A matrix-free parallel solution method for the three-dimensional heterogeneous Helmholtz equation

TL;DR

This paper introduces a matrix-free parallel iterative solver for the 3D heterogeneous Helmholtz equation, achieving high scalability and robustness for large-scale scientific applications.

Contribution

It presents a novel matrix-free parallel Krylov solver with a complex shifted Laplace preconditioner for efficient 3D Helmholtz equation solutions.

Findings

Robustness demonstrated through numerical experiments.

Outstanding strong scalability in parallel computing.

Suitable for realistic large-scale 3D heterogeneous problems.

Abstract

The Helmholtz equation is related to seismic exploration, sonar, antennas, and medical imaging applications. It is one of the most challenging problems to solve in terms of accuracy and convergence due to the scalability issues of the numerical solvers. For 3D large-scale applications, high-performance parallel solvers are also needed. In this paper, a matrix-free parallel iterative solver is presented for the three-dimensional (3D) heterogeneous Helmholtz equation. We consider the preconditioned Krylov subspace methods for solving the linear system obtained from finite-difference discretization. The Complex Shifted Laplace Preconditioner (CSLP) is employed since it results in a linear increase in the number of iterations as a function of the wavenumber. The preconditioner is approximately inverted using one parallel 3D multigrid cycle. For parallel computing, the global domain is partitioned blockwise. The matrix-vector multiplication and preconditioning operator are implemented in a matrix-free way instead of constructing large, memory-consuming coefficient matrices. Numerical experiments of 3D model problems demonstrate the robustness and outstanding strong scaling of our matrix-free parallel solution method. Moreover, the weak parallel scalability indicates our approach is suitable for realistic 3D heterogeneous Helmholtz problems with minimized pollution error.