shallow unconfined aquifers, providingan alternative to the classical but less tractable 3d-Richards model. Its derivation is guided by twoobjectives: to obtain a model that has low computational cost and yields relevant results on everytime scale.

Thus, we keep track of two types of flow that occur in such a context and are dominant whenthe ratio of thickness to longitudinal length is small: the first is dominant on a small time scaleand is described by a vertical 1d-Richards problem; the second corresponds to a large time scale,when the evolution of the hydraulic head becomes independent of the vertical variable. Thesetwo types of flow are appropriately modeled by a one-dimensional and two-dimensional system ofPDE boundary value problems, respectively. They are coupled at an artificial level below whichthe Dupuit hypothesis holds true (i.e., the vertical flow is instantaneous) so that the global modelis mass conservative. Tuning the artificial level, which caneven depend on an unknown of theproblem, we obtain the new class of models. Using asymptoticexpansions, we prove that the3d-Richards model and each model in the class behave identically on every considered time scale(short, intermediate, and large) in thin aquifers. The results are illustrated by numerical simulations,and it is demonstrated that they fit well with those obtained by the original 3d-Richards model evenin non-thin aquifers.

References[1] Christophe Bourel; Catherine Choquet; Carole Rosier; Munkhgerel Tsegmid. Modelling of shallow aquifers

in interaction with overland water.Applied Mathematical Modeling, (submitted 2019).

22

Richardson, Romberg, and others

Claude Brezinski1, Michela Redivo-Zaglia2

1University of Lille, France2University of Padua, Italy

AbstractIn this talk, I show the development of extrapolation techniques through the ages. I begin by theancient estimations forπ and the improvement techniques used up to the 19th century. Then Idiscuss the method of Lewis Fry Richardson who gave the first modern treatment of extrapolation.His very rich life is described with his many different achievements. After that, I come to themethod of Werner Romberg for improving the trapezoidal rulefor computing a definite integral.His difficult life is discussed. Richardson’s and Romberg’smethods come out, in fact, from theAitken-Neville scheme for recursively computing the interpolation polynomial, a result establishedby Pierre-Jean Laurent in 1963. Laurent also gave necessaryand sufficient conditions for havingconvergence acceleration.

References[1] C. Brezinski, M. Redivo-Zaglia,Extrapolation and Rational Approximation. The Works of the Main Contrib-

utors since 1910, Springer, Heidelberg, to appear.

23

A Phase Shift Deep Neural Network Algorithmfor High Frequency Wave Equations

Wei Cai1, Xiaoguang Li1, Lizuo Liu1

1Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA.

AbstractIn this talk, we will present a new phase shift deep neural network (PhaseDNN) algorithm for afrequency uniform convergence in approximating high frequency functions and solutions of waveequations. The PhaseDNN takes advantage of fast convergence of common DNNs in low frequen-cies, and constructs a series of moderately-sized DNNs for selected high frequency ranges. Withthe help of phase shifts in the frequency domain, each of the trained DNNs can approximate afunction’s specific high frequency range at the speed of low frequency learning. As a result, theproposed PhaseDNN is able to convert high frequency learning to low frequency one, allowing auniform learning for wideband functions. The PhaseDNN is then applied to learn the solution ofhigh frequency wave problems in inhomogeneous media through the least square residuals of eitherdifferential or integral equations. Numerical results demonstrate the capability of the PhaseDNNin learning high frequency functions and oscillatory solutions of interior and exterior Helmholtzproblems.

References[1] Wei Cai, Xiaoguang Li, Lizuo Liu, A phase shift deep neural network for high frequency approximation and

wave problems , arXiv:1909.11759, Sept. 23, 2019.

24

Equilirium chemistry modelling : from Newtonto combined algorithms

Jerome Carrayrou1

1Universite de Strasbourg, LHyGeS, CNRS, UdS, ENGEES, Strasbourg, France

AbstractIn 1972, the works of Morel and Morgan [1] has founded the equilibrium aqueous chemistry mod-elling research field. They proposed the decomposition of the chemical system using species andcomponents leading to a set of nonlinear algebraic equations. They proved the efficiency of Newton-Raphsons method for this kind of system by solving (in 1972 !)a chemical system composed of 782chemical species ! Following these pioneers, many researchers developed numerical codes [2, 3, 4]solving: aqueous equilibrium; aqueous-solid interface exchange; reactive transport that include theideas proposed by these authors. Even if some weaknesses of the Newton-Raphsons method havebeen pointed, it is still used as the major routine for the most modern chemical codes [5, 6]. Never-theless, many authors has drawn attention to some convergence problems for the Newton-Raphsonsmethod and developed rules to overpass these problems without any definitive solution. Recentworks [7, 8, 9] proposed an explanation to these difficulties: chemical systems generate jacobianmatrix with huge condition numbers. According to this new point of view, we propose a rereadingof this research fields history to reveal some new progression ways.

References[1] Morel F, Morgan J (1972) Numerical method for computing equilibriums in aqueous chemical systems.

Environ Sci Technol 6:5867.

[2] Parkhurst DL, Appelo CAJ (1999) Users guide to PHREEQC (version 2)- A computer program for speciation,batch-reaction, one-dimensional transport, and inverse geochemical calculations. Denver, CO, USA.

[3] van der Lee J, De Windt L, Lagneau V, Goblet P (2002) Presentation and application of the reactive trans-port code HYTEC. Proceedings of the XIVth International Conference on Computational Methods in WaterResources (CMWR XIV), Volume 47. pp 599606

[4] Carrayrou J, Mos R, Behra P (2002) New efficient algorithmfor solving thermodynamic chemistry. AIChE J48:894904

[5] Steefel CI, Appelo CAJ, Arora B, et al (2015) Reactive transport codes for subsurface environmental simu-lation. Comput Geosci 19:445478.

[6] Carrayrou J, Hoffmann J, Knabner P, et al (2010) Comparison of numerical methods for simulating stronglynonlinear and heterogeneous reactive transport problems-the MoMaS benchmark case. Comput Geosci14:483502

[7] Erhel J, Sabit S (2017) Analysis of a global reactive transport model and results for the MoMaS benchmark.Math Comput Simul 137:286298.

[8] Machat H, Carrayrou J (2017) Comparison of linear solvers for equilibrium geochemistry computations.Comput Geosci 21:131150.

[9] Marinoni M, Carrayrou J, Lucas Y, Ackerer P (2017) Thermodynamic equilibrium solutions through a mod-ified Newton Raphson method. AIChE J 63.

25

Multidimensional Laplace transform inversionusing multivariate rational approximants

Y. Chakir1, J. Abouir1

1University Hassan II- Casablanca, Faculty of Science and Technology Mohammedia, Laboratory LMCMAN, B.P. 146 Mohamme-dia, Morocco

AbstractIn this work, we present a new method for obtaining multidimensional Laplace transform inversion.This approach is based on multivariate homogeneous two-point Pade approximants defined recently.Some numerical examples are given to illustrate our results.keyword: Multidimensional Laplace transform, Multivariate homogeneous two-point Pade approx-imants, continued fractions.

References[1] J. Abouir, B. Benouahmane, Multivariate homogeneous two-point Pade approximants, Jaen J. Approx., 10(1-

2) (2018), 29-48.

[2] Y. CHAKIR , J. ABOUIR, B. BENOUAHMANE, Multivariate homogeneous two-point Pade approximants andcontinued fractions, Comp. App. Math. (2020) 39: 15.

[3] A. M. COHEN, Numerical methods for Laplace transform inversion. Springer Science+Business Media,LLC, New York (2007).

[4] R.E. Grundy, Laplace transform inversion using two-point rational approximants, J. Inst. Maths. Applics., 20(1975), 299-306.

26

Existence and uniqueness of a renormalizedperiodic solution of a parabolic equation withvariable exponents and L

1 data

Abderrahim Charkaoui1, Mariam Zirhem1, Nour Eddine Alaa1

1Laboratory LAMAI, Faculty of Science and Technology, University Cadi Ayyad, Marrakesh-Morocco

AbstractIn this work, we study the following periodic nonlinear parabolic problem

∂tu− div(|∇u|p(x)−2∇u) = f in QT

u(0, .) = u(T, .) in Ωu(t, x) = 0 on ΣT

(1)

whereΩ is an open regular bounded subset ofRN , N ≥ 2, with smooth boundary∂Ω, T > 0 is the

period,QT =]0, T [×Ω, ΣT =]0, T [×∂Ω, f a periodic function with periodT belonging toL1(QT )andp : Ω →]1,+∞[ is a continuous function. Using a suitable fixed point theorem, we establishthe existence and uniqueness of a renormalized periodic solution for the equation (1).

References[1] N. Alaa and M.Iguernane,Weak periodic solutions of some quasilinear parabolic equations with data mea-

sure, J. of Inequalities in Pure and Applied Mathematics 3 (2002), no.3, Article 46.

[2] S.N. Antontsev, S.I. Shmarev, A model porous medium equation with variable exponent of nonlinearity:Existence, uniqueness and localization properties of solutions, Nonlinear Anal.60 (2005) 515-545.

[3] H. Amann,Periodic solutions of semilinear parabolic equations, Nonlinear Analysis, Academic Press, NewYork, 1978, pp. 1-29.

[4] D.W. Bange,Periodic Solution of a Quasilinear Parabolic Differential Equation, J. Differential Equation,17(1975), pp. 61-72.

[5] P. Benilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre, J.L. Vazquez, AnL1-theory of existence anduniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci.22 (1995)241-273.

[6] D. Blanchard, F. Murat, Renormalised solutions of nonlinear parabolic problems withL1 data: Existence anduniqueness, Proc. Roy. Soc. Edinburgh Sect.A 127 (6) (1997) 1137-1152.

27

Times schemes with high-modes stabilizationtechniques for nonlinear parabolic equations

Matthieu Brachet1, Jean-Paul Chehab2

1Laboratoire LJK (UMR CNRS 5224) and INRIA Project AIRSEA- Batiment IMAG, Universite Grenoble Alpes 700 Avenue Centrale,Campus de Saint Martin d’Heres 38401 Domaine Universitaire de Saint-Martin-d’Heres2Laboratoire LAMFA (UMR CNRS 7352), Universite de Picardie Jules Verne 33 rue Saint Leu, 80039 Amiens Cedex, France

AbstractWe propose a unified framework to develop stabilized time marching schemes for Nonlinear parabolicequations. These schemes are based on a stabilization matrix whose the spectrum allows to dampthe high frequency components (associated to a fluctuent part of the solution) while maintainingmainly unchanged the low frequency ones (associated to the mean part of the solution); dampingonly the high mode components is a way to prevent the instabilities generated by their propagationwithout deteriorating the consistency in time of the scheme. We consider different approaches andfocus on preconditioning, numerical filtering and projections for separating the different compo-nents. The resulting schemes are semi-implicit and give rise to a fast resolution . We give illustra-tions on Reaction-Diffusion systems in several physical situations. We use Finite differences butthe approach presented remains valid for other discretization techniques such as Finite ElementsMethod.

References[1] M. Brachet and J.-P. Chehab, Stabilized Times Schemes for High Accurate Finite Differences Solutions of

Nonlinear Parabolic Equations, Journal of Scientific Computing, 69(3), 946-982, 2016

[2] M. Brachet and J.-P. Chehab, Fast and Stable Schemes for Phase Fields Models, HAL-02301006, 2019

[3] T. Dubois, F. Jauberteau, R. Temam.Incremental Unknowns, Multilevel Methods and the Numerical Sim-ulation of Turbulence. Comp. Meth. in Appl. Mech. and Engrg. (CMAME), Elsevier Science Publishers(North-Holland).

[4] Jie Shen, Jie Xu and Jiang Yang, A New Class of Efficient andRobust Energy Stable Schemes for GradientFlows, SIAM Rev., 61(3), 474–506 (2019).

28

On the numerics of convolution flows in freeprobability

Reda Chhaibi1

1Institut de Mathmatiques de Toulouse, Toulouse.

AbstractMotivated by questions in operator algebras, Dan Voiculescu initiated the topic of free probabilityin 1986, see e.g. [1]. Nowadays, free probability is the reference method to understand the spectraof large random matrices hence its importance in probability, statistics (sample covariance matrices)and signal processing. In this framework, additive and multiplicative convolutions of matrices takea different flavor than the abelian setting: convolution flows are expressed in terms of the complex(generalized) Burgers equation. We will tackle the problemof effective numerical solutions in casesof interest. This is work in progress.

References[1] Free probability theory: random matrices and von Neumann algebras, Proceedings International Congress

of Mathematicians, Zurich: 227–241, 1994.

29

On a fast hierarchical sparse grid quadratureand applications.

Moulay Abdellah Chkifa1

1Universite Mohammed VI polytechnique, Ben Guerrir, Morocco.

AbstractThe motivation of this work is the computation of quadratures of multivariate functions which arisein parametrized physical and engineering models, the simulation of which depends ond input pa-rametersy1, . . . , yd varying in a parameter domainP. The main challenge is the curse of dimen-sionality arising ford ≫ 1.

We present a quadrature based on the Smolyak approach using sequences with certain “binary”symmetry properties which yield simple and straight-forward computation of such quadratures. Inparticular, these quadratures are hierarchical with the cost of enrichment as low as one additionalcollocation. We present numerical experiments which compares such quadratures with QMC meth-ods and Chebyshev-Frolov lattice quadratures.

References[1] A. Chkifa, A. Cohen, and C. Schwab.High-dimensional adaptive sparse polynomial interpolation and ap-

plications to parametric PDEs Foundations of Computational Mathematics 14 (4): 601-633,2013.

[2] C. Kacwin , J. Oettershagen, and T. Ullrich.On the orthogonality of the Chebyshev?Frolov lattice and appli-cations. Monatsh Math (184): 425-441, 2017.

30

Extrapolation methods for Nonlinear Problems

C. Brezinski1, S. Cipolla2, M. Redivo-Zaglia2, Y. Saad3, F. Tudisco4

1Universite Lille, CNRS, UMR 8524 - Laboratoire Paul Painleve, F-59000 Lille, France, Claude.Brezinski@univ-lille.fr.2Universita degli Studi di Padova, Dipartimento di Matematica “Tullio Levi-Civita”, Via Trieste 63, 35121–Padova, Italy,stefano.cipolla@unipd.it, Michela.RedivoZaglia@unipd.it.3Dept. of Computer Science and Engineering, University of Minnesota, Mississippi National River and Recreation Area, Min-neapolis, MN 55455, USA, saad@cs.umn.edu.4GSSI − Gran Sasso Science Institute, 67100 L’Aquila, Italy, francesco.tudisco@gssi.it.

AbstractIn numerical analysis and in applied mathematics one often has to deal with sequences of numbers,vectors, matrices or even tensors. When the sequence is slowly converging, or even diverging, andwhen one has no access to it (when it is given by a “black box”), one can transform it, by a sequencetransformation, into a new sequence which, under some assumptions, converges faster to the samelimit. In a series of recent papers, extrapolation methods for sequence acceleration are increasinglyattracting attention for their impressive performance when applied to optimization problems [5, 6].In this talk, we present different extrapolation techniques based on the framework introduced in[1] and compare their numerical performance on a series of nonlinear problems ranging from thesolution of PDE [2] to the computation of the Perron eigenvector of tensors [3, 4] .

