Block Krylov Space Methods for Linear Systems with Multiple Right-hand Sides: an Introduction

In a number of applications in scientific computing and engineering one has to solve huge sparse linear systems of equations with several right-hand sides that are given at once. Block Krylov space solvers are iterative methods that are especially designed for such problems and have fundamental advantages over the corresponding methods for systems with a single right-hand side: much larger search spaces and simultaneously computable matrix-vector products. But there are inherent difficulties that make their implementation and their application challenging: the possible linear dependence of the various residuals, the much larger memory requirements, and the drastic increase of the size of certain small (“scalar”) auxiliary problems that have to be solved in each step. We start this paper with a quick introduction into sparse linear systems, their solution by sparse direct and iterative methods, Krylov subspaces, and preconditioning. Then we turn to systems with multiple right-hand sides, block Krylov space solvers, and the deflation of right-hand sides in the case where the residuals are linearly dependent. As a model we discuss in particular a block version of the General Minimum Residual (GMRes) algorithm.