On restarted and deflated block FOM and GMRES methods for sequences of shifted linear systems
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On restarted and deflated block FOM and GMRES methods for sequences of shifted linear systems Lakhdar Elbouyahyaoui1 · Mohammed Heyouni2 Saberi-Movahed4
· Azita Tajaddini3 · Farid
Received: 7 September 2019 / Accepted: 17 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The problem of shifted linear systems is an important and challenging issue in a number of research applications. Krylov subspace methods are effective techniques for different kinds of this problem due to their advantages in large and sparse matrix problems. In this paper, two new block projection methods based on respectively block FOM and block GMRES are introduced for solving sequences of shifted linear systems. We first express the original problem explicitly by a sequence of Sylvester matrix equations whose coefficient matrices are obtained from the shifted linear systems. Then, we show the restarted shifted block FOM (rsh-BFOM) method and derive some of its properties. We also present a framework for the restarted shifted block GMRES (rsh-BGMRES) method. In this regard, we describe two variants of rshBGMRES, including (1) rsh-BGMRES with an unshifted base system that applies a fixed unshifted base system and (2) rsh-BGMRES with a variable shifted base system in which the base block system can change after restart. Furthermore, we consider the use of deflation techniques for improving the performance of the rsh-BFOM and rsh-BGMRES methods. Finally, some numerical experiments are conducted to demonstrate the effectiveness of the proposed methods. Keywords Block Arnoldi process · Block Krylov subspace methods · Deflation · Sequences of shifted linear systems Mathematics Subject Classification (2010) MSC 65F
Mohammed Heyouni
[email protected]
Extended author information available on the last page of the article.
Numerical Algorithms
1 Introduction In this work, we consider the solution of sequences of shifted linear systems that have the form: (i)
(i)
(A − σj In ) xj = b(i) , (i)
for i = 1, . . . , s, and j = 1, . . . , k, (i)
where A ∈ Cn×n , xj , b(i) ∈ Cn , σj (i) A − σj In
(1)
∈ C and In is the identity matrix of size n. (i)
The coefficient matrices are assumed to be nonsingular for all σj . The solution of linear systems having the form (1) is of great importance in a variety of practical applications and scientific fields. Shifted linear systems are encountered in several contexts such as control theory [1, 2], quantum chromodynamics problems [3–7], time-dependent partial differential equations [8–11], large-scale eigenvalue problems [12, 13], and other engineering problems [14–19]. Here, we point out that our interest in the problem (1) comes from the description of a method based on the use of the block Arnoldi process solving the multi-input Sylvester-observer equations [20]. It is clear that direct methods based on the LU decomposition are not recommended for the solution of (1) since the LU decomposition must be recalculated for each shift σj(i) . In addition, in ma
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