Une classe de préconditionneurs efficaces construits localement basés sur les espaces grossiers

In this paper we present a class of robust and fully algebraic two-level preconditionersfor SPD matrices. We introduce the notion of algebraic local SPSD splitting of an SPD matrixand we give a characterization of this splitting. This splitting leads to constructalgebraically and locally a class of efficient coarse spaces which bound the spectral condition number of thepreconditioned system by a number defined a priori. We also introduce the τ-filtering subspace.This concept helps compare the dimension minimality of coarse spaces. Some PDEs-dependantpreconditioners correspond to a special case. The examples of the algebraic coarse spaces in thispaper are not practical due to expensive construction. We propose a heuristic approximationthat is not costly. Numerical experiments illustrate the efficiency of the proposed method.Ce papier présente une classe de deux-niveaux préconditioneurs totalement algébriqueet efficaces pour les matrices SPD. Nous introduisons la notion de la séparation local et algébriqueSPSD d’une matrice SPD et nous characterisons tout les séparations possibles. Cette séparationaide à construire algéebriquement et localement une classe d’espaces grossiers efficaces qui bornentle conditionement spectral du système préconditioné par un numbre défini a priori. Nous introdui-sons ainsi les espaces de τ-filtering. Ces derniers permettent à comparer les dimensions des espacesgrossiers. Certains préconditioneurs qui dépendendent de l’EDP font l’objet d’un cas particulierde la classe intorduite. Les exemples des espace grossiers algébriques dans ce papier ne sont paspratiques suite à la construction chère de la séparation algébrique. Nous proposon une approxima-tion heuristique qui n’est pas chèr. Les résultats numériques montrent l’efficacité de la méthodeproposée

An extended Krylov-like method for the solution of multi-linear systems

In the present work, numerical methods for the solution of multi-linear system are presented. Most large-scale multi-linear solvers rely on either the alternating leastsquares or low rank Krylov methods. The approach we use to develop our methods lies somehow in between and can be considered as a generalisation of an alternated direction method. Given the multi-linear operator in the form of a sum of Kronecker product of matrices, we solve at each iteration a linear system for each summand. The approximate solution is then defined to be the best linear combination of these solutions, as well as the previous solution and the residual. Some convergence results are proved. Numerical experiments on two problems arising from parametric PDEs show the effectiveness of the proposed method

Two-level Nystrom-Schur preconditioner for sparse symmetric positive definite matrices

Randomized methods are becoming increasingly popular in numerical linear algebra. However, few attempts have been made to use them in developing preconditioners. Our interest lies in solving large-scale sparse symmetric positive definite linear systems of equations where the system matrix is preordered to doubly bordered block diagonal form (for example, using a nested dissection ordering). We investigate the use of randomized methods to construct high quality preconditioners. In particular, we propose a new and efficient approach that employs Nystrom's method for computing low rank approximations to develop robust algebraic two-level preconditioners. Construction of the new preconditioners involves iteratively solving a smaller but denser symmetric positive definite Schur complement system with multiple right-hand sides. Numerical experiments on problems coming from a range of application areas demonstrate that this inner system can be solved cheaply using block conjugate gradients and that using a large convergence tolerance to limit the cost does not adversely affect the quality of the resulting Nystrm-Schur two-level preconditioner

Enlarged GMRES for reducing communication

We propose a variant of the GMRES method for solving linear systems of equations with one or multiple right-hand sides.Our method is based on the idea of the enlarged Krylov subspace to reduce communication.It can be interpreted as a block GMRES method.Hence, we are interested in detecting inexact breakdowns. We introduce a strategy to perform the test of detection.Furthermore, we propose an eigenvalues deflation technique aiming to have two benefits.The first advantage is to avoid the plateau of convergence after the end of a cycle in the restarted version.The second is to have a very fast convergence when solving the same system with different right-hand sides, each given at a different time (useful in the context of CPR preconditioner).With the same memory cost, we obtain a saving of up to 50 % in the number of iterations to reach convergence with respect to the original method