References[1] C. Brezinski, M. Redivo-Zaglia, Y. Saad, Shanks sequence transformations and Anderson acceleration, SIAM

Rev., 60(3), 646–669, 2018.

[2] C. Brezinski, S. Cipolla, M. Redivo-Zaglia, Y. Saad, Anderson-type transformations for systems of non-linearequations, In Preparation.

[3] S. Cipolla, M. Redivo-Zaglia, F. Tudisco, Extrapolation Methods for fixed-point Multilinear PageRank com-putations, Numer. Linear Algebra Appl., Accepted, 2019.

[4] S. Cipolla, M. Redivo-Zaglia, F. Tudisco, Shifted and extrapolated power sequences for tensor `p-eigenpairs,Electron. Trans. Numer. Anal., Accepted, 2019.

[5] D. Scieur, A. D’Aspremont, F. Bach, Regularized nonlinear acceleration, Math. Program, 2018.

[6] Zhang, J., O’Donoghue, B., Boyd, S., Globally convergent type-I Anderson acceleration for non-smoothfixed-point iterations. arXiv preprint arXiv:1808.03971, 2018.

31

Optimal hierarchical sampling andreconstruction using weighed least squares

Albert Cohen1

1Laboratoire Jacques Louis Lions, Sorbonne Universites, Paris

AbstractMotivated by non-intrusive approaches for high-dimensional parametric PDEs, we consider thegeneral problem of approximating an unknown arbirary function in any dimension from the dataof point samples. The approximants are picked from given or adaptively chosen finite dimensionalspaces. One principal objective is to obtain an approximation which performs as good as the bestpossible using a sampling budget that is linear in the dimension of the approximating space. Wewill show that this objectif can is met by taking a random sample distributed according to a wellchosen probability measure, and reconstructing by weighted least-squares methods. We also showthat this method has good potential for adaptivity in the sense that previous samples can be recycledas the spaces are enriched.

References[1] Albert Cohen, Giovanni Migliorati.Optimal weighted least-squares methods. SMAI Journal of Computa-

tional Mathematics(3):181-203, 2017.

32

Interpolation, approximation and prediction

Azzouz Dermoune1, Mohammed Es.Sebaiy2, Jabrane Moustaaid2

1Laboratoire Paul Painleve, USTL-UMR-CNRS 8524, UFR de Sciences, Villeneuve dAscq Cdex, France2National School of Applied Sciences-Marrakech, Cadi Ayyad University, Marrakesh, Morocco

AbstractInterpolation, approximation and prediction using the data intervene in various applications. Theyare based on the knowledge of a sequencey1 ∈ R, x1 ∈ D, yn ∈ R, xn ∈ D of n pointsof the domainR × D and a finite dimensional vector spaceE of the spaceC(D) of continuousfunctions onD. If the interpolationy1 = f(x1), . . ., yn = f(xn) with f ∈ E has a unique so-lution f ∈ E, then knownxn+1 we predictyn+1 by f(xn+1). If not we look for f ∈ E suchthat y1 ≈ f(x1), . . ., yn ≈ f(xn) and then we predictyn+1 by f(xn+1). The most known ap-proximationsy1 ≈ f(x1), . . ., yn ≈ f(xn) are respectively the global least squares minimiza-tion f = argmin

∑n

i=1|yi − f(xi)|

2 : f ∈ E and the local least squares minimizationf = argmin

∑n

i=1|yi − f(xi)|

2w(xn+1, xi) : f ∈ E with a given kernelw. In this talkwe apply various interpolation and approximation technicsfor the prediction of the temperature ofFrance and Morocco. We select the good technic using cross-validation method.

References[1] Scattered data approximation: Holger Wendland, CAMBRIDGE MONOGRAPHS ON APPLIED AND

COMPUTATIONAL MATHEMATICS, 2005.

[2] Parametrizations, weights, and optimal prediction Azzouz Dermoune, Khalifa Es-Sebaiy, MohammedEs.Sebaiy, Jabrane Moustaaid, Communications in Statistics - Theory and Methods, 2019.

[3] Parametrizations, fixed and random effects, A. Dermoune, C. Preda, Journal of Multivariate Analysis, 2017.

[4] Interpolation and prediction of data-three kernel selection criteria, A. Dermoune, M. Es.Sebaiy, J. Moustaaid,Preprint 2018.

33

Canonical structure transitionsof system pencils

Andrii Dmytryshyn1, Stefan Johansson2, Bo Kagstrom2

1Orebro University, Orebro, Sweden2Umea University, Umea, Sweden

AbstractWe considergeneralized state-space(or descriptor) systems

Ex(t) = Ax(t) +Bu(t),

y(t) = Cx(t) +Du(t),(1)

whereA,E ∈ Cn×n andE is non-singular,B ∈ C

n×m, C ∈ Cp×n,D ∈ C

p×m, andx(t), y(t), u(t)are thestate, output,andinput (control)vectors, respectively. The system characteristics of (1) aretypically ill-posed problems,i.e., small perturbations in the matrices can lead to drastic changesin the system characteristics. Such problems can be represented and analyzed from thecanon-ical structure information(elementary divisors, column and row minimal indices) of the block-structuredsystem pencil:

S(λ) :=

[

A B

C D

]

− λ

[

E 00 0

]

, det(E) 6= 0. (2)

By constructingstratifications(i.e., closure hierarchy graphs) of system pencils (2), we investi-gate the possible changes of the canonical structure information caused by arbitrarily small pertur-bations of the original system pencilS(λ) [1]. We also explain how the closest neighbours (coverrelations) in the closure hierarchy can be obtained.

The special case of the system (1) with no direct feedforward(i.e. the matrixD is zero) isconsidered too.

References[1] A. Dmytryshyn, S. Johansson, and B. Kagstrom,Canonical structure transitions of system pencils, SIAM J.

Matrix Anal. Appl., 38(4) (2017) 1249–1267.

34

Algorithms for optimal sampling in weightedleast-squares methods on general domains

Matthieu Dolbeault11Laboratoire Jacques Louis Lions, Sorbonne Universites, Paris

AbstractIn optimal weighted least-squares approximation, the sample points are chosen according to a prob-ability measure that depends on the Christoffel function of the approximation space. For generaldomains, this function cannot be computed exactly because we lack the knowledge of an orthonor-mal basis of this space. We propose different algorithms to circumvent this difficulty by computingan approximation of the Christoffel function, and show that the resulting sampling remains optimal.In the case of an adaptative method, in which the approximation space is enlarged incrementallydepending on the previous samples, we investigate a strategy to keep the total number of samplesoptimal, even though the approximation errors accumulate.

References[1] Albert Cohen, Giovanni Migliorati. Optimal weighted least-squares methods. SMAI Journal of Computa-

tional Mathematics(3):181-203, 2017.

35

Conditioning and backward errors ofeigenvalues of matrix polynomials underMobius transformations

Luis M. Anguas1, Marıa I. Bueno2, Froilan M. Dopico3

1Departamento de Matemtica Aplicada, ICAI-Universidad Pontificia Comillas, Calle Alberto Aguilera 23, 28015 Madrid, Spain2Department of Mathematics and College of Creative Studies, University of California, Santa Barbara, CA 93106, USA3Departamento de Matematicas, Universidad Carlos III de Madrid, Avda. Universidad 30, 28911 Leganes, Spain

AbstractMobius transformations have been used in numerical algorithms for computing eigenvalues of struc-tured generalized and polynomial eigenvalue problems (PEPs). These transformations convert prob-lems with certain structures arising in applications into problems with other structures and whoseeigenvalues are easily related to the ones of the original problem. Thus, an algorithm that is efficientand stable for some particular structure can be used for solving efficiently another type of structuredproblem via an adequate Mobius transformation. A key question in this context is whether thesetransformations may change significantly the conditioningof the problem and the backward errorsof the computed solutions, since, in that case, their use might lead to unreliable results. We presentthe first general study on the effect of Mobius transformations on the eigenvalue condition numbersand backward errors of approximate eigenpairs of PEPs. By using the homogeneous formulation ofPEPs, we are able to obtain two clear and simple results. First, we show that, if the matrix inducingthe Mobius transformation is well conditioned, then such transformation approximately preservesthe eigenvalue condition numbers and backward errors when they are defined with respect to per-turbations of the matrix polynomial which are small relative to the norm of the whole polynomial.However, if the perturbations in each coefficient of the matrix polynomial are small relative to thenorm of that coefficient, then the corresponding eigenvaluecondition numbers and backward errorsare preserved approximately by the Mobius transformations induced by well-conditioned matri-ces only if a penalty factor, depending on the norms of those matrix coefficients, is moderate. Itis important to note that these simple results are no longer true if the standard non-homogeneousformulation of the PEP is used.

36

Online learning in large-scale data sets for anomalydetection

Franck Dufrenois1, Denis Hamad2

1,2LISIC- 50 rue Ferdinand Buisson- 62100 Calais France

AbstractWith the revolution of information and communication technologies, the size and the dimension-nality of the data bases have significantly increased in recent years. In this context, identifying ordetecting abnormal data in such a volume of information is became a challenging research topicin the machine learning community. Recently, the null proximal discriminant analysis has beenrecently introduced for solving the one class problem (OC-NPDA, [1]). Originally, OC-NPDAconsiders the problem of detectingabnormal objects among a set ofnormal or target objects byprojecting the training target data in a specific lower dimensional subspace, theNull space, wherethe variability of the training data set is reduced to a single point. Then a novelty score is easilyderived by computing the distance between the projection ofa test sample and this single point.However, OC-NPDA is formulated as a batch algorithm and hence is hard to scale up because ofthe high computational cost induced for computing eigen-decomposition of the kernel gram matrix.Indeed, since eigen decomposition involves a time complexity of O(n3) where n is the size of thetraining data set, computing OC-NPDA is completely excluded for very large data sets and is notformulated to deal with data requiring a sequential processing ([2]). The contribution of this workis to develop an efficient incremental implementation of OC-NPDA. More precisely, at each update,the OC-NPDA’s solution involves the call of two SVD for computing a new set of eigenvectors anda new set of null projecting directions. We show in particular that updating the null space is equiv-alent to solve an optimization problem with a constant size,i.e which only depends on the chunksize l(l < n). This property avoids us to solve an ever-growing problem byreducing significantlythe time complexity fromO((n + l)3) toO(l3). Another important point is that the size of the nullspace increases by|O| + 1, where|O| denotes the number of negative examples given in the newdata chunk. To overcome this problem, we introduce a compression strategy based on the computa-tion of the representativeness of the null projection directions for each data vectors. We show thatthis strategy significantly reduces the memory cost while preserving a good level of classification.

References[1] F. Dufrenois and J.C. Noyer. A null space based one class kernel Fisher discriminant. IEEE International

Joint Conference on Neural Networks,2016.(Vancouver, Canada)

[2] F. Dufrenois and D. Hamad. Sparse and online null proximal discriminant analysis for one class learning inlarge-scale datasets. IEEE International Joint Conference on Neural Networks, 2019.(Budapest, Hungary)

37

Jacobian-free Newton-type implementationof IFOSMONDI Co-simulation Algorithm

Yohan Eguillon1, Bruno Lacabanne2, Damien Tromeur-Dervout1

1Institut Camille Jordan, Universite de Lyon ,UMR5208 CNRS-U.Lyon 1, Villeurbanne, France2Siemens Industry Software, Roanne, France

AbstractIFOSMONDI iterative algorithm for implicit co-simulation[1] enables us to solve the nonlinearconstraint coupling function while keeping the smoothnessof interfaces without introducing a de-lay. Moreover, it automatically adapts the size of the stepsbetween data exchanges among thesubsystems according to the difficulty of the solving of the coupling constraint. The original im-plementation focuses on the fixed-point algorithm as solving method whereas this paper introducesits Newton version. The latter usually requires the computation of a jacobian matrix which canbe tricky to compute except in the case where the interfaces are represented by a Zero-Order-Hold(ZOH) [2]. As far as IFOSMONDI coupling algorithm uses Hermite interpolation for smooth-ness enhancement (up to Third-Order-Hold), we propose hereafter the formulation of the nonlinearcoupling as a jacobian-free Newton-type method [3]. Consequently, successive function evalua-tions consist in multiple simulations of the subsystems on aco-simulation time-step using rollback.Different nonlinear methods and preconditionners will be compared on an academic mass-spring-damper test-case thanks to the PETSc framework.

References[1] Y. Eguillon, B. Lacabanne and D. Tromeur-Dervout, IFOSMONDI:A Generic Co-simulation Approach

Combining Iterative Methods for Coupling Constraints and Polynomial Interpolation for Interfaces Smooth-ness, Proc. of SIMULTECH, 176-186, 2019

[2] S. Sicklinger & Al., Interface Jacobian-based Co-Simulation, International Journal for Numerical Methodsin Engineering, 98: 418–444, 2014

[3] D.A. Knoll, D.E. Keyes, Jacobian-free Newton - Krylov methods: a survey of approaches and applications,J. Comput. Phys., 193: 357-397, 2004

38

Extended Krylov subspace methods for solv-ing Sylvester and Stein tensor equations

Smahane El-Halouy1, A. H. Bentbib1, El M. Sadek2

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

AbstractTensors are multidimensional arrays, in which the order is the number of dimensions, they are usedtoday in a wide variety of applications such as image processing, machine learning and scientificcomputing. One of the popular problems in tensor based modelling is the following equations,known as Sylvester and Stein tensor equations

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

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

whereAi ∈ RIi×Ii , i = 1, 2, · · · , N, are the coefficient matrices, and the tensorB ∈ R

I1×I2×···×IN

is known andX ∈ RI1×I2×···×IN is the unknown tensor. Such structures arise, for example, from

the finite element discretization of PDEs on anN−dimensional hypercube. We are interested inthe case where the right hand sideB has a low rank representation. We propose extended Krylov-like methods for solving Sylvester and Stein tensor equations (1) and (2). We show how to extractapproximate solutions via matrix Krylov subspaces basis. Several theoretical results such as expres-sions of residual and its norm are presented. To show the performance of the proposed approaches,some numerical experiments are given.

References[1] A. Beik, F. Panjeh, F. S. Movahed,On the Krylov subspace methods based on tensor format for positive

definite Sylvester tensor equations, Numer. Linear Algebra Appl, 23, pp. 444–466. (2016).

[2] Heyouni, M. Extended Arnoldi methods for large low-rank Sylvester matrix equations. App. Num.Math.,60(11) (2010) 11711182.

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

[4] Kressner, D., Tobler, C.Krylov subspace methods for linear systems with tensor product structure. SIAMjournal on matrix analysis and applications 31(4), 1688-1714 (2010).

[5] Saad, Y.Iterative methods for sparse linear systems, vol. 82. SIAM (2003).

