Linearly scalable fast direct solver based on proxy surface method for two-dimensional elastic wave scattering by cavity

TL;DR

This paper introduces an $O(N)$ fast direct solver for 2D elastic wave scattering that extends the proxy surface method to elastodynamics, avoiding matrix inversion and enabling efficient solutions for multiple right-hand sides.

Contribution

It develops a novel $O(N)$ fast direct solver for elastodynamics using a proxy surface approach, improving efficiency and scalability over existing methods.

Findings

Achieves $O(N)$ complexity in low-frequency elastic scattering problems.

Demonstrates high parallel efficiency with 70% scaling.

Solves multiple right-hand sides with significantly reduced time after the first.

Abstract

This paper proposes an $O(N)$ fast direct solver for two-dimensional elastic wave scattering problems. The proxy surface method is extended to elastodynamics to obtain shared coefficients for low-rank approximations from discretized integral operators. The proposed method is a variant of the Martinsson-Rokhlin-type fast direct solver. Our variant avoids the explicit computation of the inverse of the coefficient matrix, thereby reducing the required number of matrix-matrix multiplications. Numerical experiments demonstrate that the proposed solver has a complexity of $O(N)$ in the low-frequency range and has a highly parallel computation efficiency with a strong scaling efficiency of 70\%. Furthermore, multiple right-hand sides can be solved efficiently; specifically, when solving problems with 180 right-hand side vectors, the processing time per vector from the second vector onward was approximately 28,900 times faster than that for the first vector. This is a key advantage of fast direct methods.