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Enlarged GMRES for solving linear systems with one or multiple right-hand sides

Hussam Al Daas 1 Laura Grigori 1 Pascal Hénon 2 Philippe Ricoux 3
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 : We propose a variant of the generalized minimal residual (GMRES) method for solving linear systems of equations with one or multiple right-hand sides. Our method is based on the idea of the enlarged Krylov subspace to reduce communication. It can be interpreted as a block GMRES method. Hence, we are interested in detecting inexact breakdowns. We introduce a strategy to perform the test of detection. Furthermore, we propose a technique for deflating eigenvalues that has two benefits. The first advantage is to avoid the plateau of convergence after the end of a cycle in the restarted version. The second is to have very fast convergence when solving the same system with different right-hand sides, each given at a different time (useful in the context of a constrained pressure residual preconditioner). We test our method with these deflation techniques on academic test matrices arising from solving linear elasticity and convection–diffusion problems as well as matrices arising from two real-life applications, seismic imaging and simulations of reservoirs. With the same memory cost we obtain a saving of up to 50% in the number of iterations required to reach convergence with respect to the original method.
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Submitted on : Friday, December 21, 2018 - 9:42:29 AM
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Hussam Al Daas, Laura Grigori, Pascal Hénon, Philippe Ricoux. Enlarged GMRES for solving linear systems with one or multiple right-hand sides. IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2018, ⟨10.1093/imanum/dry054⟩. ⟨hal-01963032⟩



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