A simpler GMRES and its adaptive variant for shifted linear systems

Abstract

Summary A variant of the simpler GMRES method is developed for solving shifted linear systems (SGMRES‐Sh), exhibiting almost the same advantage of the simpler GMRES method over the regular GMRES method. Because the remedy adapted by GMRES‐Sh is no longer feasible for SGMRES‐Sh due to the differences between simpler GMRES and GMRES for constructing the residual vectors of linear systems, we take an alternative strategy to force the residual vectors of the add system also be orthogonal to the subspaces, to which the residual vectors of the seed system are orthogonal when the seed system is solved with the simpler GMRES method. In addition, a seed selection strategy is also employed for solving the rest non‐converged linear systems. Furthermore, an adaptive version of SGMRES‐Sh is presented for the purpose of improving the stability of SGMRES‐Sh based on the technique of the adaptive choice of the Krylov subspace basis developed for the adaptive simpler GMRES. Numerical experiments demonstrate the benefits of the presented methods.

A flexible and adaptive Simpler GMRES with deflated restarting for shifted linear systems

“…The numerical results are obtained by using MATLAB R2014a (64bit) on an PC-Intel Core i5-6200U, CPU 2.4 GHz, 8 GB RAM with machine epsilon 10 −16 in double precision floating point arithmetic. Example 3.1 We consider the same matrices used in [28]. These matrices are from the University of Florida Sparse Matrix Collection and the Example 1 in [44].…”

“…In addition, from (2.7), we can see the update of the residual vectors will also cost some time. Consequently, for solving add systems, similar to SGMRES [28], FAd-SGMRES-Sh may not faster than GMRES [17]. But fortunately, for seed system, due to without solving an upper Hessenberg least-square problem, and with inexact preconditioning, FAd-SGMRES-Sh is much faster than SGMRES, GMRES and FGMRES [13], especially for large-scale problems.…”

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

“…However, as pointed out by Simoncini et al, this will cause extra work to solve an artificial system. In this work, we use the seed selection strategy similar to other works to improve the performance of the proposed BGMRES‐DR‐Sh method for solving the multishifted systems. During each cycle, we choose the seed block system that has maximum residual norm among the nonconverged shifts.…”

Making global simpler GMRES more stable

“…Test 2. In Table 1, we choose some frequently used matrices in other works 16,18,19 to test the features of the Rb-GlSGMRES method, and B is taken as B = A * [ones(N,1), rand(N, s − 1)] with s = 3.…”

“…When s = 1, it reduces to the standard simpler GMRES and its variant method using the normalized residual as the basis was proven to be conditionally stable . Further investigation on an adaptive version was done in the work of Jiránek and Rozlo nĺk (JR) and Jing et al For s > 1, the idea in the works of JR was generalized to an (adaptive) simpler block GMRES method . However, for the GlSGMRES method, stability keeps unevaluated due to the absence of the definition of the condition number for the global method.…”