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

A Two Level Domain Decomposition Preconditionner Based on Local Dirichlet to Neumann Maps

Coarse grid correction is a key ingredient in order to have scalable domain decomposition methods. In this work we construct the coarse grid space using the low frequency modes of the subdomain DtN (Dirichlet-Neumann) maps, and apply the obtained two-level preconditioner to the linear system arising from an overlapping domain decomposition. Our method is suitable for the parallel implementation and its efficiency is demonstrated by numerical examples on problems with high heterogeneities

Overlapping Domain Decomposition Methods with FreeFem++

International audienceIn this note, the performances of a framework for two-level overlapping domain decomposition methods are assessed. Numerical experiments are run on Curie, a Tier-0 system for PRACE, for two second order elliptic PDE with highly heterogeneous coefficients: a scalar equation of diffusivity and the system of linear elasticity. Those experiments yield systems with up to ten billion unknowns in 2D and one billion unknowns in 3D, solved on few thousands cores

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

Scalable Domain Decomposition Preconditioners for Heterogeneous Elliptic Problems

International audienceDomain decomposition methods are, alongside multigrid methods, one of the dominant paradigms in contemporary large-scale partial differential equation simulation. In this paper, a lightweight implementation of a theoretically and numerically scalable preconditioner is presented in the context of overlapping methods. The performance of this work is assessed by numerical simulations executed on thousands of cores, for solving various highly heterogeneous elliptic problems in both 2D and 3D with billions of degrees of freedom. Such problems arise in computational science and engineering, in solid and fluid mechanics. While focusing on overlapping domain decomposition methods might seem too restrictive, it will be shown how this work can be applied to a variety of other methods, such as non-overlapping methods and abstract deflation based preconditioners. It is also presented how multilevel preconditioners can be used to avoid communication during an iterative process such as a Krylov method