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Journal of Computational Physics 182, 418477 (2002)doi:10.1006/jcph.2002.7176

Preconditioning Techniques for Large LinearSystems: A Survey

Michele Benzi

Mathematics and Computer Science Department, Emory University, Atlanta, Georgia 30322E-mail: benzi@mathcs.emory.edu

Received April 17, 2002; revised July 23, 2002

This article surveys preconditioning techniques for the iterative solution of largelinear systems, with a focus on algebraic methods suitable for general sparse ma-trices. Covered topics include progress in incomplete factorization methods, sparseapproximate inverses, reorderings, parallelization issues, and block and multilevelextensions. Some of the challenges ahead are also discussed. An extensive bibliog-raphy completes the paper. c 2002 Elsevier Science (USA)

Key Words: linear systems; sparse matrices; iterative methods; algebraic precondi-tioners; incomplete factorizations; sparse approximate inverses; unstructured grids;multilevel methods; parallel computing; orderings; block algorithms.

1. INTRODUCTION

The solution of large sparse linear systems of the form

Ax = b, (1)

where A = [ai j ] is an n n matrix and b a given right-hand-side vector, is central to manynumerical simulations in science and engineering and is often the most time-consuming partof a computation. While the main source of large matrix problems remains the discretization(and linearization) of partial differential equations (PDEs) of elliptic and parabolic type,large and sparse linear systems also arise in applications not governed by PDEs. Theseinclude the design and computer analysis of circuits, power system networks, chemicalengineering processes, economics models, and queueing systems.

Direct methods, based on the factorization of the coefficient matrix A into easily invertiblematrices, are widely used and are the solver of choice in many industrial codes, especiallywhere reliability is the primary concern. Indeed, direct solvers are very robust, and theytend to require a predictable amount of resources in terms of time and storage [121, 150].With a state-of-the-art sparse direct solver (see, e.g., [5]) it is possible to efficiently solve

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PRECONDITIONING TECHNIQUES 419

in a reasonable amount of time linear systems of fairly large size, particularly when theunderlying problem is two dimensional. Direct solvers are also the method of choice incertain areas not governed by PDEs, such as circuits, power system networks, and chemicalplant modeling.

Unfortunately, direct methods scale poorly with problem size in terms of operation countsand memory requirements, especially on problems arising from the discretization of PDEs inthree space dimensions. Detailed, three-dimensional multiphysics simulations (such as thosebeing carried out as part of the U.S. Department of Energys ASCI program) lead to linearsystems comprising hundreds of millions or even billions of equations in as many unknowns.For such problems, iterative methods are the only option available. Even without consideringsuch extremely large-scale problems, systems with several millions of unknowns are nowroutinely encountered in many applications, making the use of iterative methods virtuallymandatory. While iterative methods require fewer storage and often require fewer operationsthan direct methods (especially when an approximate solution of relatively low accuracy issought), they do not have the reliability of direct methods. In some applications, iterativemethods often fail and preconditioning is necessary, though not always sufficient, to attainconvergence in a reasonable amount of time.

It is worth noting that in some circles, especially in the nuclear power industry and thepetroleum industry, iterative methods have always been popular. Indeed, these areas havehistorically provided the stimulus for much early research on iterative methods, as witnessedin the classic monographs [282, 286, 294]. In contrast, direct methods have been traditionallypreferred in the areas of structural analysis and semiconductor device modeling, in mostparts of computational fluid dynamics (CFD), and in virtually all applications not governedby PDEs. However, even in these areas iterative methods have made considerable gains inrecent years.

To be fair, the traditional classification of solution methods as being either direct oriterative is an oversimplification and is not a satisfactory description of the present state ofaffairs. First, the boundaries between the two classes of methods have become increasinglyblurred, with a number of ideas and techniques from the area of sparse direct solvers beingtransferred (in the form of preconditioners) to the iterative camp, with the result that iterativemethods are becoming more and more reliable. Second, while direct solvers are almostinvariably based on some version of Gaussian elimination, the field of iterative methodscomprises a bewildering variety of techniques, ranging from truly iterative methods, likethe classical Jacobi, GaussSeidel, and SOR iterations, to Krylov subspace methods, whichtheoretically converge in a finite number of steps in exact arithmetic, to multilevel methods.To lump all these techniques under a single heading is somewhat misleading, especiallywhen preconditioners are added to the picture.

