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Parallel Helmholtz Solvers

Roldan Pozo, ACMD
Karin A. Remington, Scientific Computing Environments Division

This project has been focused on developing and analyzing the performance characteristics of fast three-dimensional elliptic solvers on parallel architectures such as the IBM SP2, Connection Machine CM-5, and Intel iPSC/860. Developing efficient implementations is challenging, due to required global ``all-to-all'' communication traffic, which occurs primarily in multidimensional array transpositions, and improvements over existing implementations is a continuing area of research.

The problem class considered in this project is elliptic partial differential equations of the form



Equations of this form frequently arise in large-scale modeling of physical and chemical processes. The communication requirements for parallel Fourier solvers such as those considered in this project are among the most demanding possible in terms of bandwidth and communication latency. The communication to computation ratio (one measure of the amount of overhead on a parallel computer) becomes increasingly unfavorable as problem size increases: for a three dimensional domain with points, the ratio is . These communication patterns are taxing for some architectures. Nevertheless, such applications serve as a valuable tool to identify bottlenecks in a physical networks and message passing layers.