Fast Directional Computation for the High Frequency Helmholtz Kernel in Two Dimensions

TL;DR

This paper presents a fast, accurate, and scalable directional multiscale algorithm for high frequency 2D Helmholtz scattering problems, leveraging low-rank approximations in a multidirectional framework to achieve optimal computational complexity.

Contribution

It introduces an improved randomized low-rank approximation method within a multidirectional multiscale algorithm for 2D Helmholtz problems, enhancing efficiency and accuracy.

Findings

Achieves O(N log N) complexity for high frequency scattering

Demonstrates high accuracy through numerical examples

Validates effectiveness for various test cases

Abstract

This paper introduces a directional multiscale algorithm for the two dimensional $N$-body problem of the Helmholtz kernel with applications to high frequency scattering. The algorithm follows the approach in [Engquist and Ying, SIAM Journal on Scientific Computing, 29 (4), 2007] where the three dimensional case was studied. The main observation is that, for two regions that follow a directional parabolic geometric configuration, the interaction between the points in these two regions through the Helmholtz kernel is approximately low rank. We propose an improved randomized procedure for generating the low rank representations. Based on these representations, we organize the computation of the far field interaction in a multidirectional and multiscale way to achieve maximum efficiency. The proposed algorithm is accurate and has the optimal $O(N\log N)$ complexity for problems from two dimensional scattering applications. We present numerical results for several test examples to illustrate the algorithm and its application to two dimensional high frequency scattering problems.