The focus of this survey is on preconditioning techniques for improving the performanceand reliability of Krylov subspace methods. It is widely recognized that preconditioning isthe most critical ingredient in the development of efficient solvers for challenging problemsin scientific computation, and that the importance of preconditioning is destined to increaseeven further. The following excerpt from the textbook [272, p. 319] by Trefethen and Bauaptly underscores this point:

In ending this book with the subject of preconditioners, we find ourselves at the philosophical center ofthe scientific computing of the future.... Nothing will be more central to computational science in thenext century than the art of transforming a problem that appears intractable into another whose solutioncan be approximated rapidly. For Krylov subspace matrix iterations, this is preconditioning.

420 MICHELE BENZI

Indeed, much effort has been put in the development of effective preconditioners, andpreconditioning has been a more active research area than either direct solution methodsor Krylov subspace methods for the past few years. Because an optimal general-purposepreconditioner is unlikely to exist, this situation is probably not going to change in theforeseeable future.

As is well known, the term preconditioning refers to transforming the system (1) intoanother system with more favorable properties for iterative solution. A preconditioner is amatrix that effects such a transformation. Generally speaking, preconditioning attempts toimprove the spectral properties of the coefficient matrix. For symmetric positive definite(SPD) problems, the rate of convergence of the conjugate gradient method depends on thedistribution of the eigenvalues of A. Hopefully, the transformed (preconditioned) matrixwill have a smaller spectral condition number, and/or eigenvalues clustered around 1. Fornonsymmetric (nonnormal) problems the situation is more complicated, and the eigen-values may not describe the convergence of nonsymmetric matrix iterations like GMRES(see [161]). Nevertheless, a clustered spectrum (away from 0) often results in rapid conver-gence, particularly when the preconditioned matrix is close to normal.

If M is a nonsingular matrix that approximates A (in some sense), then the linear system

M1 Ax = M1b (2)

has the same solution as (1) but may be easier to solve. Here M is the preconditioner. In caseswhere M1 is explicitly known (as with polynomial preconditioners or sparse approximateinverses), the preconditioner is M1 rather than M .

System (2) is preconditioned from the left, but one can also precondition from the right:

AM1 y = b, x = M1 y. (3)

When Krylov subspace methods are used, it is not necessary to form the preconditionedmatrices M1 A or AM1 explicitly (this would be too expensive, and we would losesparsity). Instead, matrixvector products with A and solutions of linear systems of the formMz = r are performed (or matrixvector products with M1 if this is explicitly known).

In addition, split preconditioning is also possible, i.e.,

M11 AM12 y = M11 b, x = M12 y, (4)

where the preconditioner is now M = M1 M2. Which type of preconditioning to use dependson the choice of the iterative method, problem characteristics, and so forth. For example,with residual minimizing methods, like GMRES, right preconditioning is often used. Inexact arithmetic, the residuals for the right-preconditioned system are identical to the trueresiduals rk = b Axk .

Notice that the matrices M1 A, AM1, and M11 AM12 are all similar and therefore

have the same eigenvalues. If A and M are SPD, the convergence of the CG method willbe the same (except possibly for round-off effects) in all cases. On the other hand, in thenonnormal case, solvers like GMRES can behave very differently depending on whether agiven preconditioner is applied on the left or on the right (see the discussion in [251, p. 255];see also [202, p. 66] for a striking example).

PRECONDITIONING TECHNIQUES 421

In general, a good preconditioner M should meet the following requirements:

The preconditioned system should be easy to solve. The preconditioner should be cheap to construct and apply.The first property means that the preconditioned iteration should converge rapidly, while

the second ensures that each iteration is not too expensive. Notice that these two requirem