A distributed-memory parallel algorithm for discretized integral equations using Julia

Abstract

Boundary value problems involving elliptic PDEs such as the Laplace and the Helmholtz equations are ubiquitous in physics and engineering. Many such problems have alternative formulations as integral equations that are mathematically more tractable than their PDE counterparts. However, the integral equation formulation poses a challenge in solving the dense linear systems that arise upon discretization. In cases where iterative methods converge rapidly, existing methods that draw on fast summation schemes such as the Fast Multipole Method are highly efficient and well established. More recently, linear complexity direct solvers that sidestep convergence issues by directly computing an invertible factorization have been developed. However, storage and compute costs are high, which limits their ability to solve large-scale problems in practice. In this work, we introduce a distributed-memory parallel algorithm based on an existing direct solver named ``strong recursive skeletonization factorization.'' The analysis of its parallel scalability applies generally to a class of existing methods that exploit the so-called strong admissibility. Specifically, we apply low-rank compression to certain off-diagonal matrix blocks in a way that minimizes data movement. Given a compression tolerance, our method constructs an approximate factorization of a discretized integral operator (dense matrix), which can be used to solve linear systems efficiently in parallel. Compared to iterative algorithms, our method is particularly suitable for problems involving ill-conditioned matrices or multiple right-hand sides. Large-scale numerical experiments are presented to demonstrate the performance of our implementation using the Julia language