Multigrid methods on parallel computers—A survey of recent developments

Abstract

Multigrid methods have been established as being among the most efficient techniques for solving complex elliptic equations. We sketch the multigrid idea, emphasizing that a multigrid solution is generally obtainable in a time directly proportional to the number of unknown variables on serial computers. Despite this, even the most powerful serial computers are not adequate for solving the very large systems generated, for instance, by discretization of fluid flow in three dimensions. A breakthrough can be achieved here only by highly parallel supercomputers. On the other hand, parallel computers are having a profound impact on computational science. Recently, highly parallel machines have taken the lead as the fastest supercomputers, a trend that is likely to accelerate in the future. We describe some of these new computers, and issues involved in using them. We describe standard parallel multigrid algorithms and discuss the question of how to implement them efficiently on parallel machines. The natural approach is to use grid partitioning. One intrinsic feature of a parallel machine is the need to perform interprocessor communication. It is important to ensure that time spent on such communication is maintained at a small fraction of computation time. We analyze standard parallel multigrid algorithms in two and three dimensions from this point of view, indicating that high performance efficiencies are attainable under suitable conditions on moderately parallel machines. We also demonstrate that such performance is not attainable for multigrid on massively parallel computers, as indicated by an example of poor efficiency on 65,536 processors. The fundamental difficulty is the inability to keep 65,536 processors busy when operating on very coarse grids. This example indicates that the straightforward parallelization of multigrid (and other) algorithms may not always be optimal. However, parallel machines open the possibility of finding really new approaches to solving standard problems. In particular, we present an intrinsically parallel variant of standard multigrid. This “PSMG” (parallel superconvergent multigrid) method allows all processors to be used at all times. even when processing on the coarsest grid levels. The sequential version of this method is not a sensible algorithm