Algebraic domain decomposition methods for hybrid (iterative/direct) solvers

Louis Poirel 1
1 HiePACS - High-End Parallel Algorithms for Challenging Numerical Simulations
LaBRI - Laboratoire Bordelais de Recherche en Informatique, Inria Bordeaux - Sud-Ouest
Abstract : The solution of large linear problems is one of the most time consuming kernels in many numerical simulations. On the one hand, the computational linear algebra community has developed several high performance linear solvers that only require algebraic information (the matrix K and its associated right-hand side f) to compute the solution x such that Ku = f. On the other hand, the Domain Decomposition (DD) community has developed many efficient and robust methods in the last decades, that take into account the underlying partial differential equation and the geometry to accelerate the solution of such problems. In this thesis, both approaches are combined: an analysis of coarse correction for abstract Schwarz (aS) DD solvers is proposed, leading to a new methodology for building robust preconditioners for Symmetric Positive Definite (SPD) matrices based on an algebraic generalization of the Generalized Eigenvalue in the Overlap (GenEO) approach. The only requirement is that the SPD matrix K is provided as a sum of local symmetric positive semi-definite (SPSD) matrices K_i. A robust preconditioner following this methodology was developed for a sparse hybrid parallel distributed solver and applied on several test cases. A new algebraic parallel DD toolbox in python was developed to facilitate the development of new DD solvers relying on state-of-the-art high performance solvers. This ddmpy module is exposed in this document using a literate programming approach for reproducible science.
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Louis Poirel. Algebraic domain decomposition methods for hybrid (iterative/direct) solvers. Numerical Analysis [math.NA]. Université de Bordeaux, 2018. English. ⟨tel-02070618⟩

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