On representations of the Helmholtz Green's function

TL;DR

This paper analyzes the Helmholtz Green's function by decomposing it into oscillatory and non-oscillatory parts, enabling efficient computation and multiresolution representation, which advances numerical methods for wave problems.

Contribution

It introduces a novel decomposition of the Helmholtz Green's function into oscillatory and singular components with efficient application methods.

Findings

Oscillatory component can be differentiated finitely many times at the origin.

Non-oscillatory component admits a multiresolution Gaussian-based representation.

Application complexity for the oscillatory part is $ ext{O}(k^{d} ext{log}k)$ operations.

Abstract

We consider the free space Helmholtz Green's function and split it into the sum of oscillatory and non-oscillatory (singular) components. The goal is to separate the impact of the singularity of the real part at the origin from the oscillatory behavior controlled by the wave number k. The oscillatory component can be chosen to have any finite number of continuous derivatives at the origin and can be applied to a function in the Fourier space in $\mathcal{O}\left(k^{d}\log k\right)$ operations. The non-oscillatory component has a multiresolution representation via a linear combination of Gaussians and is applied efficiently in space.