Fast Directional Computation of High Frequency Boundary Integrals via Local FFTs

TL;DR

This paper introduces a fast, kernel-independent algorithm for evaluating high-frequency boundary integrals in acoustic scattering, utilizing directional low-rank approximations, oscillatory Chebyshev interpolation, and local FFTs to achieve quasi-linear complexity.

Contribution

It develops a novel, efficient algorithm that combines directional low-rank approximations with local FFTs for high-frequency boundary integral evaluations in 2D.

Findings

Achieves quasi-linear computational complexity.

Demonstrates effectiveness through numerical experiments.

Kernel-independent and simple to implement.

Abstract

The boundary integral method is an efficient approach for solving time-harmonic acoustic obstacle scattering problems. The main computational task is the evaluation of an oscillatory boundary integral at each discretization point of the boundary. This paper presents a new fast algorithm for this task in two dimensions. This algorithm is built on top of directional low-rank approximations of the scattering kernel and uses oscillatory Chebyshev interpolation and local FFTs to achieve quasi-linear complexity. The algorithm is simple, fast, and kernel-independent. Numerical results are provided to demonstrate the effectiveness of the proposed algorithm.