39

Slurry transport in pipelines using machine learn-ing techniques

Khalid El Asnaoui1, Ali Idri1,2, Fayssal Benkhaldoun1,3

1Complex Systems Engineering and Human Systems, Mohammed VI Polytechnic University, Lot 660, Hay Moulay Rachid, BenGuerir, 43150, Moroccokhalid.elasnaoui@um6p.ma2 Software Project Management Research Team, (RITC), ENSIAS, Mohammed V University, BP 713, Rabat, Morocco,idri.ali123@gmail.com/ali.idri@um6p.ma3LAGA UMR7539, Paris13 University, France,fayssal.benkhaldoun@um6p.ma

AbstractTransport of slurry through pipe is widely utilized in the industry. This slurry eventually can causeinjury persons or environmental damage, it is fundamental to monitor these pipes in order to preventand avoid disasters and losses. Despite sophisticated and advanced models exist for slurry flow,models for modeling slurry flow present limitations and enormous drawbacks. These problemshave been predominately caused by the operating parametersthat are related with water chemistry,flow attributes, the material properties of the pipeline, and microbiological activities. In order toget more insight into the slurry flow process, a combination of machine learning techniques isnecessary. The techniques utilized are mainly based on artificial intelligence in order to, improvethe accuracy of identification of leaks or erosion-corrosion in pipes, or predict pressure, velocity,etc. This paper reviews current limitations and future opportunities for the application of machinelearning techniques in slurry pipeline.

Keyword: Slurry Transport; Slurry Flow; Artificial Intelligence; Machine Learning

40

Extended-rational global Arnoldi methodfor the matrix function approximation

M. El Ghomari1, A.H. Bentbib1, K. Jbilou2

1Laboratory LAMAI, Faculty of Sciences and Technologies, Cadi Ayyad University, Marrakech, Morocco.E-mail: m.elghomari10@gmail.com; a.bentbib@uca.ac.ma2Laboratory LMPA, 50 Rue F. Buisson, ULCO Calais cedex, France.Laboratory CSEHS , University UM6P, Bengurir Morocco.E-mail: khalide.jbilou@univ-littoral.fr

AbstractThe numerical computation of matrix functions such asf(A)V , whereA is n × n large andsquare matrix,V is ann × s block with s ≪ n andf is a nonlinear matrix function, arises invarious applications such as network analysis (f(t) = exp(t) or f(t) = t3), machine learning(f(t) = log(t)), theory of Quantum Chromodynamics(f(t) = t1/2), electronic structure compu-tation, and others. In this work, we propose the extended-rational global Arnoldi method for com-puting approximations of such expressions. The derived method projects the initial problem onto

an extended-rational global Krylov subspaceKe,σm (A,V ) = span(

m∏

i=1

(A − σiIn)−1V, . . . , (A −

σ1In)−1V, V,AV, . . . , Am−1V ) of a low dimension. When shift parameters are zero, this method

reduces to the extended Arnoldi method. An adaptive procedure for the selection of shift parametersσj is given. The proposed method is also applied to solve parameter dependent systems. Numericalexamples are presented to show the performance of the extended-rational global Arnoldi for theseproblems.

References[1] O. ABIDI , M. HACHED AND K. JBILOU, A global rational Arnoldi method for model reduction, J. Comput.

Appl. Math., 325, 175–187 (2017).

[2] L. K NIZHNERMAN AND V. SIMONCINI , A new investigation of the extended Krylov subspace methodformatrix function evaluations, Numer. Linear Algebra Appl.,17, 615–638 (2010).

[3] V. SIMONCINI , Extended Krylov subspace for parameter dependent systems, App. Num. Math., 60, 550–560(2010).

41

Efficient numerical solvers for the RBF solu-tion of Helmoltz equations in three space dimensions

M. El Guide1, K. Jbilou2, D. Ouazar3, M. Seaid2

1University Mohammed VI Polytechnic, Benguerir, Morocco2LMPA, University ULCO, France3Department of Engineering, University of Durham, South Road, Durham DH1 3LE, UK4University Mohammed VI Polytechnic, Benguerir, Morocco

AbstractIn this paper, we discuss different algorithms for the solution of linear discrete ill-posed problemsarising in the application of meshless method for solving Helmhotlz equation in three-dimensionalspace using multiquadrics radial basis function. It is well known that the truncated singular valuedecomposition (TSVD) is the most common effective solver for ill-conditioned systems, but unfor-tunately the operation count for solving a linear system with the TSVD is computationally expensivefor large-scale matrices. The proposed TSVD algorithm overcome this problem by computing aninexpensive TSVD using Golub-Kahan bidiagonalization algorithm. Another derived algorithmcan be provided by combinig Tikhonov regularization method and Golub-Kahan bidiagonalizationalgorithm to get regularized solutions in a few iterations.

References[1] G. H. GOLUB AND W. KAHAN, Calculating the singular values and pseudo-inverse of a matrix, J. SIAM

Ser. B Numer. Anal., 2(1965) 205-224.

[2] S. SARRA AND E. KANSA , Multiquadric Radial Basis Function Approximation Methods for the NumericalSolution of Partial Differential Equations (Advances in Computational Mechanics vol 2) ed Atluri S N (TechScience Press) 2009.

[3] H. WENDLAND,Scattered Data Approximation, Cambridge University Press, 2005.

42

Extended symbolic Gaussian cubature overconcurrent squares

A. Elidrissi1, J. Abouir1, B. Benouahmane1

1Hassan II University of Casablanca, Faculty of Sciences and Techniques, Laboratory of Mathematics Cryptography,Mechanics and Numerical Analysis

AbstractThe connection between orthogonal polynomials and Gaussian quadrature has been studied in[4, 8]. Several generalizations of these two concept to the multivariate case have been suggested[1, 2, 3, 5, 6, 7]. The so-called multivariate homogeneous two-point orthogonal polynomials is usedfor the appearance of the multivariate homogeneous two-point Pade approximant by using orthogo-nality conditions and related problems [1].In this work we will show how this multivariate homogeneous two-point orthogonal polynomialsleads to extend the m-point Gaussian cubature rules introduced by B. Benouahmane and A. Cuyt in[3] to two-point case. The theoretical expositions are clearly illustrated by some numerical exam-ples.

References[1] J. Abouir, B. Benouahmane, Multivariate homogeneous two-point Pade approximants, Jaen J. Approx. Vol-

ume 10, Number 1-2, 29-48 (2018).

[2] B. Benouahmane, A. Cuyt, Properties of multivariate homogeneous orthogonal polynomials, J. Approx.Theory 113 (2001) 1-20.

[3] B. Benouahmane, A. Cuyt, Multivariate orthogonal polynomials, homogeneous Pade approximants andGaussian cubature, Numerical Algorithms 24(1-2), pp. 1-15, (2000).

[4] C. Brezinski, Pade-type Approximation and General Orthogonal Polynomials, ISNM 50, Birkhauser Verlag,Basel, 67-105, (1980).

[5] R. Cools, Constructing cubature formulas: The science behind the art, Acta Numerica (1997) 1-53.

[6] A. Cuyt, B. Benouahmane, Hamsapriye, I. Yaman, Symbolic-numeric gaussian cubature rules, Applied Nu-merical Mathematics 61(8), (2011) pp. 929-945

[7] A. Cuyt, How well can the concept of Pade approximant be generalized to the multivariate case, J. Comput.Appl. Math. 105 (1999) 25-50.

[8] C. Diaz-Mendoza, P. Gonzalez-Vera and M. Jimenez-Paiz,Strong Stieltjes distributions and orthogonal Lau-rant polynomials with applications to quadrature and Padeapproximation, Mathematics of ComputationVolume 74, Number 252, (2005), Pages 1843-1870 .

43

2D meso modelling of crowd motion with thekinetic theory approach : influence of modelparameters on the evacuation time

A. El Mousaoui1,2, A. Jebrane2,3, P.Argoul4,5, M. EL Rhabi2,6, A. Hakim2

1EMINES - School of Industrial Management Universit Mohammed VI Polytechnique2LAMAI, FST Marrakech, Universit Cadi Ayyad, Morocco3quipe Systmes Complexes et Interactions, cole centrale Casablanca4Universit Paris-Est, LVMT (UMRT 9403), Ecole des Ponts ParisTech, IFSTTAR, UPEMLV, F-77455 Marne la Valle, France5Universit Paris-Est, MAST, SDOA, IFSTTAR, F-77447 Marne-la-Valle, France6INTERACT Research Unit, Institut PolyTechnique UniLaSalle, France,

AbstractIn crowd motion models, model parameters can very significantly. Therefore a large number ofsimulations is necessary to study the average effects of these parameters. Hence, in this paper thekinetic model developed in the previous our work is considered [1]. The evacuation time of theroom is calculated by our technique. Then, the influence of domain quality, size of the exit and therole of initial conditions on this time is studied. Afterwards, the shape, position and dimensions ofan obstacle allowing an optimal evacuation time are determined. Finally, the calculated evacuationtime is compared with the time obtained by a deterministic approach by randomly varying some ofits parameters.Keywords: Discrete kinetic theory; Crowd dynamics; Panic situation;Complex system; Evacuationtime.

References[1] A. Elmoussaoui, P. Argoul, M. El Rhabi, A. Hakim. Discrete kinetic theory for 2d modeling of a moving

crowd: Application to the evacuation of a non-connected bounded domain. Computers & Mathematics withApplications, 75(4):11591180, 2018.

[2] J.P. Agnelli, F. Colasuonno and D. Knopoff. A kinetic theory approach to the dynamics of crowd evacuationfrom bounded domains. Mathematical Models and Methods in Applied Sciences, 25(01) :109129, 2015.

[3] N. Bellomo and L. Gibelli. Behavioral crowds : Modeling and Monte Carlo simulations toward validation.Computers and Fluids, 141 :13 21, 2016. Advances in Fluid-Structure Interaction.

[4] G.A. Frank and C.O. Dorso. Room evacuation in the presence of an obstacle. Physica A : Statistical Mechan-ics and its Applications, 390(11) :2135 2145, 2011.

44

Preconditioned RBF meshless method appliedin free boundary identification problem

Youness El Yazidi1, Abdellatif Ellabib1

1Faculty of Sciences and Technology, Cadi Ayyad University, Marrakech, Morocco

AbstractThe aim of this work is to adapt the preconditioning techniques to the meshless RBF method forthe identification free boundary problem. First, we manipulate the shape optimization techniqueto transform the free boundary problem into a minimization problem. The meshless RBF methodapplied to the state problem involve a bad conditioned linear system, which is expansive compu-tationally if we use the Gaussian elimination method. A simple preconditioning strategy is use toconvert the linear system to a well conditioned, that can reduce the number of variables and it-erations. At the end we establish some numerical results to show the robustness of the proposedscheme, also we compare the results with and without the preconditioning.

References[1] G. Allaire, Shape Optimization by the Homogenization Method, Springer-Verlag New York, 2002.

[2] L. Ling and E. J. Kansa, A least-squares preconditioner for radial basis functions collocation methods, Ad-vances in Computational Mathematics,23:3154, 2005.

45

Bivariate polynomial interpolation of Lagrangeand Hermite : A general recursive resolution

M. Errachid1, A. Essanhaji2, A. Messaoudi3

1LabMIA-SI, Centre regional des metiers de l’education et de la formation Avenue Allal Al Fassi, 10000 Rabat, e-mail: er-rachid.m@gmail.com2LabMIA-SI, Centre regional des metiers de l’education et de la formation Avenue Allal Al Fassi, 10000 Rabat, e-mail: abdel-hakessanhaji@gmail.com3LabMIA-SI, Ecole Normale Suprieure, University Mohammed V in Rabat, e-mail: abderrahim.messaoudi@gmail.com

AbstractIn this talk, we will study the bivariate polynomial interpolation problem of Lagrange or Hermite onany finite configuration of the plan. In particular, we will define an interpolation polynomial spaceover which we will prove the existence and uniqueness of the interpolant polynomial associatedwith a given configuration. Algorithms for computing recursively the interpolant will be proposed.Examples and particular cases will be studied.

key words: Lagrange Multivariate Polynomial Interpolation Problem, Recursive Algorithms,General Recursive Polynomial Interpolation Algorithm, Recursive Multivariate Polynomial Inter-polation algorithm.

References[1] A. Messaoudi, M. Errachid, K. Jbilou and H. Sadok, GRPIA:a new algorithm for computing interpolation

polynomials. Numerical Algorithms Journal 80(2019) 253-278.

[2] M. Errachid, A.Messaoudi et A. Essanhaji, A new algorithm for computing the Lagrange multivariate poly-nomial interpolation. (Soumis) (2020).

[3] M. Gasca et T. Sauer, On the history of multivariate polynomial interpolation. JCAM (2003) 23-35.

[4] R.A Lorentz, Multivariate Hermite interpolation by algebraic polynomials: A survey. JCAM (2003) 167-201.

[5] R. D. Neidinger, Multivariate Polynomial Interpolation in Newton Forms, Siam Review 61(2019) 361-381.

46

On the numerical solution of fractionalstochastic differential equations with jumps

Adil Ez-zetouni1, Mehdi Zahid1, Khadija Akdim1

1Department of Mathematics, Faculty of Sciences and Technology, Cadi Ayyad University, PO Box 549, 40000 Marrakesh, Mo-rocco

AbstractIn this work we give an overview of numerical methods for the solution of fractional stochasticdifferential equations with jumps. Then we present discrete time strong and weak approximationmethods that are suitable for different applications.

References[1] Z.-Q. Chen, K.-H. Kim and P. Kim, Fractional time stochastic partial differential equations. Stochastic Pro-

cess. Appl. 125 (2015) 1470-1499.

[2] J. Jacod and P. Protter (1998), ’Asymptotic error distribution for the Euler method for stochastic differentialequations’, Ann. Probab. 26, 267-307.

[3] A. Janicki, Z. Michna and A. Weron (1996), ’Approximation of stochastic differential equations driven bya-stable Levy motion’, Applicationes Mathematicae 24, 149-168.

[4] R. Janssen (1984), ’Discretization of the Wiener process in difference methods for stochastic differentialequations’, Stochastic Process. Appl. 18, 361-369.

[5] Mishura, Y., and Nualart, D. (2004). Weak solutions for stochastic dierential equations with additive frac-tional noise. Statistics and Probability Letters 70:253261.

47

An efficient identification of red blood cell equi-librium shape in 2D using shape optimizationand neural networks

Houda Fahim1, Nour Eddine Alaa2

1 LAMAIL Laboratory, FST MARRAKECH, Cadi Ayyad University2 LAMAIL Laboratory, FST MARRAKECH, Cadi Ayyad University

AbstractThis work is connected about the numerical modelisation of red blood cells in 2 dimensions. Sincethe membrane represents the fundamental architecture of the cell with its shapes determined by theHelfrich curvature bending energy. The pattern of Canham and Helfrich is adopted in this work todescribe the behaviour of red blood cells. Concerning the static equilibrum of the red blood cell,a mechanical equilibrum equation (Euler-Lagrange equation) of the membrane under a generalizedelastic bending energy with constraints is obtained and theapproach is based on shape optimizationtools.

The methodology is based on the Lagrange multiplier theory in optimization (to ensures theaccurate conservation of volume and area constraints) and neural network algorithm which seeksto provide solutions satisfying the necessary conditions of optimality. The equilibrium point of thenetwork satisfies the Kuhn-Tucker condition for the problem. No explicit restriction is imposed onthe form of the cost function apart from some general regularity and convexity conditions.

References[1] A. Laadhari, C. Misbaha, P. Saramito. On the equilibriumequation for a generalized biological membrane

energy by using a shape optimization approach. Physica D 239(2010) 1567-1572.

