Journée DDM - parallélisation du jeudi 11 juin 2015

Preconditioning iterative methods for the solution of linear systems goes in principle back to Jacobi (1845), just after the invention of the classical iterative method by Gauss (1823). Today, preconditioning of linear systems for their solution by Krylov methods has become a major field of research, and there are two main approaches for constructing preconditioners : either one has very good intuition and can propose directly a preconditioner which leads to a favorable spectrum of the preconditioned system, or one uses the splitting matrix of an effective stationary iterative method like multigrid or domain decomposition as the preconditioner.

Much less is known about the preconditioning of non-linear systems of equations. The standard iterative solver in that case is Newton’s method (1671) or a variant thereof, but what would it mean to precondition the non-linear problem ? An important contribution in this field is ASPIN (Additive Schwarz Preconditioned Inexact Newton) by Cai and Keyes (2002), where the authors use their intuition about domain decomposition methods to propose a transformation of the non-linear equations before solving them by an inexact Newton method. Using the relation between stationary iterative methods and preconditioning for linear systems, we show in this presentation how one can systematically obtain a non-linear preconditioner from classical fixed point iterations, and present as an example a new two level non-linear preconditioner called RASPEN (Restricted Additive Schwarz Preconditioned Exact Newton).

L’intérêt pour la simulation numérique est grandissant et les phénomènes physiques considérés sont de plus en plus complexes. Pour augmenter la précision de la solution tout en ayant un coût de calcul raisonnable, il est utile de paralléliser l’algorithme : on décompose le domaine de calcul en $n$ sous-domaines ; la solution globale est alors obtenue par itération en échangeant des informations à l’interface entre les sous-domaines.
Ici on propose d’utiliser ce découpage pour adapter le modèle physique à la région de l’espace (on résout une équation différente sur chaque sous-domaine). Pour le cas d’un couplage de modèles visqueux/non visqueux, on fera un bref historique des méthodes existantes et on proposera un premier critère de comparaison. On introduira ensuite une nouvelle méthode plus performante. Nous donnerons une analyse d’erreur et montrerons des simulations numériques.
Ce travail est un travail commun avec M.J. Gander et L. Halpern.

The aim of this talk is to present recent advances in the construction of robust coarse spaces for overlapping and nonoverlapping methods as well as their implementation inside a C++ open-source framework (https://github.com/hpddm). A broad spectrum of applications will be covered, ranging from Poisson to Helmholtz equation, and including incompressible linear elasticity and Stokes equation. Numerical results with thousands of processes will be provided clearly showing the effectiveness and the robustness of the proposed approach. It will also be shown how HPDDM can be efficiently used to outperform state-of-the-art algebraic multigrid solvers.