A PARALLEL FAST MULTIPOLE METHOD FOR THE HELMHOLTZ EQUATION

Presented is a parallel algorithm based on the fast multipole method (FMM) for the Helmholtz equation. This variant of the FMM is useful for computing radar cross sections and antenna radiation patterns. The FMM decomposes the impedance matrix into sparse components, reducing the operation count of the matrix-vector multiplication in iterative solvers to O(N3/2) (where N is the number of unknowns). The parallel algorithm divides the problem into groups and assigns the computation involved with each group to a processor node. Careful consideration is given to the communications costs. A time complexity analysis of the algorithm is presented and compared with empirical results from a Paragon XP/S running the lightweight Sandia/University of New Mexico operating system (SUNMOS). For a 90,000 unknown problem running on 60 nodes, the sparse representation fits in memory and the algorithm computes the matrix-vector product in 1.26 seconds. It sustains an aggregate rate of 1.4 Gflop/s. The corresponding dense matrix would occupy over 100 Gbytes and, assuming that I/O is free, would require on the order of 50 seconds to form the matrix-vector product.