Enlarged GMRES for reducing communication

We propose a variant of the GMRES method for solving linear systems of equations with one or multiple right-hand sides.Our method is based on the idea of the enlarged Krylov subspace to reduce communication.It can be interpreted as a block GMRES method.Hence, we are interested in detecting inexact breakdowns. We introduce a strategy to perform the test of detection.Furthermore, we propose an eigenvalues deflation technique aiming to have two benefits.The first advantage is to avoid the plateau of convergence after the end of a cycle in the restarted version.The second is to have a very fast convergence when solving the same system with different right-hand sides, each given at a different time (useful in the context of CPR preconditioner).With the same memory cost, we obtain a saving of up to 50 % in the number of iterations to reach convergence with respect to the original method

Recyclage de Sous-Espaces de Krylov et Troncature de Sous-Espaces de Déflation pour Résoudre Séquence de Systèmes Linéaires

This paper presents deflation strategies related to recycling Krylov subspace methods for solving one or a sequence of linear systems of equations. Besides well-known strategies of deflation, Ritz- and harmonic Ritz-based deflation, we introduce an SVD-based deflation technique. We consider the recycling in two contexts, recycling the Krylov subspace between the cycles of restarts and recycling a deflation subspace when the matrix changes in a sequence of linear systems. Numerical experiments on real-life reservoir simulations demonstrate the impact of our proposed strategy.Ce papier présente plusieures stratégies de déflation liées aux méthodes de recyclage de sous-espaces de Krylov pour résoudre une séquence de systèmes linéaires. À côté de stratégies de déflation très connues qui sont basées sur la déflation des vecteurs de Ritz et Ritz harmonique, on introduit une technique de déflation basée sur la décomposition en valeurs singulières. On considère deux contextes du recyclage, le recyclage de l’espace de Krylov entre les cycles de resart et le recylcage de l’espaces de déflation quand la matrice change dans la séquence. L’efficacité de la méthode proposée est étudiée sur des séquence de systèmes linéaires issues de la modélisation de réservoirs

Tight Memory-Independent Parallel Matrix Multiplication Communication Lower Bounds

International audienceCommunication lower bounds have long been established for matrix multiplication algorithms. However, most methods of asymptotic analysis have either ignored the constant factors or not obtained the tightest possible values. Recent work has demonstrated that more careful analysis improves the best known constants for some classical matrix multiplication lower bounds and helps to identify more efficient algorithms that match the leading-order terms in the lower bounds exactly and improve practical performance. The main result of this work is the establishment of memory-independent communication lower bounds with tight constants for parallel matrix multiplication. Our constants improve on previous work in each of three cases that depend on the relative sizes of the aspect ratios of the matrices

Communication Lower Bounds and Optimal Algorithms for Multiple Tensor-Times-Matrix Computation

Multiple Tensor-Times-Matrix (Multi-TTM) is a key computation in algorithms for computing and operating with the Tucker tensor decomposition, which is frequently used in multidimensional data analysis. We establish communication lower bounds that determine how much data movement is required to perform the Multi-TTM computation in parallel. The crux of the proof relies on analytically solving a constrained, nonlinear optimization problem. We also present a parallel algorithm to perform this computation that organizes the processors into a logical grid with twice as many modes as the input tensor. We show that with correct choices of grid dimensions, the communication cost of the algorithm attains the lower bounds and is therefore communication optimal. Finally, we show that our algorithm can significantly reduce communication compared to the straightforward approach of expressing the computation as a sequence of tensor-times-matrix operations

