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

## TL;DR

This paper develops a parallel algorithm for the inverse fast multipole method (IFMM), improving efficiency in solving large dense linear systems and demonstrating significant speedups in boundary element method applications.

## Contribution

It introduces a parallelization scheme for IFMM using a greedy coloring algorithm and applies it to boundary element methods, achieving notable speedups over existing methods.

## Key findings

- Achieved up to 4x speedup over multi-threaded linear algebra approaches.
- Achieved up to 11x speedup over block-diagonal preconditioners.
- Effective parallelization for large-scale boundary element problems.

## 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 $\sigma$ nodes. We proved that when $\sigma \ge 6$, the workload associated with one color is embarrassingly parallel. However, the number of nodes in a group (color) may be small when $\sigma = 6$. Therefore, we also explored $\sigma = 3$, 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 $4\times$ and $11\times$ speedups over the reference method (a) and (b), respectively, in realistic examples involving more than one million variables.

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Source: https://tomesphere.com/paper/1905.10602