In this paper two well-known robust multigrid solvers for anisotropic operators on structured grids are compared: alternating-plane smoothers combined with full coarsening and plane smoothers combined with semi-coarsening. The study has taken into account not only numerical properties but also architectural ones, focusing on cache memory exploitation and parallel characteristics. Experimental results for the sequential algorithms have been obtained on two different systems based on the MIPS R10000 processor, but with different L2 cache sizes and memory bandwidths (an SGI O2 workstation and an SGI Origin 2000 system). Although the alternating-plane approach is the best choice for sequential implementations, experimental estimations show poor parallel efficiencies. For the semicoarsening alternative two different parallel implementations have been considered. The first one has optimal parallel characteristics but due to deterioration of the convergence properties its realistic efficiency is not satisfactory. In the second one, some processors remain idle during a short period of time on every multigrid cycle. However, the second parallel algorithm is more efficient since it preserves the numerical properties of the sequential version. Parallel experiments have also been taken on a Cray T3E system.

The aim of this paper is to present an easy and efficient method to implement alternating-line processes on current parallel computers. First we show how data locality has an important impact on global efficiency, which leads us to the conclusion that one-dimensional compositions are the most convenient ones for 2D problems. Once this is asserted, a parallel algorithm is presented for the solution of the distributed tridiagonal systems along the partitioned direction. The key idea is to pipeline the simultaneous resolution of many systems of equations, not parallelising each resolution separately. This approach presents good numerical and architectural properties, in terms of memory usage and data locality, and high parallel efficiencies are obtained. For the case of alternating-line processes, the election of the optimal decomposition is studied. The experimental results have been obtained on a Cray T3E.

We present the results of different communication tests on some current parallel computers, the Cray T3E and the SGI Origin 2000. The aim of this paper is to study the effect of the local memory use and the communication network exploitation on message sending. For this purpose, we have first designed experiments without network contention to establish the achievable bandwidths. We have then modified this base experiment by increasing the contention of the network and by decreasing the spatial locality properties of the messages. We analyse these results taking into account the underlying architectures and we conclude with some hints for regular applications.