A Sparse Stochastic Collocation Technique for High-Frequency Wave Propagation with Uncertainty

TL;DR

This paper introduces a combined high-frequency Gaussian beam and sparse stochastic collocation method to efficiently estimate statistical properties of wave solutions with uncertain parameters, even when solutions are highly oscillatory.

Contribution

It develops a novel approach integrating Gaussian beams with sparse stochastic collocation to handle high-frequency wave equations with uncertainty, providing theoretical and numerical validation.

Findings

Quantities of interest are smooth in stochastic space despite oscillatory solutions.

The method outperforms Monte Carlo in efficiency for high-frequency wave problems.

Numerical tests confirm the theoretical advantages of the proposed approach.

Abstract

We consider the wave equation with highly oscillatory initial data, where there is uncertainty in the wave speed, initial phase and/or initial amplitude. To estimate quantities of interest related to the solution and their statistics, we combine a high-frequency method based on Gaussian beams with sparse stochastic collocation. Although the wave solution, $u^\varepsilon$, is highly oscillatory in both physical and stochastic spaces, we provide theoretical arguments and numerical evidence that quantities of interest based on local averages of $|u^\varepsilon|^2$ are smooth, with derivatives in the stochastic space uniformly bounded in $\varepsilon$, where $\varepsilon$ denotes the short wavelength. This observable related regularity makes the sparse stochastic collocation approach more efficient than Monte Carlo methods. We present numerical tests that demonstrate this advantage.