Some observations on preconditioning for non-self-adjoint and time-dependent problems

Abstract

Numerical Linear Algebra—specifically the computational solution of equations—forms a significant part of Computational Methods for Partial Differential Equations. Here we discuss the contrast between the solution of symmetric systems of equations that arise from self-adjoint problems and non-symmetric systems that arise from non-self-adjoint problems when iterative methods are employed; such methods are the only feasible methods for very large scale computation with PDEs. We then go on to consider non-symmetric all-at-once systems that arise in approximation of time-dependent problems, discuss causality and the parallel-in-time paradigm, suggesting an approach that involves preconditioning initial value problems with time-periodic problems.

Introduction

Computational Partial Differential Equations is a large and vibrant research area. The computational solution of linear(ized) equations that arise from whatever approximation scheme is employed is commonly a major task involving methods of numerical linear algebra. Sparse direct methods are applicable for many problems, but, in particular for 3-dimensional domain problems, iterative methods usually present the only effective solution approach in combination with preconditioning [32] [8]. Almost always, very slow convergence is observed when an effective preconditioner (matrix approximation) is not used. Simple stationary iterations, including multigrid cycles, can be effectively used for some problems, but more generally applicable are Krylov subspace methods [27] [30]. (Multigrid cycles are really effective parts of many preconditioners: see for example [10, chapter 4]).

For self-adjoint problems, symmetric (or Hermitian) matrices generally arise and a common approach is to use the Conjugate Gradient method (cg) [16] (for definite problems) or the minres method [24] (for indefinite problems) with some appropriate symmetric preconditioner. In this situation, a priori descriptive convergence bounds for convergence of the iterative method depend solely on the eigenvalue spectrum of the preconditioned matrix: thus establishing estimates of the eigenvalues is all that is required to reliably predict the number of iterations needed for solution. Fewer iterations than predicted by these bounds can occur, but never more. The mathematics thus indicates what is required of a good preconditioner for a self-adjoint problem (see [32]).

By contrast, for non-self-adjoint problems when non-symmetric (non-Hermitian) matrices arise from approximation, though iterative solution methods are widely used, no generally descriptive convergence bounds are known. There are situations where special techniques can be used, but preconditioning is generally necessarily heuristic for non-symmetric matrices except in rare cases when preconditioning induces symmetry [32] or where self-adjointness in a non-standard inner-product exists [25].

Time-dependent PDE problems which are first order in time necessarily are non-self-adjoint and in model situations even give rise to matrices of the form $I+F$ where F is nilpotent of high index. Thus non-diagonalisable matrices and Jordan structure are centrally relevant for such problems. In some sense, such time-dependent problems give rise to matrices that are furthest from the ideal cases that arise for self-adjoint problems as described above. Nevertheless, some effective preconditioners for such problems have been suggested and even theory guaranteeing fast convergence of certain iterations for such problems has been established. We will describe the relevant structures in this short paper in which we also seek to clarify the symmetric/non-symmetric dichotomy.