Parallel multigrid methods: Implementation on SUPRENUM-like architectures and applications

Abstract

Multigrid (MG) methods for partial differential equations (and for other important mathematical models in scientific computing) have turned out to be optimal on sequential computers. Clearly, one wants to apply them also on vector and parallel computers in order to exploit both, the high MG-efficiency (compared to classical methods) and the full computational power of modern supercomputers. For this purpose, parallel MG methods are needed. It turns out that certain well-known standard MG methods (with RB and zebra-type relaxation, as described in [25]) already contain a sufficiently high degree of parallelism.

Among innovative supercomputer architectures, MIMD multiprocessor computers with local memory and a vector unit in each processor are particularly promising. A software approach that corresponds to such architectures in a natural way is the abstract SUPRENUM concept. It is characterized by a dynamical process system, where each process has its own data space and communicates with other processes by message-passing.

In this paper, we show how such architectures and software concepts are used for the solution of large scale grid problems (discrete PDEs, etc.). Grid partitioning and blockstructuring — with communication only along the subgrid or block boundaries — are the natural approches in this context. Any grid oriented method, in particularly any MG method can be efficiently parallelized using these approaches. In the SUPRENUM project, powerful software tools (e.g. a mapping library for the process-processor mapping and a communication library for the intergrid data exchanges) are developed that make it very easy to implement single grid and MG methods on local memory multiprocessor systems. Parallel MG programs have been run on the SUPRENUM simulator [16], the SUPRENUM pre-prototype [22] and some other local memory machines like the Intel iPSC and the CalTech hypercube.