[2] U. Seifert. Configuratons of fluid membranes and vesiclesAdvances in Phys.Vol.46,(1997) No.1,13-137.

[3] Ou-Yang Zhong-Can and Wolfgang Helfrich. Bending energy of vesicle membranes: General expressionsfor the first, second, and third variation of the shape energyand applications to spheres and cylinders.Physical Review A, 39(10):5280, 1989

[4] Wolfgang Helfrich. Elastic properties of lipid bilayers: theory and possible experiments. Zeitschrift frNaturforschung C, 28(11-12):693703, 1973.

48

Systems of multidimensional BSDEs arising inthe balance sheet optimal switching problem

Rachid Belfadli1, M’hamed Eddahbi2, Imade Fakhouri3, Youssef Ouknine4,5

1Cadi Ayyad University, Faculty of Sciences and Techniques, Department of Mathematics, B.P. 549,Marrakesh, 40000, Morocco. E-mail: belfadli@gmail.com2King Saud University, College of Sciences, Mathematics Department, P.O. Box 2455, Riyadh, Z.C. 11451, Riyadh, Kingdom ofSaudi Arabia. E-mail: meddahbi@ksu.edu.sa3Mohammed VI Polytechnic University, Complex Systems Engineering and Human Systems, Ben Guerir, 43150, Morocco. E-mail:imade.fakhouri@um6p.ma4Cadi Ayyad University, Faculty of Sciences Semlalia, Department of Mathematics, B.P. 2390, Marrakesh, 40.000, Morocco. E-mail: ouknine@uca.ac.ma5Mohammed VI Polytechnic University, Africa Business School, Ben Guerir, 43150, Morocco.E-mail: youssef.ouknine@um6p.ma

AbstractIn this paper, we study a system of multidimensional coupled backward stochastic differential equa-tions (BSDEs) with interconnected generators and barriers and mixed reflections, i.e. oblique andnormal reflections. This system of coupled reflected BSDEs (RBSDEs) which is living in specialunbounded convex domains, is arising in the context of optimal switching problem when both sidesof the balance sheet are considered. This problem incorporates both the action of switching betweeninvestment modes and the action of abandoning the investment project before its maturity once itbecomes unprofitable. Pricing such real options (switch option and abandon option) is equivalent tosolve the system of coupled RBSDE considered in the paper, for which we show the existence of acontinuous adapted minimal solution via a Picard iteration method.

References[1] S. Aazizi, T. El Mellali, I. Fakhouri and Y. Ouknine. Optimal switching problem and related system of

BSDEs with left-Lipschitz coefficients and mixed reflections. Statistics and Probability Letters, 137:70–78,(2018).

[2] M. Eddahbi, I. Fakhouri and Y. Ouknine. A Balance Sheet Optimal Multi-Modes Switching Problem. AfrikaMatematika, (2019). DOI:10.1007/s13370-019-00719-7

[3] M. Eddahbi, I. Fakhouri and Y. Ouknine. Mean-field optimal multi-modes switching problem: a balancesheet. Stochastics and Dynamics, 19(4):1950026, (2019).

49

Benchmark of various multi-level linear solverson different hardware configurations with ALIEN,an open generic and extensible linear algebraframework

Cedric Chevalier1, Sylvain Desroziers2, Jean-Marc Gratien2, Pascal Have3,Xavier Tunc2

1CEA, DAM, DIFF, F-91297 Arpajon, France2IFPEN, 1 et 4, avenue du Bois-Preau, 922 Rueil-Malmaison Cedex, France2HAVENEER, www.heveneer.com

AbstractNowadays frameworks like Arcane [1], Dune[2], Fenics[3], Feel++[4] provide high level abstrac-tions to write efficient applications to solve large and complex partial derivative equation systems.They usually rely on frameworks like PETSc [5] or Trilinos [6] to manage matrices and vectors andto have access to a wide range of algorithms. In this paper, wepresent ALIEN, a C++ frameworkthat provides a high level and unified API to handle large distributed matrices and vectors, to per-form BLAS 1 and 2 operations and to solve linear systems or eigenproblems. Its differ from thepreviously cited frameworks in its design based on light weight structures that encapsulate multi-ple coexisting internal representations of algebraic objects. These representations are dedicated tothe algebra operations or linear solver algorithms available through different linear solver packagesor libraries. A mechanism is provided to manage efficiently and transparently conversion betweenthe different object representations. A plugin mechanism based on converter objects ensures theextensibity of framework to any kind of external linear solver libraries.

We have used the ALIEN framework to compare popular multi-level algorithms like AlgebraicMulti-Grids solvers (Hypre BoomerAMG[7], Trilinos ML and MueLU solvers[8], Nividia NVAMGsolver[9]), multi level domain decomposition algorithms (HPDDM[10] available in PETSc, DDML[11]provided in IFPEN solver library). A parallel synthetic simulator has been developed to solve theadvection-diffusion equation :

∇ · (vu−D∇u) = f onΩ

whereu, v, D and f are respectively the unknown, the velocity, the diffusivity and the sourceterm. The equation is discretized on a regular grid with an upwind Finite Volume scheme leadingto a sparse linear system solved with the ALIEN framework. The linear system size depends onthe mesh size and the matrix spectral properties depend on parameters controlling the variationof the diffusion coefficientD(x) and the velocity fieldv(x). We study the numerical robustnessof the tested algorithms, their scalability, and extensibility regarding the number of cores and theproblem sizes. We analyze the capability of each algorithm implementation to take advantage ofnew hardware features such as SIMD instructions or GP-GPU accelerators and to use the hybridMPI and threads parallel paradigm to handle large multi-node cluster with many-core processors.

50

References[1] The Arcane Development Framework,Proceedings of the 8th Workshop on Parallel/High-Performance

Object-Oriented Scientific Computing, Grospellier, Gilles and Lelandais, Benoit, POOSC09,2009,978-1-60558-547-5

[2] BASTIAN, Peter, HEIMANN, Felix, et MARNACH, Sven. Generic implementation of finite element meth-ods in the distributed and unified numerics environment (DUNE). Kybernetika, 2010, vol. 46, no 2, p. 294-315.

[3] ALNAES, Martin, BLECHTA, Jan, HAKE, Johan, et al. The FEniCS project version 1.5. Archive of Numer-ical Software, 2015, vol. 3, no 100.

[4] PRUDHOMME, Christophe, CHABANNES, Vincent, DOYEUX, Vincent, et al. Feel++: A computationalframework for galerkin methods and advanced numerical methods. In : ESAIM: Proceedings. EDP Sciences,2012. p. 429-455.

[5] PETSc Web page, url = http://www.mcs.anl.gov/petsc, Balay, Satish and Abhyankar, Shrirang and Adams,Mark F. and Brown, Jed and Brune, Peter and Buschelman, Kris and Dalcin, Lisandro and Dener, Alp andEijkhout, Victor and Gropp, William D. and Kaushik, Dinesh and Knepley, Matthew G. and May, Dave A.and McInnes, Lois Curfman and Mills, Richard Tran and Munson, Todd and Rupp, Karl and Sanan, Patrickand Smith, Barry F. and Zampini, Stefano and Zhang, Hong and Zhang, Hong, 2018

[6] Heroux, Michael A and Bartlett, Roscoe A and Howle, VickiE and Hoekstra, Robert J and Hu, JonathanJ and Kolda, Tamara G and Lehoucq, Richard B and Long, Kevin R and Pawlowski, Roger P and Phipps,Eric T and Salinger, Andrew G and Thornquist, Heidi K and Tuminaro, Ray S and Willenbring, JamesM and Williams, Alan, An Overview of the Trilinos Project, 31,0098-350,10.1145/1089014.1089021 ACMTransactions on Mathematical Software

[7] Falgout, Robert D. and Yang, Ulrike Meier, hypre: A Library of High Performance Preconditioners, 978-3-540-47789-1, Computational Science ICCS 2002,Springer Berlin Heidelberg

[8] HU, Jonathan Joseph et PROKOPENKO, Andrey. MueLu: Multigrid Framework for Advanced Architec-tures. Sandia National Lab.(SNL-CA), Livermore, CA (United States); Sandia National Lab.(SNL-NM),Albuquerque, NM (United States), 2015.

[9] EATON, Joe. GPU-Accelerated algebraic multigrid for commercial applications. Manager NVAMG CUDALibrary. NVIDIA, 2013.

[10] JOLIVET, Pierre et NATAF, Frederic. Hpddm: High-Performance Unified framework for Domain Decom-position methods, MPI-C++ library. 2014.

[11] GRATIEN, Jean-Marc., A robust and scalable multi-level domain decomposition preconditioner for multi-core architecture with large number of cores. Journal of Computational and Applied Mathematics, 2019, p.112614.

2

51

Spectral clustering: incremental and evolution-ary algorithms

P. El Alam1,2, J. Constantin2, D. Hamad1

1LISIC-ULCO, 50 rue F. Buisson, BP 719, 62228 Calais Cedex, France2Research Laboratory in Networks, Computer Science and Security (LaRRIS), Lebanese University, Campus Fanar, BP 90656Jdeidet, Lebanon

AbstractSpectral clustering is a successful approach due to its rigorous foundation based on graph theoryand matrix computation. It relies on the computation of the spectrum and the corresponding eigen-vectors of the Laplacian matrix. It has been used in many applications such as image segmentation,signal processing, data clustering, web ranking and analysis, etc. However, standard spectral clus-tering algorithms are off-line algorithms, and hence they cannot be directly applied to dynamicdata. Therefore, to handle evolving data, there is a need to develop efficient algorithms for induc-tive spectral clustering to avoid computation of the eigen-system solution from the scratch. Forthat, two approaches are used: incremental spectral clustering and evolutionary spectral cluster-ing. Incremental spectral clustering, initialized by a standard spectral clustering, handles evolvingdata by incrementally updating the eigen-system in order to generates instant cluster labels as thedata is evolving. In evolutionary spectral clustering, a good clustering result should fit the currentdata well, while simultaneously not deviate too much from the recent history. More precisely, itshould provide more stable and consistent clustering results that are less sensitive to short-termnoises while at the same time are adaptive to long-term cluster drifts. Incremental and evolutionaryspectral approaches are applied to dynamic partitioning of transportation network.

References[1] Von Luxburg, U.: A tutorial on spectral clustering. Statistics and computing 17(4), 395-416, 2007.

[2] Ning, H., Xu, W., Chi, Y., Gong, Y., Huang, T.: Incremental spectral clustering with application to monitoringof evolving blog communities. In: Proceedings of the SIAM International Conference on Data Mining, pp.261-272, 2007.

[3] Chi, Y., Song, X., Zhou, D., Hino, K., Tseng, B.L.: Evolutionary spectral clustering by incorporating tempo-ral smoothness. In: Proceedings of the 13th ACM SIGKDD international conference on Knowledge discoveryand data mining, pp. 153-162, 2007.

[4] Lopez, C., Krishnakumari, P., Leclercq, L., Chiabaut, N., Van Lint, H.: Spatiotemporal partitioning of trans-portation network using travel time data. Transportation, Research Record 2623(1), 98-107, 2017.

52

Stabilization of incompressible flow problemsusing Riccati feedback approach based on Krylovsubspace methods

M.A Hamadi1,2, K. Jbilou1,2, A. Ratnani2

1Laboratory LMPA, 50 rue F. Buisson, ULCO Calais cedex, France. E-mail: khalide.jbilou@univ-littoral.fr2Laboratory CSEHS, Lot 660, Hay Moulay Rachid, UM6P Ben Guerir, Morocco. E-mail: amine.hamadi@um6p.ma. ahmed.ratnani@um6p.ma

AbstractThe Navier-Stokes system is the basis for computational modeling of the flow of an incompressibleNewtonian fluid, such as air or water. We can encounter this system in many technical fields asairplane design, nuclear reactor safety evaluation, or microfluids in micro- and nanosystems. Astable and controlled velocity field are considered as the basis for an ongoing reaction or productionprocesses of many applications. One of the robust method that guarantee us a stable and controlledvelocity is the one based on a Riccati feedback approach. During the process we need to solve a largescale Riccati equation which is considered as the key to compute what we call a feedback matrix.For solving this equation we investigate numerical methodsusing Krylov subspaces methods.

References[1] E. Bansch, P. Benner, J. Saak and H.-K. Weichelt, Riccati-based boundary feedback stabilization of incom-

pressible Navier-Stokes flows. Siam J. Sci. Comput. Vol. 37,No. 2, pp A832-A858.

[2] P. Benner, J. Saak, M.Stoll and H.-K Weichelt, Efficient solution of large-scale scale saddle point systemsarising in Riccati-based boundary feedback stabilizationof incompressible stokes flows. Siam J. Sci. Comput.Vol. 35, No. 5, pp S150-S170.

[3] M. Heinkenschloss, D.-C Sorensen and K. Sun, Balanced trubcation model reduction for a class of descriptorsystems with application to the oseen equations. Siam J. Sci. Comput. Vol. 30, No. 2, pp 1038-1063.

[4] M. Heyouni and K. Jbilou An extended block arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation. Elctronic Transactions on Numerical Analysis. Vo 33, pp. 53-62, 2009.

53

Modeling and optimization of an energy distri-bution system

Jean-Paul Chehab, Vivien Desveaux, Marouan HandaLAMFA, UMR 7352 CNRS, University of Picardie Jules Verne, Amiens, France

AbstractThis work is concerned with optimization problems arising in an energy distribution system withstorage. We start from the derivation of a simplified networktopology model around four nodes:load aggregator, the external grid, consumption and storage with the charge and the discharge, takinginto account the charging (resp. discharging) efficiencies. The imported power from the externalgrid should balance the consumption and the storage variation. We define the merit function wewant to minimize as the total price to pay in a given time interval using the external power load.The first mathematical problem we derived is a discrete coupled linear optimization problem withsome constrainst that includes bounds of storage capacity,and of charge and of discharge. Wepropose a simplex method as well as an interior point method to compute an effective numericalsolution; we establish mathematical properties of the model. Next, we introduce a second modelfrom the first one by taking into account power subscription possibilities. The merit function is thennonlinear and non-differentiable; we discuss two approaches to avoid a non-differentiability pointand to solve numerically the problem with a non-linear method (SQP algorithm or interior pointalgorithm).Finally, we introduce a sliding window algorithm that allows to reduce the computation time and tomake real time simulations. Numerical results are presented on real data to highlight both modelsand to illustrate the performance of the sliding window algorithm.

References[1] Y. Xu, L. Xie, and C. Singh, Optimal scheduling and operation of load aggregators with electric energy

storage facing price and demand uncer- tainties, in 2011 North American Power Symposium, pp. 17, IEEE,2011.

[2] J.-F. Bonnans, J. C. Gilbert, C. Lemarechal, and C. A. Sagastizabal, Numerical optimization: theoretical andpractical aspects. Springer Science & Business Media, 2006.

