On short recurrence Krylov type methods for linear systems with many right-hand sides

Abstract

Block and global Krylov subspace methods have been proposed as methods adapted to the situation where one iteratively solves systems with the same matrix and several right hand sides. These methods are advantageous, since they allow to cast the major part of the arithmetic in terms of matrix-block vector products, and since, in the block case, they take their iterates from a potentially richer subspace. In this paper we consider the most established Krylov subspace methods which rely on short recurrencies, i.e. BiCG, QMR and BiCGStab. We propose modifications of their block variants which increase numerical stability, thus at least partly curing a problem previously observed by several authors. Moreover, we develop modifications of the "global" variants which almost halve the number of matrix-vector multiplications. We present a discussion as well as numerical evidence which both indicate that the additional work present in the block methods can be substantial, and that the new "economic" versions of the "global" BiCG and QMR method can be considered as good alternatives to the BiCGStab variants.

Preconditioned Bminpert Algorithms for Matrix Equation <i>AX</i> = <i>B</i> and Their Applications in Color Image Restoration

“…GsGMRES [9], Bl-BiCG-rQ [10] and BGMRES are illustrated with numerical experiments in Section 6. Finally, the conclusions appear in Section 7.…”

Block Krylov subspace methods for functions of matrices

“…Regarding B as a collection of columns b i , B := [b 1 |...|b s ], one might consider applying methods for a single vector, such as those described in [12,27] or the newly proposed restarted Arnoldi methods [24,25,26], to each problem f (A)b i . It is well known for linear systems, however, that block Krylov approaches treating all columns b i at once can be computationally advantageous; see, e.g., [3,33,46,50,52,59,60,61,63]. It is therefore reasonable to consider block Krylov methods for computing f (A)B.…”

“…Global GMRES and global FOM were first introduced in 1999 in [40] for matrix equations. Additional global methods can be found in [5,9,22,36,52,67].…”

“…Rather, they have been proposed as an implementation alternative to the non-block approach in which the systems are solved sequentially, one column after another. While it is likely that such methods have been considered before, they are first formally named and developed in [52].…”