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

TL;DR

This paper enhances short recurrence Krylov subspace methods for solving linear systems with multiple right-hand sides by improving numerical stability and reducing matrix-vector multiplications, offering more efficient alternatives.

Contribution

It introduces modifications to block BiCG, QMR, and global variants that improve stability and efficiency, addressing previous limitations of these methods.

Findings

Modified block methods show increased numerical stability.

Global variants nearly halve matrix-vector multiplications.

Proposed methods are competitive alternatives to BiCGStab.

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.