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

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.

A new preconditioning approach for an interior point‐proximal method of multipliers for linear and convex quadratic programming

“…Usually columns of V are chosen as the (approximate) eigenvectors of P −1 NE,kM NE,k corresponding to the smallest eigenvalues of this matrix. 37,38 However, this choice would not produce a significant reduction in the condition number of the preconditioned matrix as the spectral analysis of Theorem 1 suggests a possible clustering of smallest eigenvalues around 1. We choose instead, as the columns of V, the rightmost eigenvectors of P −1 NE,kM NE,k , approximated with low accuracy by the function eigs of MATLAB.…”

Accelerating the solution of linear systems appearing in two-phase reservoir simulation by the use of POD-based deflation methods

“…where Δρ = ρ w − ρ nw . With this approach, the system is expressed in terms of the non wetting phase pressure, (27), and the saturation of the wetting phase, (31). In the pressure equation, the coupling to saturation is present via the phase mobilities, and the derivative of the capillary function.…”

“…In recent years, deflation techniques have been developed to accelerate the convergence of Krylov subspace methods [8,9,22,26,49]. Especially useful when a sequence of linear systems with constant or slightly varying matrices has to be solved [31]. For this technique to be effective, a deflation subspace needs to be found.…”

Preconditioners for Krylov subspace methods: An overview

“…In the particular, but important, case that these systems are all of the form , with changing, modifying a given preconditioner for such a sequence is considered in, for example, [306‐308]. We refer to [309] for a recent survey of low‐rank updates of preconditioners, which considers a number of the above applications.…”