Solving large sparse linear systems with a variable s-step GMRES preconditioned by DD

• Main Authors: Imberti, David, Erhel, Jocelyne
• Other Authors: Fluid Flow Analysis, Description and Control from Image Sequences (FLUMINANCE), Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), European Project: 610741,EC:FP7:ICT,FP7-ICT-2013-10,EXA2CT(2013)
• Format: Conference Object
• Language: English
• Published: HAL CCSD 2017
• Subjects: [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]; [INFO.INFO-DC]Computer Science [cs]/Distributed; Parallel; and Cluster Computing [cs.DC]; Longyearbyen; Norway
• Online Access: https://hal.inria.fr/hal-01528636; https://hal.inria.fr/hal-01528636/document; https://hal.inria.fr/hal-01528636/file/fibgmres.pdf

International audience Krylov methods such as GMRES are efficient iterative methods to solve large sparse linear systems, with only a few key kernel operations: the matrix-vector product, solving a preconditioning system, and building the orthonormal Krylov basis. Domain Decomposition methods allow parallel computations for both the matrix-vector products and preconditioning by using a Schwarz approach combined with deflation (similar to a coarse-grid correction). However, building the orthonormal Krylov basis involves scalar products, which in turn have a communication overhead. In order to avoid this communication, it is possible to build the basis by a block of vectors at a time, sometimes at the price of a loss of orthogonality. We define a sequence of such blocks with a variable size. We show through some theoretical results and some numerical experiments that increasing the block size as a Fibonacci sequence improves stability and convergence.