Multigrid methods are among themost important algorithms forcomputational scientists becausethey’re the most efficient solvers for

a wide range of problems. The modern era of multi-grid methods began more than 30 years ago withthe publication of Achi Brandt’s seminal papers.1,2

Although it was originally successful for solvingelliptic partial differential equations (PDEs) and thelinear systems that arise when they’re discretized,the basic multigrid principle of coupling multiplescales has much wider applicability.

Why a Special Issue?At first glance, it might seem strange that such a fun-damental task as the solution of linear systems is stillworth a special issue in a computational sciencemagazine in 2006. In fact, the solution algorithmsfor linear systems are usually hidden from the com-putational scientist—say, behind Matlab’s backslashoperator or within state-of-the art commercial finiteelement packages or computational fluid dynamicstools. Ideally, computational scientists shouldn’t have

10 COMPUTING IN SCIENCE & ENGINEERING

Multigrid Computing

G U E S T E D I T O R S ’ I N T R O D U C T I O N

STEVE MCCORMICK

University of Colorado, BoulderULRICH RÜDE

Universität Erlangen-Nürnberg

1521-9615/06/$20.00 © 2006 IEEE

Copublished by the IEEE CS and the AIP

NOVEMBER/DECEMBER 2006 11

to worry how a linear system is being solved, so why,then, did we bring together a special issue on it?

The answer is that the linear solver is still thecomputational bottleneck in terms of complexity formany mesh-based numerical simulations. Althoughwe can usually perform data pre- and postprocess-ing linearly in the number of mesh points and thenparallelize it with standard partitioning techniques,the complexity of most linear system solvers—director iterative—is worse than linear. Multigrid meth-ods are among the few exceptions. In practice, thesolver’s complexity might not dominate small- ormoderate-sized problems, but using multigrid meth-ods can offer tremendous gains in efficiency forlarge-scale computing. One more aspect to the com-plexity argument concerns asymptotically optimalsolvers—their overall cost is linear or close to linearin the number of unknowns. A classical example is afast Fourier transform- (FFT-) based algorithm withNlogN complexity.

An important issue is the constant in the com-plexity estimates because it ultimately decideswhich algorithm is best. Although much of themathematical analysis involved doesn’t exhibitthese constants, we can get a solution for simpleenough problems by using multigrid methods withfewer than 30 elementary arithmetic operations perunknown. A well-implemented multigrid algo-rithm is thus likely to be more competitive in timeto solution, the ultimate metric for a computationalscientist. Furthermore, unlike FFT-based methods,we can generalize multigrid methods to more com-plex domains, boundary conditions, differential op-erators, and so on.

But if multigrid methods are so good, whydoesn’t Matlab’s backslash simply run a multigridsolver? Unfortunately, the multigrid algorithmdraws its strength from exploiting the underlyingdifferential equation’s structure. Thus, we can’t eas-ily use it as a generic linear solver—rather, we oftenhave to customize it for each individual problem.

Moreover, multigrid algorithms can’t solve alllinear systems efficiently (probably not even allsparse systems), so we can’t easily connect them tostandard linear algebra interfaces. Multigrid algo-rithms are unforgiving in the sense that optimalityimmediately disappears if the developer makes asingle mistake, whether a coding error or a mis-conception (for example, how a fine-scale featurein the problem is transferred to the coarser resolu-tions). Developing a truly efficient multigridmethod for a given problem is far from trivial.

The ArticlesIrad Yavneh presents a basic introduction to multi-

grid methods that illustrates the fundamental prin-ciples in a very readable way. The article is wellsuited as basic material both for teaching multigridmethods and for nonspecialists in engineering orscience classes that deal with numerical methods.

Robert Falgout describes the basics of algebraicmultigrid (AMG) solvers, a variant of multigridmethods that attempts to reconstruct a grid-like hi-erarchy when given only the linear system. BecauseAMG tries to make multigrid usable via the usuallinear algebra interfaces, this development has highpractical relevance. Indeed, most multigrid users arelikely to employ some form of an algebraic multigridmethod eventually.

The three other articles in the issue explore com-mon extensions of multigrid methods: a discussionof multigrid methods for systems by Cornelis Oost-erlee and Francisco José Gaspar Lorenz; an excur-sion into multigrid-based adaptive finite elementmethods by Peter Bastian and Christian Wieners;and an article on very large-scale computing byBenjamin Bergen, Tobias Gradl, Frank Hülsemann,and Ulrich Rüde.

Computational scientists who must solvevery large problems might have to un-derstand multigrid methods; in somecases, they might even have to develop

their own multigrid solvers. This special issueshould help CiSE readers understand the potentialof multigrid methods and start multigrid develop-ments of their own.

References1. A. Brandt, “A Multi-Level Adaptive Technique (MLAT) for Fast

Numerical Solution to Boundary Value Problems,” Proc. 3rd Int’lConf. Numerical Methods in Fluid Mechanics, LNP 18, H. Cabannesand R. Temam, eds., Springer-Verlag, 1973, pp. 82–89.

2. A. Brandt, “Multi-Level Adaptive Solutions to Boundary-Value Prob-lems,” Mathematics of Computation, vol. 31, 1977, pp. 333–390.

Steve McCormick is a professor at the University of Col-orado, Boulder. His research interests include multigridmethods, adaptive mesh refinement, and least-squaresfinite element methods. McCormick has a PhD in mathe-matics from the University of Colorado. He is a memberof SIAM. Contact him at stevem@colorado.edu.

Ulrich Rüde is a professor at the Universität Erlangen-Nürnberg. His research interests include computational sci-ence and high-performance computing. Rüde has a Dr.habil. in computer science and applied mathematics fromTechnische Universität München. He is a member of SIAMand editor in chief of the SIAM Journal on Scientific Com-puting. Contact him at ruede@informatik.uni-erlangen.de.