# A Class of Efficient Locally Constructed Preconditioners Based on Coarse Spaces

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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Document type :
Journal articles

https://hal.inria.fr/hal-01963067
Contributor : Laura Grigori <>
Submitted on : Monday, December 30, 2019 - 3:21:10 PM
Last modification on : Saturday, April 11, 2020 - 1:51:43 AM

### Citation

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-01963067v2⟩

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