54

Improvemments of some iterative methods fora special common solution of split equilibriumproblems and fixed points problems

Ihssane Hay1, Abdellah Bnouhachem2

1Equipe MAISI, Ibn Zohr University, ENSA, BP 1136, Agadir, Morocco.ihssane.hay@gmail.com2Equipe MAISI, Ibn Zohr University, ENSA, BP 1136, Agadir, Morocco.babedallah@yahoo.com

AbstractThroughout this paper we suggest and study a new iterative method for solving a split equilibriumproblem and fixed point of a finite family of nonexpansive mapping in a real Hilbert space. Thenwe prove the strong convergence of our algorithm under some suitable assumptions, to the commonelement of the solution set of split equilibrium problems and the set of fixed points problems inthe setting of Hilbert spaces. Furthermore we give some numerical experiments to illustrate theefficiency of our proposed iterative method. Our results presented in this paper improve someexisting methods in the earlier and recent literature.

References[1] A. B NOUHACHEM , A modified projection method for a common solution of a systemof variational inequal-

ities, a split equilibrium problem and a hierarchical fixed-point problem; Journal of Numerical Mathematics,volume 2014, Issue 22, pp 1-25, (2014).

[2] F. E. BROWDER , Convergence of approximants to fixed points of nonexpansivenonlinear mappings inBanach spaces; Archive for Rational Mechanics and Analysis, volume 24, Issue 1, pp 82-90, (1967).

[3] H. K. X U , Iterative algorithms for nonlinear operators; Journal of the London Mathematical Society, vol-ume 66, Issue 1, pp 240-256, (2002).

55

Credit risk management by artificial intelligence

Hourri Maryem1, Bailoul Charaf Eddine1, Alaa Nour Eddine1

1Laboratory LAMAI, University Cadi Ayyad, Marrakesh, Morocco

AbstractThe rapid development of deep learning and machine learningtechniques has sparked much inter-est in its application to financial problems. Here, we develop a deep learning algorithm that canaccurately detect customers who have a higher risk of default by using an approach who uses thetechniques of artificial intelligence and a deep history of data. In this approach, some informationthat depends on the client’s financial profile is used in our model to get an output which is the grant-ing or not of the credit. Our model with deep learning classifies customers who have a higher riskof default of those who have not. On set of test and validationof our database, our new model givesvery good results in terms of precision and performance. We also show that our model based ondeep learning is more efficient than the classical method which is scoring. These results show thatautomated deep learning methods can be easily trained to achieve high accuracy, and are extremelypromising for improving the bank’s tools to reduce risk.

References[1] Anderson, R. (2007),The Credit Scoring Toolkit, Oxford University Press.

[2] Deepshikha, K. (2015),Supervised and unsupervised document classification-a survey, International Journalof Computer Science and Information Technologies.

[3] Hornik, K. (1991),Approximation capabilities of multilayer feedforward networks, Neural Networks.

[4] McNelis, P. (2004),Neural networks in Finance: gaining predictive edge in the market.

[5] Refaat, M. (2006),Data Preparation for Data Mining Using SAS, Morgan Kaufmann.

56

A modification and extension ofthe fast algorithm for computingthe mock-Chebyshev nodes

B. Ali Ibrahimoglu

Department of Mathematical Engineering, Yıldız Technical University, Davutpasa Campus, 34210 Istanbul, Turkey

Abstract

Polynomial interpolation on an equispaced is notoriously unreliable due to the Runge phenomenon,

and also numerically ill-conditioned. By taking advantage of the optimality of the interpolation

processes on Chebyshev-Lobatto nodes, the mock-Chebyshev subset interpolation is one of the best

strategies to defeat the Runge phenomenon [1].

In the recent paper [2], we have presented a fast algorithm to compute the mock-Chebyshev

nodes using the distance between each pair of consecutive points. In this study, we propose a

modification and extension of the algorithm by changing the function to compute the quotient of the

distance and show that this modified algorithm is also fast and stable; and always gives a satisfactory

grid with the complexity of the algorithm being O(n). We also give some numerical experiments

using the points obtained by the procedure for some different non-uniform grids.

References

[1] J.P. Boyd, F. Xu, Divergence (Runge Phenomenon) for least-squares polynomial approximation on an equis-

paced grid and Mock-Chebyshev subset interpolation. Appl. Math. Comput. 210 (2009) 158–168.

[2] B.A. Ibrahimoglu, A fast algorithm for computing the mock-Chebyshev nodes. J. Comput. Appl. Math.

(2019).

57

Molecular dynamics modelling in ultra-high pre-cision manufacturing

Khaled Abou-El-Hossein1

1Department of Mechatronics, Nelson Mandela University, Summerstrand, Port Elizabeth 6031, South Africa; Khaled.abou-el-hossein@mandela.ac.za

AbstractOptical materials such as single-crystalline silicon and rapidly solidified aluminium are intensivelyused for making optical components used in a number of critical applications. Ultra-high precisionmachining (UHPM) based on diamond turning is used in shapingthese components. The process isrealised using a complex mechanical system that is based on the mechanical interaction between anatural diamond cutting tool, workpiece material and machine tool to shape a component by remov-ing shavings. The success of precision diamond turning largely depends on a tremendous numberof factors including the microstructure of the machined optical materials. Understanding the effectof the material microstructure on the performance of UHPM, in terms of surface finish and diamondtool wear, is usually realised using extensive experimental observations and modelling that is basedon the experimental findings. However, it is extremely difficult to experimentally predict the be-haviour of the UHPM as a result of nanometric variation of thematerials microstructure. A methodthat has been successfully used in predicting the nanometric behaviour of UHPM is widely calledmolecular dynamics (MD). MD is a physics-based modelling method that provides detailed infor-mation on the fluctuations and conformational changes of atoms and molecules in materials duringmachining. The information obtained at this nanometric level can be generalised with confidence todescribe the UHPM systems behaviour. This presentation will attempt to give the audience a hinton the recent developments of the use of MD in ultra-high precision manufacturing when makingoptical components. The presenter will share their contribution in this area of diamond machiningof optical components made from single-crystalline silicon and rapidly solidified aluminium.

58

Model reduction for large scale first and sec-ond order dynamical systems via an adaptivetangential Lanczos-type method

A. H. Bentbib1, K. Jbilou2, Y. Kaouane3

1Universite Cadi Ayyad, Laboratoire de Mathematiques Appliquees et Informatique, Marrakech, Maroc.2Laboratoire de Mathematiques Pures et Appliquees, Universite du Littoral Cote d’Opale, Calais, France.3Universite Mohammed VI Polytechnique, Ben Guerir, Maroc: yassine.kaouane@um6p.ma

AbstractLarge-scale simulations play a crucial role in the study of agreat variety of complex physical phe-nomena, leading often to overwhelming demands on computational resources. Managing thesedemands constitutes the main motivation for model reduction: produce simpler reduced-order mod-els, which allow for faster and cheaper simulation while accurately approximating the behaviourof the original model. The presence of multiple inputs and outputs (MIMO) systems, makes thereduction process even more challenging. In this paper, we present a new approach to treat largescale first and second order dynamical systems with multipleinputs and multiple outputs (MIMO),named: Adaptive Tangential Lanczos Algorithm (ATLA). We give some algebraic properties andpresent some numerical examples to show the effectiveness of the proposed algorithm.

References[1] A. C. Antoulas, C. A. Beattie and S. Gugercin, Interpolatory Model Reduction of Large-scale Dynamical

Systems. Efficient Modeling and Control of Large-Scale Systems, (2010), 3–58.

[2] A. H. Bentbib, K. Jbilou and Y. Kaouane. An adaptive blocktangential method for multi-input multi-outputdynamical systems. J. of Compu. Appl. Math., 358(2019), 190–205.

[3] V. Druskin, V. Simoncini, and M. Zaslavsky. Adaptive Tangential Interpolation in Rational Krylov Subspacesfor MIMO Dynamical Systems. SIAM. J. Matrix Anal. and Appl.,35(2014), 476–498.

59

Approximation of pseudospectra of blocktriangular matrices

Gorka Armentia1, Shreemayee Bora2, Michael Karow3, Nandita Roy4

1Department of Applied Mathematics and Statistics, University of the Basque Country,Spain2Department of Mathematics, IIT Guwahati, India3Department of Mathematics, TU Berlin, Germany4Department of Mathematics, IIT Guwahati, India

AbstractThe ε-pseudospectrum of A ∈ Cn,n is defined as the union of spectra of all matrices A + E withE ∈ Cn,n and ‖E‖2 ≤ ε. The pseudospectra satisfy Λε(A) = z ∈ C | σmin(zI −A) ≤ ε , whereσmin denotes the smallest singular value. We assume that A is given in block Schur form

A = U

[L C0 M

]U∗ with L ∈ C`,`, M ∈ Cm,m, C ∈ C`,m, U unitary.

Our goal is to give inner and outer bounds for the pseudospectra of A in terms of the pseudospectraof L and M . A trivial inner bound of this kind is Λε(L)∪Λε(M) ⊆ Λε(A), with equality if C = 0.First, we discuss several old and new outer approximations of the form

Λε(A) ⊆ Λg(ε) ε(L) ∪ Λg(ε) ε(M).

for some function g(ε). By analyzing the 2× 2 case we show that our bounds are all sharp in somesituation. Next, we consider inner approximations of the form Λf(ε) ε(L) ⊆ Λε(A) for the case thatthe size of L does not exceed the size of M . If A has the simple eigenvalue λ = L then the factorsf and g approach its condition number as ε tends to 0.

References[1] L. Grammont, A. Largillier. On ε-spectra and stability radii. Journal of Computational and Applied Mathe-

matics, 147 (2002), 453–469.

[2] L.N. Trefethen, M. Embree. Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Opera-tors. Princeton University Press, 2005.

60

Numerical solution of second kind Volterra in-tegral systems

A. Kouibia and M. Pasadas1

1Department of Applied Mathematics, University of Granada, 18071 Granada, Spain

AbstractThe theory of integral equations has close contacts with many different areas of mathematics. Manyproblems in the fields of differential equations can be recast as integral equations and they areencountered in various fields of science and numerous applications as elasticity, plasticity, heat andmass transfer, oscillatory theory, fluid dynamics among others.

A second kind Volterra system can be expressed in the form

f(t) = x(t)−

∫ t

0

k(t, s)x(s)ds, 0 ≤ s ≤ t ≤ T, (1)

wheref(t) = (f1(t), . . . , fm(t))T , the matrix-valued functionk will be given byk(t, s) = (kij(t, s))1≤i,j≤m andx is the unknown vector-valued function of the system.It is known that ifk andf are continous (1) has a unique solution.We propose a new numerical method to approximate the solution of this type of Volterra integralsystems. This method is based on the minimization of a suitable quadratic functional in a finite-dimensional space generated by a compactly supported radial basis functions set. We study itsconvergence under adequate hypotheses and we present some numerical examples in order to showthe validity of the proposed method.

61

Algebraic optimized Schwarz methods for Black-Scholes models

Lahcen Laayouni1

1School of science and enegineering, Al Akhawayn University, Morocco

AbstractBlack-Scholes Models are know to predict price options of the market. There are mainly two ap-proaches to solve these equations. One approach is based on Monte Carlo simulations solvingstochastic differential equations, and the other approachis based on solving partial differential equa-tions using FEM or FD methods. In this talk we will consider the latter approach. We will explorethe applicability and the efficiency of Algebraic OptimizedSchwarz Methods (AOSMs) for solvingBlack-Scholes models. We will consider the European Vanilla Call and Put Options. Both semi andfully implicit schemes in time will be considered. At each time step we need to solve a large-scalelinear system. The main idea of AOSMs is based on modifying the classical transmission blocksinto optimized blocks obtained from the neighboring sub-domains. The convergence of the optimalAOSM in the case of a two sub-domain decomposition is in two iterations for the present model.Some numerical examples will be presented at the end.

62

An efficient algorithm to compute pedestriansdesired directions in a dense crowd

A. Lamghari1, A. Jebrane2, A. Hakim3

1LAMAI FST Marrakech Email: a.hakim@uca.ma2COMPLEX SYSTEMS AND INTERACTIONS, ECOLE CENTRALE CASABLANCA Email: aissam.jebrane@centrale-casablanca.ma3LAMAI-FST Marrakech Email: abdelkarimsm@gmail.com

AbstractIn this work, we develop an algorithm for optimally solving the short path problem in case of adense crowd which is a good example of a complex system. The proposed algorithm is an originaltechnique to compute pedestrians’ desired directions for a dense crowd regardless of the used model.and consequently, reduce the calculation time of the differents pedestrians movement simulators asPTV VisWalk, Legion, Anylogic, MassMotion and SimWalk. the relevance of the proposed methodis proven by comparing it with the Dijkstra algorithm and the Fast Marching method.

References[1] X. DESQUESNES, A. ELMOATAZ, O. LEZORAY, Eikonal equation adaptation on weighted graphs: fast

geometric diffusion process for local and non-local image and data processing . Journal of MathematicalImaging and Vision, 2013.

[2] BUCKLEY, F. AND HARARY, F., Distance in Graphs, Addison-Wesley (1990).

[3] CHARTRAND, G., On Hamiltonian line graphs, Trans. Amer. Math. Soc. 134 (1968) 559-566.

63

Preconditioners to compute the least squaressolution of overdetermined linear systems withdense rows

J. Mas1, J. Marın1, M. Tuma3, R. Bru1

1Instituto de Matematica Multidisciplinar, Universitat Politecnica de Valencia, Camino de Vera, 14, 46022–Valencia(SPAIN). Email:jmasm@imm.upv.es, jmarinma@upv.es, rbru@imm.upv.es2Department of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University and Institute of Computer Sci-ence, Academy of Sciences of the Czech Republic, Pod vodarenskou vezı 2, 182 07 Prague 8, Czech Republic, Email: mirek-tuma@karlin.mff.cuni.cz

AbstractThe least squares solution of a large and sparse overdetermined inconsistent linear system

Ax = b, (1)

can be computed using iterative methods. One of the most used iterative methods for LS problemsis CGLS [1]. CGLS is equivalent to applying the Conjugate Gradient method (CG) to the normalequations ATAx = AT b. Preconditioners are used to improve the convergence of the iterativemethod. A natural choice is to compute an incomplete Cholesky factorization of the matrix ATA,see [2] for instance. However ATA tends to be much denser than A –observe if A has one rowwhose entries are all different from 0, then ATA is structurally full – and it is difficult to computegood and sparse preconditioners for dense matrices.

We will propose some techniques to solve the LS problems when A contains a substruture thatmakes impossible to compute an efficient preconditioner. To do this we propose to compute anapproximate factorization of ATA.

If we have more dense rows than only a few and they can be described as a submatrix B of A,we propose to reduce the block approximating the range of BT by a scaled orthogonal matrix Qwith a small number of columns.

References[1] A. Bjorck, Numerical methods for Least Squares Problems. SIAM, Philadelphia (1996)

[2] R. Bru, J. Marın, J. Mas, M. Tuma, Preconditioned Iterative Methods for Solving Linear Least SquaresProblems SIAM J. Sci. Comput., 36(4), A2002–A2022, 2014.

64

On the Gauss-Radau upper bounds for the A-norm of the error in the Conjugate Gradientalgorithm

Gerard Meurant

Paris, France

This is a joint work with Petr Tichy (Charles University, Prague)

Abstract

The connection between the Conjugate Gradient (CG) algorithm for solving Ax = b (with A sym-

metric and positive definite) and Gauss quadrature has been known since the seminal paper of

Hestenes and Stiefel [1] in 1952. This link has been exploited by G.H. Golub and his collaborators

to bound or estimate the A-norm of the error in CG; see [2, 3, 4, 5].

