A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

Luca Bergamaschi

doi:10.3390/a13040100

Abstract

The aim of this survey is to review some recent developments in devising efficient preconditioners for sequences of symmetric positive definite (SPD) linear systems A k x k = b k , k = 1 , … arising in many scientific applications, such as discretization of transient Partial Differential Equations (PDEs), solution of eigenvalue problems, (Inexact) Newton methods applied to nonlinear systems, rational Krylov methods for computing a function of a matrix. In this paper, we will analyze a number of techniques of updating a given initial preconditioner by a low-rank matrix with the aim of improving the clustering of eigenvalues around 1, in order to speed-up the convergence of the Preconditioned Conjugate Gradient (PCG) method. We will also review some techniques to efficiently approximate the linearly independent vectors which constitute the low-rank corrections and whose choice is crucial for the effectiveness of the approach. Numerical results on real-life applications show that the performance of a given iterative solver can be very much enhanced by the use of low-rank updates.

Highlights

The solution of sequences of large and sparse linear systems Ak xk = bk, k = 1, . . ., is a problem arising in many realistic applications including time-dependent solution of Partial DifferentialEquations (PDEs), computation of a part of the spectrum of large matrices, solution to nonlinear systems by the Newton methods and its variants, evaluation of a matrix function on a vector, by rational Krylov methods
We will analyze a number of techniques of updating a given Incomplete Cholesky (IC) preconditioner by a low-rank matrix with the aim of further improving this clustering
To give experimental evidence of the efficiency of this approach we report some results in solving a sequence of 30 linear systems arising from the Finite Element/Backward Euler discretization of the branched transport equation [35] which gives raise to a sequence of (2 × 2) block nonlinear systems in turn solved by the Inexact Newton method

Summary

Introduction

Equations (PDEs), computation of a part of the spectrum of large matrices, solution to nonlinear systems by the Newton methods and its variants, evaluation of a matrix function on a vector, by rational Krylov methods. The large size and sparsity of the matrices involved make iterative methods as the preeminent solution methods for these systems and call for robust preconditioners. The low-rank correction is based on a (small) number of linearly independent vectors whose choice is crucial for the effectiveness of the approach. In many cases these vectors are approximations of eigenvectors corresponding to the smallest eigenvalues of the preconditioned matrix P0 A [8,9]

Objectives

Results

Conclusion