Parallelization of the inverse fast multipole method with an application to boundary element method

Abstract

We present an algorithm to parallelize the inverse fast multipole method (IFMM), which is an approximate direct solver for dense linear systems. The parallel scheme is based on a greedy coloring algorithm, where two nodes in the hierarchy with the same color are separated by at least σ nodes. We proved that when σ ≥ 6, the workload associated with one color is embarrassingly parallel. However, the number of nodes in a group (color) may be small when σ = 6. Therefore, we also explored σ = 3, where a small fraction of the algorithm needs to be serialized, and the overall parallel efficiency was improved. We implemented the parallel IFMM using OpenMP for shared-memory machines. Successively, we applied it to a fast-multipole accelerated boundary element method (FMBEM) as a preconditioner, and compared its efficiency with (a) the original IFMM parallelized by linking a multi-threaded linear algebra library and (b) the commonly used parallel block-diagonal preconditioner. Our results showed that our parallel IFMM achieved at most 4× and 11× speedups over the reference method (a) and (b), respectively, in realistic examples involving more than one million variables.

Overlapping Domain Decomposition Preconditioner for Integral Equations

“…The second group employs the so-called strong admissibility and compresses offdiagonal blocks corresponding to two sufficiently distant regions [11,24,30]. The resulting numerical ranks are constant regardless of the sizes of the regions according to standard fast multipole estimates [16,17].…”

Distributed-memory $\mathcal{H}$-matrix Algebra I: Data Distribution and Matrix-vector Multiplication

“…Many of these fast algorithms and algebraic representations have been parallelized on either shared-memory or distributed-memory setting to be applicable to practical problems of interest [10,16,18,35,40,41,42,43,44,46,47,50]. Here we focus on the parallelization of H-matrix.…”