Uncertainty quantification for phase-space boundary integral models of ray propagation

Abstract

Vibrational and acoustic energy distributions of wave fields in the high-frequency regime are often modeled using flow transport equations. This study concerns the case when the flow of rays or non-interacting particles is driven by an uncertain force or velocity field and the dynamics are determined only up to a degree of uncertainty. A boundary integral equation description of wave energy flow along uncertain trajectories in finite two-dimensional domains is presented, which is based on the truncated normal distribution, and interpolates between a deterministic and a completely random description of the trajectory propagation. The properties of the Gaussian probability density function appearing in the model are applied to derive expressions for the variance of a propagated initial Gaussian density in the weak noise case. Numerical experiments are performed to illustrate these findings and to study the properties of the stationary density, which is obtained in the limit of infinitely many reflections at the boundary.

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