Stochastic regularity of general quadratic observables of high frequency waves

TL;DR

This paper investigates the statistical behavior of quadratic functionals of high-frequency wave solutions with uncertain parameters, demonstrating $ ext{ε}$-independent derivative estimates using Gaussian beam approximations.

Contribution

It identifies non-oscillatory quadratic QoIs and proves $ ext{ε}$-independent derivative estimates, advancing the analysis of high-frequency wave uncertainties.

Findings

Established $ ext{ε}$-independent bounds for derivatives of quadratic QoIs

Demonstrated the effectiveness of Gaussian beam approximation in high-frequency regimes

Provided a framework for analyzing statistical properties of wave functionals

Abstract

We consider the wave equation with uncertain initial data and medium, when the wavelength $\varepsilon$ of the solution is short compared to the distance traveled by the wave. We are interested in the statistics for quantities of interest (QoI), defined as functionals of the wave solution, given the probability distributions of the uncertain parameters in the wave equation. Fast methods to compute this statistics require considerable smoothness in the mapping from parameters to the QoI, which is typically not present in the high frequency case, as the oscillations on the $\varepsilon$ scale in the wave field is inherited by the QoIs. The main contribution of this work is to identify certain non-oscillatory quadratic QoIs and show $\varepsilon$-independent estimates for the derivatives of the QoI with respect to the parameters, when the wave solution is replaced by a Gaussian beam approximation.