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A class of efficient locally constructed preconditioners based on coarse spaces

Hussam Al Daas 1 Laura Grigori 1
1 ALPINES - Algorithms and parallel tools for integrated numerical simulations
INSMI - Institut National des Sciences Mathématiques et de leurs Interactions, Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
Abstract : In this paper we present a class of robust and fully algebraic two-level preconditioners for symmetric positive definite (SPD) matrices. We introduce the notion of algebraic local symmetric positive semidefinite splitting of an SPD matrix and we give a characterization of this splitting. This splitting leads to construct algebraically and locally a class of efficient coarse spaces which bound the spectral condition number of the preconditioned system by a number defined a priori. We also introduce the tau-filtering subspace. This concept helps compare the dimension minimality of coarse spaces. Some PDEs-dependant preconditioners correspond to a special case. The examples of the algebraic coarse spaces in this paper are not practical due to expensive construction. We propose a heuristic approximation that is not costly. Numerical experiments illustrate the efficiency of the proposed method.
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Submitted on : Friday, December 21, 2018 - 9:57:34 AM
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Hussam Al Daas, Laura Grigori. A class of efficient locally constructed preconditioners based on coarse spaces. SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 2019, 40 (1), pp.66-91. ⟨10.1137/18M1194365⟩. ⟨hal-01963067⟩



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