Deflation and augmentation techniques in Krylov subspace methods for the solution of linear systems

TL;DR

This paper reviews deflation and augmentation techniques that enhance the convergence of Krylov subspace methods for solving various types of linear systems, including non-Hermitian and Hermitian matrices.

Contribution

It provides a comprehensive review of numerical approaches for applying deflation and augmentation in Krylov methods across different matrix types.

Findings

Improved convergence rates demonstrated for non-Hermitian systems.

Enhanced efficiency of conjugate gradient method for Hermitian positive definite problems.

Comparison of techniques within Arnoldi and conjugate gradient frameworks.

Abstract

In this paper we present deflation and augmentation techniques that have been designed to accelerate the convergence of Krylov subspace methods for the solution of linear systems of equations. We review numerical approaches both for linear systems with a non-Hermitian coefficient matrix, mainly within the Arnoldi framework, and for Hermitian positive definite problems with the conjugate gradient method.