If a lower bound µ of the smallest eigenvalue λmin of A is known, the Gauss-Radau quadrature

rule provides an upper bound of the A-norm of the error which can be computed cheaply. The closer

µ is to λmin, the sharper is the upper bound.

However, in finite precision arithmetic, below a certain level of the A-norm, the bound can

become less accurate. Moreover, the bound is somehow independent of the value of µ when it is

close to λmin.

In this talk we will discuss heuristic explanations of this problem as well as possible remedies.

References

[1] M.R. HESTENES AND E. STIEFEL, Methods of conjugate gradients for solving linear systems,

J. Nat. Bur. Standards, v 49 n 6 (1952), pp. 409–436.

[2] G. MEURANT, The computation of bounds for the norm of the error in the conjugate gradient algorithm,

Numer. Algorithms, v 16 (1997), pp. 77–87.

[3] G. MEURANT, The Lanczos and conjugate gradient algorithms, from theory to finite precision computations,

SIAM, (2006).

[4] G. MEURANT AND P. TICHY, On computing quadrature-based bounds for the A-norm of the error in con-

jugate gradients, Numer. Algorithms, v 62 n 2 (2012), pp. 163–191.

[5] G. MEURANT AND P. TICHY, Approximating the extreme Ritz values and upper bounds for the A-norm of

the error in CG, Numer. Algorithms, v 82, n 3 (2019), pp. 937-968.

65

Variable selection in statistical modelling via anumerical linear algebra approach

Marilena Mitrouli11Department of Mathematics, National and Kapodistrian University of Athens, Panepistemiopolis 15784, Athens, Greece, e-mail:mmitroul@math.uoa.gr

AbstractVariable selection requires the minimization of ||Xβ − y||2 with respect to β, where X ∈ Rn×p isthe design matrix, β ∈ Rp is a vector of predictors and y ∈ Rn is the response of the model. Theidentification of predictors is important for statistical modelling and numerical analysis can bringto statistics community advanced linear algebra techniques for handling this issue. In this work, fora given model y = Xβ + ε, where ε ∈ Rn is the vector of random errors, we study the followingproblems:

(P1) Regularization and condition estimation. It is crucial to decide whether the given modelneeds regularization or not for the derivation of the vector β. The notion of the effective conditionnumber is introduced, which provides a measure for the stability of β due to a perturbation in y.

(P2) Fast GCV estimates for correlated matrices. When regularization is applied and thereforethe minimization of ||Xβ − y||2 + λ||β||2 is needed, the specification of appropriate values ofthe tuning parameter λ is an important issue. When the design matrix has correlated columns itseigenvalue structure leads to a fast estimate for the generalized cross validation (GCV) functionwhich can provide a good value for the parameter λ.

(P3) Numerical methods for high dimensional data. When the design matrix has much morecolumns than rows we deal with high dimensional data. In such cases, it is of great interest thecomputation of β since the sparsity and the stability of the solution must be effectively treated.

References[1] P. Craven and G. Wahba. Smoothing noisy data with spline functions. Numerische Mathematik, 31 (4):377–

403, 1978.

[2] C. Koukouvinos, K. Jbilou, M. Mitrouli and O. Turek. An eigenvalue approach for estimating the generalizedcross validation function for correlated matrices. Electronic Journal of Linear Algebra, 35: 482–496, 2019.

[3] J. R. Winkler and M. Mitrouli. Condition estimation for regression and feature estimation. J. Comp. Appl.Math, to appear, 2020.

66

Space-time multipatch discontinuous galerkinisogeometric analysis for parabolic evolutionproblems

Stephen E. Moore1

1Department of Mathematics, University of Cape Coast, Ghana | stephen.moore@ucc.edu.gh

AbstractWe present and analyze a stable space-time multipatch discontinuous Galerkin isogeometric anal-ysis (dGIGA) scheme for the numerical solution of parabolicevolution equations in deformingspace-time computational domains. Following [1, 2], we usea time-upwind test function and applymultipatch dGIGA methodology for discretizing the evolution problem both in space and in time.This yields a discrete bilinear form which is elliptic on theIGA space with respect to a space-timedG norm. This property together with a corresponding boundedness property and consistency andapproximation results for the IGA spaces yields an a priori discretization error estimate with respectto the space-time dG norm. The theoretical results are confirmed by several numerical experimentswith low- and high-order IGA spaces, see [3].

References[1] U. Langer, S. E. Moore, and M. Neumuller. Space–time isogeometric analysis of parabolic evolution equa-

tions. Comput. Methods Appl. Mech. Engrg., 306:342–363, 2016.

[2] S. E. Moore. A stable space–time finite element method forparabolic evolution problems.Calcolo, 55(2):18,Apr 2018.

[3] S. E. Moore. Space–time multipatch discontinuous galerkin isogeometric analysis for parabolic evolutionproblems.SIAM Journal on Numerical Analysis, 57(3):1471–1493, 2019.

67

State estimation with reduced models

Olga Mula1

1CEREMADE, Universite Paris Dauphine, Paris.

Abstract

In this talk, we present recent results on the problem of estimating the state of a physical system

with measurement observations and reduced models. After defining the notion of what optimal

algorithms mean in this context, we show how to compute in practice an optimal affine algorithm

and how to use the whole methodology to address the problem of sensor placement. Since this task is

generally np-hard, we present a greedy strategy to iteratively find appropriate sensor measurements

from a complete dictionary. The articles related to the content of this presentation are [2, 1].

References

[1] A. Cohen, W. Dahmen, R. DeVore, J. Fadili, O. Mula, and J. Nichols. Optimal reduced model algorithms for

data-based state estimation. Submitted, 2019.

[2] P. Binev, A. Cohen, O. Mula, and J. Nichols. Greedy Algorithms for Optimal Measurements Selection in

State Estimation Using Reduced Models, SIAM/ASA Journal on Uncertainty Quantification (6-3):1101-

1126, 2018.

68

Benchmark of various multi-level linear solverswith ALIEN, an open generic and extensiblelinear algebra framework

Cedric Chevalier1, Sylvain Desroziers2, Jean-Marc Gratien2, Pascal Have3,Xavier Tunc2

1CEA, DAM, DIFF, F-91297 Arpajon, France2IFPEN, 1 et 4, avenue du Bois-Preau, 922 Rueil-Malmaison Cedex, France2HAVENEER, www.heveneer.com

AbstractNowadays frameworks providing high level abstractions to write efficient applications to solve largeand complex partial derivative equation systems usually rely on frameworks like PETSc or Trilinosto manage matrices and vectors and to have access to a wide range of algorithms. In this paper,we present ALIEN, a C++ framework that provides a high level and unified API to handle largedistributed matrices and vectors, to perform BLAS 1 and 2 operations and to solve linear systems oreigenproblems. Its differ from the previously cited frameworks in its design based on light weightstructures that encapsulate multiple coexisting internal representations of algebraic objects. Wehave used the ALIEN framework to compare popular multi-level algorithms like Algebraic Multi-Grids solvers (Hypre BoomerAMG[1], Trilinos ML and MueLU solvers, Nividia NVAMG solver,multi level domain decomposition algorithms (HPDDM available in PETSc, DDML provided inIFPEN solver library). The advection-diffusion equation has been discretized on a regular grid withan upwind Finite Volume scheme leading to a sparse linear system solved with the ALIEN frame-work. The linear system size depends on the mesh size and the matrix spectral properties depend onparameters controlling the variation of the diffusion coefficient D(x) and the velocity field v(x).We study the numerical robustness of the tested algorithms, their scalability, and extensibility re-garding the number of cores and the problem sizes. We analyze the capability of each algorithmimplementation to take advantage of new hardware features such as SIMD instructions or GP-GPUaccelerators and to use the hybrid MPI and threads parallel paradigm to handle large multi-nodecluster with many-core processors.

References[1] Falgout et al,hypre: A Library of High Performance Preconditioners, 978-3-540-47789-1, Computational

Science ICCS 2002,Springer Berlin Heidelberg

[2] HU at al, MueLu: Multigrid Framework for Advanced Architectures, 2015.

[3] EATON, GPU-Accelerated algebraic multigrid for commercial applications.NVIDIA, 2013.

69

Nonlinear reduced modeling and stateestimation of parametric PDEs

James Nichols1

1Australia National University

AbstractWe examine the problem of state estimation, that is, reconstructing the solution of a known paramet-ric PDE from m linear measurements. When linear reduced models are used, well known results inreconstruction stability and approximation errors can be used to give bounds of overall error of stateestimation. We present some results and schemes for the deployment of nonlinear reduced mod-els for this task, specifically models that are locally linear for disjoint partitions of the parameterdomain. One challenge in this task is sensing which locally linear model to apply, given some spe-cific measurements. Our strategy for this is to consider the residuals, and chose local linear modelsaccording to which minimizes the residual associated with the PDE. We discuss results and someinteresting dual-minimization strategies for parameter estimation that arise, and present a numericalstudy of this strategy.

References[1] A. Cohen, W. Dahmen, R. DeVore, J. Fadili, O. Mula, and J. Nichols.Optimal reduced model algorithms for

data-based state estimation. Submitted, 2019.

[2] P. Binev, A. Cohen, O. Mula, and J. Nichols.Greedy Algorithms for Optimal Measurements Selection inState Estimation Using Reduced Models, SIAM/ASA Journal on Uncertainty Quantification (6-3):1101-1126, 2018.

70

On the convergence of Robin-Robin methodfor solving an inverse problem

Abdellatif Ellabib1, Abdeljalil Nachaoui2, Abdessamad Ousaadane1

1Laboratoire de Mathematiques appliquees et informatique, Faculte des Sciences et Techniques, Universite Cadi Ayyad, Mar-rakech, Maroc.2Laboratoire de Mathematiques Jean Leray, Universite de Nantes, Nantes, France.

AbstractThis work deals with a new method for solving the Cauchy inverse problem in linear elasticity. Theproposed method based on Robin conditions on inaccessible boundary is described in details, andthe algebraic system corresponding to the proposed method is obtained using the integral equationformulation. The convergence of the proposed algorithm is studied. Finally, a numerical study usingthe boundary element method is presented which shows the efficiency of the proposed algorithm.The numerical stability of the algorithm is also studied.

References[1] Ellabib, A., Nachaoui, A.: A domain decomposition method for boundary element approximations of the

elasticity equations. Annals of the University of Craiova, Mathematics and Computer Science Series Volume42(1), Pages 211-225 (2015)

[2] Ellabib, A., Nachaoui, A.: An iterative approach to the solution of an inverse problem in linear elasticity.Mathematics and Computers in Simulation 77, 189-201 (2008)

[3] Jourhmane, M., Nachaoui, A.: An alternating method for an inverse Cauchy problem. Numer. Algorithms,21(1-4), 247-260 (1999)

[4] Kozlov, V. A., Maz’ya, V. G., Fomin, A. V.: An iterative method for solving the Cauchy problem for ellipticequation, Computational Mathematics and Mathematical Physics, 31, 4552 (1991)

[5] Nachaoui, A.: Iterative solution of the drift-diffusion equations. Numer. Algorithms 21(1-4), 323-341 (1999)

[6] Quarteroni, A., Valli, A.: Domain decomposition methods for partial differential equations. Numerical Math-ematics and Scientific Computation. The Clarendon Press, Oxford University Press, New York, Oxford Sci-ence Publications (1999)

71

Generalized Gauss-Kronrod quadrature rulesfor the approximation of matrix functionals

Miroslav Pranic1

1Department of Mathematics and Informatics, University of Banja Luka, Banja Luka, Bosnia and Herzegovina2Address of author B

AbstractThe need to compute expressions of the form u∗f(A)v, where A is a large nonsymmetric matrix,u and v are vectors, and f is a function, arises in many applications, including network analysisand the solution of linear discrete ill-posed problems. Commonly used approach first reduces Ato a small matrix by a few steps of the non-Hermitian Lanczos process and then evaluate the re-duced problem. In this talk we describe a new method to determine error estimates for computedquantities. Our method is based on generalization of the Gauss-Kronrod quadratures.

72

Householder sets for matrix polynomials

Thomas R. Cameron1, Panayiotis J. Psarrakos2

1Department of Mathematics and Computer Science, Davidson College, USA (thcameron@davidson.edu)2Department of Mathematics, National Technical University of Athens, Greece (ppsarr@math.ntua.gr)

AbstractWe present a generalization of Householder sets for matrix polynomials. After defining these sets,we analyze their topological and algebraic properties, which include containing all of the eigenval-ues of a given matrix polynomial. Then, we use instances of these sets to derive the Gershgorin set,weighted Gershgorin set, and weighted pseudospectra of a matrix polynomial. Finally, we show thatHouseholder sets are intimately connected to the Bauer-Fike theorem by using these sets to deriveBauer-Fike-type bounds for matrix polynomials.

73

A free diminishing finite volume scheme fornonlinear diffusion-convection problems

El Houssaine Quenjel1

1quenjel@unice.fr

Abstract

In this talk, we will first present the construction of vertex finite volume schemes on general finite

element meshes dedicated to the discretization of an anisotropic linear diffusion equation with drift

extending the approaches provided in [1, 3]. To this end, a nonlinear transformation of the elliptic

term is required. A particular emphasis is next placed on the approximation of the mobility function

on the interface between two neighbored control volumes. The goal is to reinforce the existence

of positive solutions to the numerical scheme and the decay of the free energy function as fast as

possible [2, 4]. This is ensured thanks to the established uniform estimate on the gradient of the

squared function of the discrete unknown. Additional results concerning the energy dissipation na-

ture of the approach will be also exposed. These ingredients allow the establishment of compactness

estimations. Under general assumptions on the data and the mesh, the convergence of the proposed

finite volume scheme is carried out. Numerical tests provide and compare the efficiency and the ro-

bustness of two centered approximations of the mobility function in the case of various anisotropic

ratios. At the end, we will illustrate that the scheme is free diminishing in the sense that the free

energy is decreasing to the steady state as the time becomes large.

References

[1] E-H. QUENJEL (2019), Enhanced positive vertex-centered finite volume scheme for anisotropic

convection-diffusion equations; Accepted in ESAIM Mathematical Modelling and Numerical Analysis.

[2] C. CANCS, C. CHAINAIS-HILLAIRET, S. KRELL (2018), Numerical analysis of a nonlinear free-energy

diminishing Discrete Duality Finite Volume scheme for convection diffusion equations; Computational Meth-

ods in Applied Mathematics, 18(3), pp. 407-432.

[3] C. CANCS, C. GUICHARD (2017), Numerical analysis of a robust free energy diminishing finite volume

scheme for degenerate parabolic equations with gradient structure; Foundations of Computational Mathe-

matics, 17(6), pp. 1525-1584.

[4] M. BESSEMOULIN-CHATARD (2012), A finite volume scheme for convection-diffusion equations with non-

linear diffusion derived from the Scharfetter-Gummel scheme; Numerische Mathematik, Volume 121, Issue

4, pp 637–670.

