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

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.