Gaussian Beam Methods for the Helmholtz Equation

TL;DR

This paper develops Gaussian beam methods to approximate high-frequency Helmholtz solutions with localized sources, providing error estimates that decay with increasing wave number, applicable in complex geometries.

Contribution

It introduces error estimates for Gaussian beam approximations of the Helmholtz equation that are valid in non-trapping scenarios and are independent of dimension and caustics.

Findings

Error in Gaussian beam approximations decays as $k^{-N/2}$ with increasing wave number.

Error estimates are valid for both single beams and superpositions.

Results are independent of dimension and presence of caustics.

Abstract

In this work we construct Gaussian beam approximations to solutions of the high frequency Helmholtz equation with a localized source. Under the assumption of non-trapping rays we show error estimates between the exact outgoing solution and Gaussian beams in terms of the wave number $k$, both for single beams and superposition of beams. The main result is that the relative local $L^2$ error in the beam approximations decay as {$k^{-N/2}$ independent of dimension and presence of caustics, for $N$-th order beams.