74

FETI method in topology optimization

O. Raibi1, A. Makrizi2

1raibi.ouafae@gmail.com2makrizi@gmail.com

AbstractThe need for efficient use of material is prevalent within several scientific and engineering appli-cations. As such, determining the optimal layout of structures is one of the important tasks withinthe aerospace and automotive industries and which can be obtained through the use of topologyoptimization. Given an amount of material, an external loadand boundary conditions, the topologyoptimization aims to determine the optimal distribution ofthe material subject to an appropriatelydefined objective function [1]. Yet, topology optimizationproblems are, from computational pointof view, large scale. In this context, domain decompositionmethods DDM [2] can be very effectiveto overcome this difficulty, and in addition, DDM are better suited for parallel computing. We split,indeed, the domain into subdomains in order to resolve for each subdomain a smaller problem, andconsequently interface unknowns should be treated. In thiswork a topology optimization problemof an isotropic, linear elastic solid is considered, we divide the domain into two non-overlappingsubdomains, an elliptic problem with Neumann boundary condition is defined for each subdomainand the field continuity is enforced via Lagrange multipliers, then the FETI method [3] with finiteelement method [4] are applied on a cantilever beam in order to find the optimal solution from localsolutions.

References[1]: M.P. BENDSE et O. SIGMUND : Material interpolations in topology optimization. Arch. Appl. Mech,69:635654, 1999.[2] : A. QUARTERONI et A. VALLI : Domain Decomposition Methods for Partial Differential Equations.Clarendon Press, Oxford, 1999.[3] : J. Kruis, Domain Decomposition Methods for Distributed Computing. Kippen, Stirling, Scotland, Saxe-Coburg Publication, 2006.[4] : F. BREZZI, J. RAPPAZ et P. RAVIART : Finite-dimensionalapproximation of nonlinear problems. parti : Branches of non singular solutions. Numer.Math, (36):125, 1980.

75

Gauss-Newton approach for large-scaleRiccati equations

Khalide Jbilou1, Marcos Raydan2

1Laboratoire LMPA, 50 rue F. Buisson, ULCO, 62228, Calais, France (jbilou@univ-littoral.fr)2Centro de Matematica e Aplicacoes, FCT, UNL, Caparica 2829-516, Portugal (m.raydan@fct.unl.pt)

AbstractWe describe a nonlinear least-squares approach for solving large-scale continuous-time algebraicRiccati equations with a low-rank right hand side. We project the problem onto a sequence ofnested Krylov-type low-dimensional subspaces. Then, instead of forcing the orthogonality con-ditions related to the Galerkin strategy, we minimize the residual at each subspace to get a lowdimensional nonlinear matrix least-squares problem that will be solved to obtain an approximatefactorized solution of the initial Riccati equation. To solve the low-order minimization problems,we propose a globalized Gauss-Newton matrix approach that exhibits a smooth convergence be-havior, and that guarantees global convergence to stationary points. This procedure involves thesolution of a symmetric matrix problem per iteration that will be solved by preconditioned iterativematrix methods. To illustrate the behavior of the combined scheme, we present numerical resultson some test problems.

76

Treatment of breakdown and near-breakdownin Lanczos-type algorithms (the MRZ gang!)

Michela Redivo-ZagliaDepartment of Mathematics “Tullio Levi-Civita”University of Padua, ItalyEmail: michela.redivozaglia@unipd.it

AbstractLanczos-type algorithms are recursive procedures for implementing methods for solving systems oflinear equations. They are based on Krylov subspaces. In these algorithms, when the denominatorof one of the coefficients vanishes a breakdown occurs and the procedure has to be stopped. Whena denominator is close to zero, because of cancelation, we have a near-breakdown which results inan important propagation of rounding errors.

This talk proposes a retrospective overview of some relevant results to cure these phenomena,obtained from 1989 by the MRZ gang (Claude Brezinski, Hassane Sadok and me). The cases of theBiCG, the BiCGStab, and the CGS are also considered.

77

Numerical Methods for Ill-Posed Problems witha Sparse and Nonnegative Solution

Lothar Reichel11Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA

AbstractMany problems in science and engineering require the determination of the cause of an observedeffect. After possibly linearization and discretization, these problems often can be formulated as alinear system of algebraic equations or a linear least-squares problem with a matrix, whose singularvalues “cluster” at the origin. These problems therefore are numerically underdetermined and thismakes their solution very sensitive to errors in the data (right-hand side) and to round-off errorsintroduced during the computations. This talk presents an overview of solution methods for thistype of problems. The quality of the computed solutions often can be improved by requiring that thecomputed solution satisfies constraints that the desired solution of the error-free problem is knownto possess, such as nonnegativity or sparsity. Several approaches for computing sparse nonnegativesolutions will be described.

78

A nonlinear optimization method applied to thehydraulic conductivity identification in uncon-fined aquifers

Aya Mourad1, Carole Rosier1

1Univ. Littoral Cote d’Opale, LMPA, F- 62228 Calais, France

AbstractIn this article, we study the identification, from observations or field measurements, of the hydraulicconductivity for the saltwater intrusion problem in an unconfined aquifer. The involved model con-sists in a cross-diffusion system describing the evolutions of two interfaces: one between freshwaterand saltwater and the other one between the saturated and unsaturated zones of the aquifer. The in-verse problem is formulated as an optimization problem where the cost function is a least squarefunctional measuring the discrepancy between experimental interfaces depths and those providedby the model. Considering the exact problem as a constraint for the optimization problem and in-troducing the Lagrangian associated with the cost function, we prove that the optimality systemhas at least one solution. Moreover, we establish the first order necessary optimality conditions. Anumerical method is implemented to solve this identification problem. Some numerical results arepresented to illustrate the ability of the method to determine the unknown parameters.

References[1] Mourad A., Rosier C, A nonlinear optimization method applied to the hydraulic conductivity identification

in unconfined aquifers, JOTA, vol. 183 (2) (2019), 705-730.

79

Estimates of the determinant of a perturbedidentity matrix

Siegfried M. Rump1

1Institute for Reliable Computing,Hamburg University of Technology,Am Schwarzenberg-Campus 3, 21073 Hamburg, Germany,andVisiting Professor at Waseda University,Faculty of Science and Engineering,3–4–1 Okubo, Shinjuku-ku, Tokyo 169–8555, Japanrump@tuhh.de

AbstractRecently Brent et al. presented new estimates for the determinant of a real perturbation I + E ofthe identity matrix. They give a lower and an upper bound depending on the maximum absolutevalue of the diagonal and the off-diagonal elements of E, and show that either bound is sharp. Theirbounds will always include 1, and the difference of the bounds is at least trace(E).

We present a lower and an upper bound depending on the trace and Frobenius norm ε := ‖E‖F ofthe (real or complex) perturbationE, where the difference of the bounds is not larger than ε3+O(ε4)provided that ε < 1. Moreover, we prove a bound on the relative error between det(I + E) andexp(trace(E)) of order ε2. The computing time for the bounds of an n× n matrix in either case isO(n2) operations, the same as computing the Frobenius norm.

References[1] R. BRENT, J. OSBORN, AND W. SMITH, Bounds on determinants of perturbed diagonal matrices.

arXiv:1401.7084v7, 18pp., 2014.

[2] , Note on best possible bounds for determinants of matrices close to the identity matrix, LAA, 466(2015), pp. 21–26.

[3] S. RUMP, The determinant of a perturbed identity matrix, Linear Algebra and its Applications (LAA), 558(2018), pp. 101–107.

[4] S.M. Rump and P. Batra. Addendum to the determinant of a perturbed identity matrix. Linear Algebra andits Applications (LAA), 565:309–312, 2019.

80

A model for the behaviour of an innovativeentrepreneur.

Philippe Ryckelynck1

1LMPA, F-62100 Calais (CNRS, FR 2956) France.Universite du Littoral Cote d’Opale.E-mail address: philippe.ryckelynck@lmpa.univ-littoral.fr

AbstractWe outline, in a first part, a mathematical model of decision-making for entrepreneur choosing in-novative products. Decision random variables include prices, quantities or stocks, and give rise toa huge random vector obeying to an unknown probabilistic law. The objective of the entrepreneuris quadratic with respect to the decision variables, and more accurately, w.r.t. first and second or-der momenta of them. In some respect, this model is analogousbut far more complicated thanthe Markowitz’s model for investors. In a second part, the first order necessary equations are de-composed in various sets which are dependent but may be solved through an iterative procedure,help to a generalized eigenvalue problem. The sufficient second-order condition for an optimumto occur is also analyzed and it depends on a complicate huge block matrix involving covariancesof the decision random variables. In a third part of the paperwe seek for relevance of the modelbased on actual datas. We choose datas to simulate a complex market of thirty products which areinnovative as well as heavily correlated inside groups of similar and highly correlated products, thedifferent groups being uncorrelated. Choices done by the entrepreneur may help solving equationand/or perturb this stage, and we focus on the stability of the dynamics of the model with respectto those perturbations. In a last part, we examine the numerical problems appearing when iteratingthe previous model, especially those linked to the stability of the linear and quasi singular matrixsystem, and those linked to information obtained on random vectors at early stages of iteration.

References[1] Bouchaud, J.P., Potters M.,The Theory of Financial Risks; From Statistical Physics to Risk management,

Cambridge University Press, 2000.

[2] Korn, R., Optimal Portfolio,World Scientific Publishing, Co. Pte. Ltd, Singapore, 1997.

[3] John M Mulvey, William R Pauling, and Ronald E Madey. Advantages of multiperiod portfolio models.TheJournal of Portfolio Management, 29 (2): 35–45, 2003.

81

Extended Krylov methods for approximate ofthe matrix exponential

El M. Sadek1, A. H. Bentbib2, K. Jbilou3

1Laboratory labSIPE, ENSA, d’EL Jadida, University Chouaib Doukkali, Morocco (sadek.maths@gmail.com)2Laboratory LAMAI, University Cadi Ayyad, Marrakesh, Morocco (a.bentbib@uca.ac.ma)3Laboratory LMPA, 50 Rue F. Buisson, ULCO Calais cedex, France (Khalide.Jbilou@univ-littoral.fr )

AbstractIn this talk, we propose an extended block Arnoldi and extended global Arnoldi methods to ap-proximate the matrix exponential operation exp(A)V . The computation of exp(A)V where A is areal matrix and V a rectangular matrix, play a fundamental role in many problems. For example,when solving systems of ordinary differential equations (ODEs) or time-dependent partial differen-tial equations (PDEs). The proposed methods is based on projection onto an extended block Krylovsubspace or extended global Krylov subspace using Galerkin approach. Theoretical results andnumerical tests are reported to show the effectiveness of the proposed approaches.

References[1] S. Agoujil, A. H. Bentbib, K. Jbilou, El. M Sadek, A minimal residual norm method for large-scale Sylvester

matrix equations. Elect. Trans. Numer. Anal. 2014, 43, pp. 45–59.

[2] A. Archid and A. H. Bentbib, Approximation of the matrix exponential operator by a structure-preservingblock Arnoldi-type method. Applied Numerical Mathematics, 2013.

[3] A.H. Bentbib, K. Jbilou and El. M. Sadek, On Some Extended Block Krylov Based Methods for Large ScaleNonsymmetric Stein Matrix Equations. Mathematics V. 5, 2017.

[4] T. Penzl, LYAPACK A MATLAB toolbox for Large Lyapunov and Riccati Equations, Model ReductionProblems, and Linear-quadratic Optimal Control Problems. http://www.tu-chemintz.de/sfb393/lyapack.

[5] V. Druskin and L. Knizhnerman. Extended Krylov subspaces: approximation of the matrix square root andrelated functions. SIAM J. Matrix Anal. Appl. 19 (3) (1998), pp. 755–771.

[6] Y. Saad, Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J.Numer. Anal. 29 (1992), pp. 209–228.

82

Convergence analysis of some preconditionedBlock Krylov subspace methods

Hassane Sadok1

1LMPA, 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 linear sys-tems of equations with one or several right hand sides or for solving nonsymmetric eigenvalueproblems.

In this talk, we will describe some convergence results for the GMRES method of Saad andSchultz, for solving linear system. We use techniques from constrained optimization rather thansolving the classical min-max problem (problem in polynomial approximation theory) .

The main class of symmetric and indefinite problems come from minimization of quadraticfunctionnals subject fo linear constraints. We obtain the called saddle point systems:(

A BT

B O

)︸ ︷︷ ︸

A

(XY

)︸ ︷︷ ︸X

=

(FG

)︸ ︷︷ ︸B

, (1)

where A ∈ Rn×n is a symmetric matrix and B ∈ Rm×n

Different challenges arise for sadle point problem (1) with multiple right hand side.We propose a preconditioned global approach as a new strategy to solve (1). The preconditioner

is obtained by replacing a (2, 2) block in the original saddle-point matrix A by another well cho-sen block. We apply the global GMRES method to solve this new problem with several right-handsides, and give some convergence results for the global GMRES method. Moreover, we analyzethe eigenvalue-distribution and the eigenvectors of the proposed preconditioner when the first blockis positive definite. We also compare different preconditioned global Krylov subspace algorithms(CG, MINRES, FGMRES, GMRES) with preconditioned block (CG, GMRES) algorithms. Nu-merical results show that our preconditioned global GMRES method is competitive with other pre-conditioned global subspace Krylov and preconditioned block Krylov subspace methods for solvingsaddle point problems with several right hand sides.

This is joint work with with A. Badahmane and A.H. Bentbib from university of ULCO and UCArespectively).

83

On the kernel of the vector Epsilon-algorithm

Claude Brezinski1, Michela Redivo-Zaglia2, Ahmed Salam3

1Universite de Lille, CNRS, UMR 8524 - Laboratoire Paul Painleve, F-59000 Lille, France.E-mail: Claude.Brezinski@univ-lille.fr.2 Universita degli Studi di Padova, Dipartimento di Matematica “Tullio Levi-Civita”, Via Trieste 63, 35121-Padova, Italy.E-mail: Michela.RedivoZaglia@unipd.it3Universite du Littoral, LMPA, CS 80699. 50 F. Buisson, 62228 Calais, France.E-mail: Ahmed.Salam@univ-littoral.fr.

AbstractThe vector Epsilon-algorithm introduced by P. Wynn is a powerful method for accelarating theconvergence of vector sequences. The algorithm is an extension of the scalar Epsilon algorithm,obtained by replacing the inverse of a real number in the scalar case, by the psoeudo-inverse of avector in the vector case. The kernel of the vector Epsilon is the set of sequences transformed bythe algorithm to stationnary sequences (the constant is a limit or anti-limit of the sequence). It iswell-known that the kernel contains sequences satisfying some difference equations. In this paper,we show that this condition is only sufficient and that the kernel contains other kind of sequnces. Weshow also how the use of Clifford algebra, can be very helpful for understanding and deriving newresults of the algorithm. In particular, we give necessary and sufficient condition for caracterizingthe kernel. Examples for illustrations as well as geometrical interpretations are given.

References[1] C. Brezinski, A. Salam, Matrix and vector sequence transformations revisited, Proc. Edinburgh Math. Soc.,

38 (1995) 495-510.

[2] A. Salam, An algebraic approach to the vector ε-algorithm, Numer. Algorithms, 11 (1996) 327- 337.

