Linear System Solvers: Sparse iterative solvers

Abstract

In this chapter we will present an overview of a number of related iterative methods for the solution of linear systems of equations. These methods are so-called Krylov projection type methods and they include popular methods as Conjugate Gradients, Bi-Conjugate Gradients, LSQR and GMRES. We will sketch how these methods can be derived from simple basic iteration formulas, and how they are interrelated. Iterative schemes are usually considered as an alternative for the solution of linear sparse systems, like those arising in, e.g., nite element or nite dierence approximation of (systems of) partial dierential equations. The structure of the operators plays no explicit role in any of these schemes, and the operator may be given even as a rule or a subroutine. Although these methods seem to be almost trivially parallellizable at rst glance, this is sometimes a point of concern because of the inner products involved. We will consider this point in some detail. Iterative methods are usually applied in combination with so-called preconditioning operators in order to further improve convergence properties. This aspect will receive more attention in a separate chapter in the same volume.