A block GMRES method with deflated restarting for solving linear systems with multiple shifts and multiple right‐hand sides

Abstract

Summary The restarted block generalized minimum residual method (BGMRES) with deflated restarting (BGMRES‐DR) was proposed by Morgan to dump the negative effect of small eigenvalues from the convergence of the BGMRES method. More recently, Wu et al. introduced the shifted BGMRES method (BGMRES‐Sh) for solving the sequence of linear systems with multiple shifts and multiple right‐hand sides. In this paper, a new shifted block Krylov subspace algorithm that combines the characteristics of both the BGMRES‐DR and the BGMRES‐Sh methods is proposed. Moreover, our method is enhanced with a seed selection strategy to handle the case of almost linear dependence of the right‐hand sides. Numerical experiments illustrate the potential of the proposed method to solve efficiently the sequence of linear systems with multiple shifts and multiple right‐hand sides, with and without preconditioner, also against other state‐of‐the‐art solvers.

Flexible and deflated variants of the block shifted GMRES method

“…In addition, it is well known that the convergence of Krylov subspace methods depends to a large degree on the distribution of eigenvalues. A different technique, also called deflation, that can greatly improve the convergence of a Krylov method computes spectral information of the linear system from the Arnoldi decomposition at restart, and use this information to ''remove'' the smallest eigenvalues [24].…”

“…The difficulty of preconditioning shifted linear systems is that the shift-invariant property of the Krylov subspace may not be preserved under transformation. Therefore some special preconditioners need to be used, like polynomial preconditioners [23], shift-and-invert preconditioners [24,26] and multi-shifted preconditioners [27][28][29]. Additionally, in some computational settings the use of a constant preconditioner may be restrictive; for example in domain decomposition theory approximate solvers are often applied in the interior of the domain [39,Sect.…”

“…A similar approach is used in other shifted GMRES-type methods [14,23,24,33]. At this stage we can use the search space generated for the base shifted block system to approximate the solutions of the additional shifted block systems.…”

“…Instead of applying shifted Krylov subspace methods to the solution of each of the p multi-shifted linear systems independently [13][14][15][16][17][18][19][20], or block Krylov subspace methods to the solution of the L linear systems with multiple right-hand sides in sequence [10,12,21,22], it is much more efficient to use a shifted block method to solve all of the multi-shifted linear systems simultaneously [23,24]. In this context we refer to the system (A − σ 1 I)X 1 = B as the base shifted block system, and to (A − σ i I)X i = B (i = 2, .…”

“…The results clearly establish the need for preconditioning. The problem is not trivial as only special preconditioners can maintain the shift invariance property of Krylov subspaces, like polynomial preconditioner [23], shift-and-invert preconditioner [24,26] and multi-shifted preconditioners [27][28][29]. In Section 3 we present a preconditioned DBGMRES-Sh method that can accommodate both fixed and variable preconditioning, and we report on numerical experiments in Section 3.2.…”