Generalized Filtering Decomposition

This paper introduces a new preconditioning technique that is suitable for matrices arising from the discretization of a system of PDEs on unstructured grids. The preconditioner satisfies a so-called filtering property, which ensures that the input matrix is identical with the preconditioner on a given filtering vector. This vector is chosen to alleviate the effect of low frequency modes on convergence and so decrease or eliminate the plateau which is often observed in the convergence of iterative methods. In particular, the paper presents a general approach that allows to ensure that the filtering condition is satisfied in a matrix decomposition. The input matrix can have an arbitrary sparse structure. Hence, it can be reordered using nested dissection, to allow a parallel computation of the preconditioner and of the iterative process

Generalized Filtering Decomposition

Abstract: This paper introduces a new preconditioning technique that is suitable for matrices arising from the discretization of a system of PDEs on unstructured grids. The preconditioner satisfies a so-called filtering property, which ensures that the input matrix is identical with the preconditioner on a given filtering vector. This vector is chosen to alleviate the effect of low frequency modes on convergence and so decrease or eliminate the plateau which is often observed in the convergence of iterative methods. In particular, the paper presents a general approach that allows to ensure that the filtering condition is satisfied in a matrix decomposition. The input matrix can have an arbitrary sparse structure. Hence, it can be reordered using nested dissection, to allow a parallel computation of the preconditioner and of the iterative process. Key-words: linear solvers, Krylov subspace methods, preconditioning, filtering property, block incomplete decomposition * INRIA Saclay -Ile de France, Laboratoire de Recherche en Informatique Universite Paris-Sud 11, France ([email protected]). † Laboratoire J.L. Lions, CNRS UMR7598, Universite Paris 6, France ([email protected]). Décompositionà base de filtrage généralisée Résumé : Ce document présente une nouvelle technique de préconditionnement adapté pour les matrices issues de la discrétisation d'un système d'équations aux dérivées partielles sur des maillages non structurés. Le préconditionneur satisfait une propriété dite de filtrage, qui signifie que la matrice d'entrée est identique au préconditionneur pour un vecteur donné de filtrage. Le choix de ce vecteur permet d'atténuer l'effet des modes de basse fréquence sur la convergence et ainsi de diminuer ou d'éliminer le plateau qui est souvent observé dans la convergence des méthodes itératives. En particulier, le document présente une approche générale qui permet d'assurer que la propriété de filtrage est satisfaite lors d'une décomposition matricielle. La matrice d'entrée peut avoir une structure creuse arbitraire. Ainsi, elle peutêtre rénumérotée en utilisant la méthode de dissection emboîtée, afin de permettre un calcul parallèle du préconditionneur et du processus itératif

Robust algebraic Schur complement preconditioners based on low rank corrections

In this paper we introduce LORASC, a robust algebraic preconditioner for solving sparse linear systems of equations involving symmetric and positive definite matrices. The graph of the input matrix is partitioned by using k-way partitioning with vertex separators into N disjoint domains and a separator formed by the vertices connecting the N domains. The obtained permuted matrix has a block arrow structure. The preconditioner relies on the Cholesky factorization of the first N diagonal blocks and on approximating the Schur complement corresponding to the separator block. The approximation of the Schur complement involves the factorization of the last diagonal block and a low rank correction obtained by solving a generalized eigenvalue problem or a randomized algorithm. The preconditioner can be build and applied in parallel. Numerical results on a set of matrices arising from the discretization by the finite element method of linear elasticity models illustrate the robusteness and the efficiency of our preconditioner

Block filtering decomposition

International audienceThis paper introduces a new preconditioning technique that is suitable for matrices arising from the discretization of a system of PDEs on unstructured grids. The preconditioner satisfies a so-called filtering property, which ensures that the input matrix is identical with the preconditioner on a given filtering vector. This vector is chosen to alleviate the effect of low-frequency modes on convergence and so decrease or eliminate the plateau that is often observed in the convergence of iterative methods. In particular, the paper presents a general approach that allows to ensure that the filtering condition is satisfied in a matrix decomposition. The input matrix can have an arbitrary sparse structure. Hence, it can be reordered using nested dissection, to allow a parallel computation of the preconditioner and of the iterative process. We show the efficiency of our preconditioner through a set of numerical experiments on symmetric and nonsymmetric matrices