Résolution de systèmes linéaires issus de la modélisation des réservoirs

This thesis presents a work on iterative methods for solving linear systems that aim at reducing the communication in parallel computing. The main type of linear systems in which we are interested arises from a real-life reservoir simulation. Both schemes, implicit and explicit, of modelling the system are taken into account. Three approaches are studied separately. We consider non-symmetric (resp. symmetric) linear systems. This corresponds to the explicit (resp. implicit) formulation of the model problem. We start by presenting an approach that adds multiple search directions per iteration rather than one as in the classic iterative methods. Then, we discuss different strategies of recycling search subspaces. These strategies reduce the global iteration count of a considerable factor during a sequence of linear systems. We review different existing strategies and present a new one. We discuss the parallel implementation of these methods using a low-level language. Numerical experiments for both sequential and parallel implementations are presented. We also consider the algebraic domain decomposition approach. In an algebraic framework, we study the two-level additive Schwarz preconditioner. We provide the algebraic explicit form of a class of local coarse spaces that bounds the spectral condition number of the preconditioned matrix by a number pre-defined.Cette thèse présente un travail sur les méthodes itératives pour résoudre des systèmes linéaires en réduisant les communications pendant les calculs parallèles. Principalement, on est intéressé par les systèmes linéaires qui proviennent des simulations de réservoirs. Trois approches, que l’on peut considérer comme indépendantes, sont présentées. Nous considérons les systèmes linéaires non-symétriques (resp. symétriques), cela correspond au schéma explicite (resp. implicite) du problème modèle. On commence par présenter une approche qui ajoute plusieurs directions de recherche à chaque itération au lieu d’une seule direction comme dans le cas des méthodes classiques. Ensuite, on considère les stratégies de recyclage des espaces de recherche. Ces stratégies réduisent, par un facteur considérable, le nombre d’itérations global pour résoudre une séquence de systèmes linéaires. On fait un rappel des stratégies existantes et l’on en présente une nouvelle. On introduit et détaille l’implémentation parallèle de ces méthodes en utilisant un langage bas niveau. On présente des résultats numériques séquentiels et parallèles. Finalement, on considère la méthode de décomposition de domaine algébrique. Dans un environnement algébrique, on étudie le préconditionneur de Schwarz additif à deux niveaux. On fournit la forme algébrique explicite d’une classe d’espaces grossiers locaux qui bornent le conditionnement par un nombre donné a priori

Solving linear systems arising from reservoirs modelling

Cette thèse présente un travail sur les méthodes itératives pour résoudre des systèmes linéaires en réduisant les communications pendant les calculs parallèles. Principalement, on est intéressé par les systèmes linéaires qui proviennent des simulations de réservoirs. Trois approches, que l’on peut considérer comme indépendantes, sont présentées. Nous considérons les systèmes linéaires non-symétriques (resp. symétriques), cela correspond au schéma explicite (resp. implicite) du problème modèle. On commence par présenter une approche qui ajoute plusieurs directions de recherche à chaque itération au lieu d’une seule direction comme dans le cas des méthodes classiques. Ensuite, on considère les stratégies de recyclage des espaces de recherche. Ces stratégies réduisent, par un facteur considérable, le nombre d’itérations global pour résoudre une séquence de systèmes linéaires. On fait un rappel des stratégies existantes et l’on en présente une nouvelle. On introduit et détaille l’implémentation parallèle de ces méthodes en utilisant un langage bas niveau. On présente des résultats numériques séquentiels et parallèles. Finalement, on considère la méthode de décomposition de domaine algébrique. Dans un environnement algébrique, on étudie le préconditionneur de Schwarz additif à deux niveaux. On fournit la forme algébrique explicite d’une classe d’espaces grossiers locaux qui bornent le conditionnement par un nombre donné a priori.This thesis presents a work on iterative methods for solving linear systems that aim at reducing the communication in parallel computing. The main type of linear systems in which we are interested arises from a real-life reservoir simulation. Both schemes, implicit and explicit, of modelling the system are taken into account. Three approaches are studied separately. We consider non-symmetric (resp. symmetric) linear systems. This corresponds to the explicit (resp. implicit) formulation of the model problem. We start by presenting an approach that adds multiple search directions per iteration rather than one as in the classic iterative methods. Then, we discuss different strategies of recycling search subspaces. These strategies reduce the global iteration count of a considerable factor during a sequence of linear systems. We review different existing strategies and present a new one. We discuss the parallel implementation of these methods using a low-level language. Numerical experiments for both sequential and parallel implementations are presented. We also consider the algebraic domain decomposition approach. In an algebraic framework, we study the two-level additive Schwarz preconditioner. We provide the algebraic explicit form of a class of local coarse spaces that bounds the spectral condition number of the preconditioned matrix by a number pre-defined