[3] P. Wynn, Acceleration techniques for iterated vector and matrix problems, Math. Comp., 16 (1962) 301322.

84

Variable elimination algorithm:an updated version

Linda Smail11Department of Mathematics and Statistics, Zayed University, Dubai, United Arab Emirates

AbstractGiven a Bayesian network [1] relative to a set of discrete random variables (Xi)i∈I , we are in-terested in computing the probability distribution P (A|B), where A and B are two subsets of I .Using Bayes’ theorem, we reduce the computation of conditional probabilities to the ratio of twojoint probabilities, then we compute each of the marginal probabilities apart. This is essentially anoptimization calculation problem, as it becomes increasingly heavy following the complexity of thegraph relative to both the number of variables and the number of values taken by these variables.One approach for eliminating variables is considered here, the Variable Elimination algorithm ofDechter [2], which appeared for the first time in Zhang and Poole [3]. So called because it elimi-nates by marginalization variables one after the other. The general idea of the Variable Eliminationalgorithm is to manage the succession of summations on all random variables out of the target., inother words, to sum over a set of variables from a list of factors one by one; an ordering of thesevariables is required as an input and is called an elimination ordering. The computation depends onthe order of elimination as different elimination orders produce different factors. In this work, Wepropose a variation of the Variable Elimination algorithm that will make intermediate computationswritten as conditional probabilities and not simple potentials. This has an advantage in storing thejoint probability as a product of conditions probabilities thus less constraining.

References[1] F. V. Jensen. Bayesian Networks and Decision Graphs. Springer, 2001.

[2] R. Dechter. Bucket Elimination: A Unifying Framework for Probabilistic Inference. In UAI, eds. E. Horvitzand F. Jensen, 211-219. San Francisco, CA, Morgan Kaufmann. 1996.

[3] N. L. Zhang and D. Poole. Exploiting causal independence in Bayesian network inference. Journal of Artifi-cial Intelligence Research, 5:301-328, 1996.

85

Asynchronous parallel subdomain methods cou-pled with Krylov methods

Thierry Garcia1, Pierre Spiteri2, Lilia Ziane Khodja3, Raphael Couturier4

1Universite de Toulouse, IRIT - INP, Toulouse, France2Universite de Toulouse, IRIT - INP-ENSEEIHT, Toulouse, France3ANEO, 122 Avenue du General Leclerc, Boulogne-Billancourt, France4FEMTO – ST Institute, University of Bourgogne Franche Comte, Belfort, France

AbstractThe present study is related to the analysis and application of mixed subdomain methods coupledwith Krylov methods to solve pseudo - linear stationary or evolution problems. The consideredproblems are defined as an affine application AU −F perturbed by an increasing diagonal operatorwhen the solution is not constrained and, when the solution is under constraints, the affine map-ping AU − F is perturbed by a multivalued monotone maximal operator. In the following A hasthe property of being a large M-matrix, F is a vector and U is the unknown vector. Note that thistype of problem occurs when solving elliptic, parabolic or hyperbolic second order boundary-valueproblems. Note also that the M-matrix property is well verified after classical spatial discretization.In the case of singlevalued problem, corresponding to the case of perturbation of the affine mappingby an increasing mapping, the problem will be solved by a specific method corresponding to a locallinearization by the iterative Newton method. In the case of problem under constraints we have toproject each new update on the convex defining the constraints. Thus, in each case, large sparse lin-ear system must be solved. This linear system is then associated with a fixed point problem and weintend to solve it by asynchronous parallel subdomain methos. Taking into account the propertiesof the matrix A and the operators monotony properties, it is proven that fixed point applications arecontractive with respect to an uniform weighted norm, which ensures, on the one hand, the existenceand uniqueness of the solution of the algebraic system to be solved and, on the other hand, the con-vergence of asynchronous parallel iterations towards the solution of the problem. Moreover, in thetheoretical context, due to the properties of the operators involved, the behavior of the parallel iter-ative algorithms can be analyzed by using the discrete maximum principle ; so, starting the iterativeprocess by an appropriate initial guess, the convergence of the iterate vectors is monotone. In fact,the considered algorithms correspond, on one hand, to sub-domain methods without overlap, or, onthe other hand, to sub-domain methods with overlap such as Schwarz’s alternating method. Thesekind of methods are then applied to solve the target considered problems. From an implementationpoint of view, we consider a mixed algorithm constituted by a two level iterative algorithm, wherethe outer iteration is constituted by parallel asynchronous or synchronous subdomain method andthe inner iteration is the Krylov based iterative method, typically the GMRES method. As applica-tions, we consider a diffusion convection problem perturbed by an increasing diagonal operator andalso submitted to constraints the problem being solved, on one hand, by a mixed Newton - subdo-main method and, on the other hand, by a projected relaxation method. Results of synchronous andasynchronous parallel experiments will be presented.

86

Finite volume simulation of a sharp-diffuse modelfor seawater intrusion in coastal aquifers

B. Amaziane1, M. El Ossmani2, K. Talali21CNRS / Univ Pau & Pays Adour /E2S UPPA, LMAP, Fdration IPRA, UMR5142, 64000, Pau, France2University Moulay Ismail, EMMACS-ENSAM, Marjane 2, 50000 Meknes, Morocco.

AbstractIn this talk, we present a finite volume scheme and numerical simulations using a sharp-diffuseinterface approach [1] for modeling a seawater intrusion problem in coastal aquifers. This process isformulated by a coupled system of two nonlinear parabolic partial differential equations describingtwo immiscible phase seawater/freshwater flow tacking into account the width of tran- sition zones.A fully-coupled, fully implicit cell-centred finite volume scheme is developed to descretize thecoupled sysytem. We use an implicit Euler scheme for the time discretization, which allows us totake large time steps and thus reduce CPU time.

In this context, we have developed and integrated a two phase diffuse module named 2pdiff-SWIin the DuMuX framework [2]. To ensure the validity of the implemented module, we proceeded tothe study of the numerical convergence of the scheme. The efficiency is investigated through 2Dsimulations with different grid resolutions. Furthermore, we have compared our numerical simula-tions with the results presented in [1]. The obtained results are promising and showed good agree-ment with those in [1]. In addition, we propose to compare the numerical results of the 2pdiff-SWImodule with those provided by the traditional three-dimensional model for miscible displacements.The numerical results have shown that this approach yields physically realistic and performant re-sults. Afterward, we apply the method to a real-scale test case with hydrogeological data. Thenumerical simulations for the long-term demonstrate the applicability of our developed approachin highly heterogeneous coastal aquifers. Future work will focus on the numerical analysis of thescheme.

References[1] C. Choquet, M. Diedhiou, C. Rosier (2016), Derivation of a sharp-diffuse interfaces model for seawater

intrusion in a free aquifer; Numerical simulations, SIAM J. Appl. Math, volume 76, pp. 138-158.

[2] DuMu X : Dune for multi- Phases, Component, Scale, Physics, ... flow and transport in porous media,http://www.dumux.org/, last accessed December 14, 2019.

87

Weak solutions of some quasilinear singularelliptic equations with data measures

L.Taourirte1, F. Aqel2, N.Alaa1

1Laboratory LAMAI, University Cadi Ayyad, Marrakesh, Morocco2Laboratory IR2M, University Hassan Ier, Settat, Morocco

AbstractThis work describes some results concerning existence and uniqueness of weak solutions for equa-tions of the form

−4u =a(x)

uγ+ b(x)| ∇u |p + f in Ω,

u > 0 in Ω,u = 0 on ∂Ω.

(1)

where Ω is a bounded open set in RN , λ and γ are non-negative real numbers, p ≥ 1, |.| denotes theRN -euclidean norm and f : Ω −→ [0,∞) a non-negative integrable function, or more generally agiven finite non-negative measure on Ω.We first state that, if p > 1 and a, b two non-negative functions in L1(Ω), then two necessaryconditions should be satisfied, f must be small enough and does not charge the set ofW 1,p′-capacityzeros.Next, if p = 1 by using the isoperimetric inequatlity, we prove an existence and uniqueness resultfor any given bounded measure f such that if 0 < γ < 1 the solution belongs to W 1,q

0 (Ω), 1 ≤ q <NN−1 and if γ > 1 the solution is only in W 1,q

loc (Ω).

References[1] Alaa, Nour Eddine, and Michel Pierre. ”Weak solutions of some quasilinear elliptic equations with data

measures.” SIAM journal on mathematical analysis 24.1 (1993): 23-35.

[2] Canino, Annamaria, Berardino Sciunzi, and Alessandro Trombetta. ”Existence and uniqueness for p-Laplaceequations involving singular nonlinearities.” Nonlinear Differential Equations and Applications NoDEA 23.2(2016): 8.

88

Efficient matrix computations for the minimumcovariance determinant estimator

Jurjen Duintjer Tebbens1, Jan Kalina2

1Institute of Computer Science, Czech Academy of Sciences, Pod Vodarenskou vezı 2, 18 207 Praha 8 - Liben and CharlesUniversity, Faculty of Pharmacy, Heyrovskeho 1203, 500 05 Hradec Kralove, Czech Republic. (duintjertebbens@cs.cas.cz)2Institute of Computer Science, Czech Academy of Sciences, Pod Vodarenskou vezı 2, 18 207 Praha 8 - Liben, Czech Republic.(kalina@cs.cas.cz)

AbstractIn statistics, the term robustness is mostly used to indicate robustness with regards to outliers inthe observed data. In the univariate case, a single outlier might be relatively easily detected bymeasuring with a norm called Mahalanobis distance. This distance is in fact the energy norm for theinverse of the symmetric positive definite covariance matrix S and scales the p-dimensional spacesuch that the variabilities of the individual properties are normalized. In a multivariate situation,with multiple outliers, the Mahalanobis distance itself is too strongly influenced by the outliers togive a reliable tool for their detection.

If the aim is to estimate the location and scatter by robust estimators (i.e. to compute a robustmean vector and robust covariance matrix), one can compute the location and scatter for a subsetof the observations which hopefully does not contain outliers. Assume we have n observationsxi ∈ Rp of p variables, given by the data matrix X = [x1, . . . , xn]

T ∈ Rn×p and look for a subsetof size h of the indices 1, 2, . . . , n, where [(n + p + 1)/2] ≤ h ≤ n, such that no index in thesubset corresponds to an outlier. A criterion to base the search of the subset on and that has beenproved to lead to highly robust estimators of location and scatter is to minimize the determinantof the covariance matrix [1]. The computation of the corresponding Minimum Covariance Deter-minant Estimator requires minimization over all h-subsets of 1, 2, . . . , n, thus has combinatorialcomplexity and becomes infeasible for very moderate numbers of observations n. In the widelyused fast MCD [2] algorithm the computational costs are reduced to O(np2) by using a compu-ational procedure called C-step, but the algorithm does not find in general the global miminumdeterminant. Our contribution consists of two cheap, O(np) permutations that can be added to theC-step to improve its power with regards to minimizing the determinant. Their efficient, low-costimplementations rely on sophisticated matrix computations as used in numerical linear algebra.

References[1] Grubel, R. A minimal characterization of the covariance matrix Metrika, vol. 35, 49–52, 1988.

[2] Rousseeuw, P. and Van Driessen, K. A fast algorithm for the minimum covariance determinant estimatorTechnometrics, vol. 34(3), 212–223, 1999.

89

Acceleration of the convergence of the Asyn-chronous RAS method

D. Tromeur-Dervout11Universite de Lyon, Universite Lyon 1, CNRS, Institut Camille Jordan UMR5208

AbstractThe today and upcoming high performance computers with several thousand of cores and with moreand more complex hierarchical communication networks lead to the need of developping algorithmswithout global communication (such as reduction operations of scalar product in GMRES to avoidperformance bottleneck) and if possible fault tolerant. In this context the asynchronous RestrictedAdditive Schwarz (RAS) in which some boundary conditions at artificial interfaces generated by thedomain decomposition can to not be updated or totally updated for some iteration, is becoming at-tractive. This talk focuses on the acceleration of the convergence of asynchronous RAS by Aitkensacceleration of convergence which is based on the property of having an error operator indepen-dent of iteration. Nevertheless, this property is lost with the asynchronous iterations. We develop amathematical model of the Asynchronous RAS allowing to set the percentage of the number of therandomly choosen local artificial interfaces where boundary conditions are not updated. Then weshow how this ratio deteriorate the convergence of the Asynchronous RAS and how some regular-ization techniques on the traces of the iterative solutions at artificial interfaces allow to acceleratethe convergence to the searched solution by the Aitkens acceleration.

References[1] F. Magoules and D. B. Szyld and C. Venet, Asynchronous Optimized Schwarz Methods with and without

Overlap,Numerische Mathematik, 137:199–227, 2017

[2] L. Berenguer and T. Dufaud and D. Tromeur-Dervout, Aitken’s acceleration of the Schwarz process usingsingular value decomposition for heterogeneous 3D groundwater flow problems,Computer & Fluids,80:320–326,2013.

90

Linearizations of polynomial and rational matrices

Paul Van Dooren1

1Catholic University of Louvain, Louvain-la-Neuve, Belgium.

AbstractWe show that the problem of linearizations of polynomial and rational matrices is closely related tothe polynomial matrix quadruples introduced by Rosenbrock in the seventies to represent rationaltransfer functions of dynamical systems. We also recall the concepts of irreducible and stronglyirreducible quadruples which were introduced in the eighties, and show how they relate to the lin-earizations that are more common in the numerical linear algebra community. We then show that thefamily of strong linearizations of matrix polynomials, called “block Kronecker pencils”, as well astheir extension to rational eigenvalue problems, nicely fit in that general framework. In the generalrational case the linearization are of the form

S(λ) :=

M(λ) KT2 C KT

2 (λ)

BK1 A− λI 0K1(λ) 0 0

.The novelty of these block Kronecker pencils is that they can be proven to be backward stable in astructured sense, for the polynomial matrix case as well as for the rational matrix case.

This is based on joint work with F. Dopico (UC3M), P. Lawrence (KULeuven), J. Perez (UMontana)and M.C. Quintana (UC3M).

91

Scattering of thermoelastic wavesby a multi-layered obstacle

E. S. Athanasiadou1, V. Sevroglou2, S. Zoi1

1National and Kapodistrian University of Athens, Department of Mathematics2University of Piraeus, Department of Statistics & Insurance Science3National and Kapodistrian University of Athens, Department of Mathematics

AbstractIn this work the scattering of time-harmonic plane thermoelastic waves by a multi-layered thermoe-lastic body is considered. The direct scattering problem isformulated in a unified four-dimensionalform. An integral representation of the thermoelastic scattered field in which the physical and geo-metrical parameters of the layers have been incorporated isderived. Using this integral representa-tion and applying the Kupradze radiation conditions as wellas a Rellich’s type lemma, the unique-ness of solution is proved. Moreover, using combinations ofsingle and double layer thermoelasticpotentials and applying the transmission conditions, the problem is transformed into a system ofintegral equations which is written in a matrix form with unknowns the densities of the potentials.Studying the compactness of the corresponding matrix operator and using the Riesz-Fredholm’stheory, the existence of solution is established. For the special case of a one-layered obstacle anapproximate solution is constructed in terms of the thermoelastic fundamental solutions.

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