Gabriela Malenová: Uncertainty Quantification for High Frequency Waves

We consider the scalar wave equation with highly oscillatory initial conditions where the wave speed and/or the initial data are stochastic. A spectral asymptotic method for the uncertainty quantification of high frequency waves subject to stochastic uncertainty is presented. The method consists of Gaussian beam superposition in the deterministic space and collocation on sparse grids in the stochastic space. In the presence of stochastic regularity, the method exhibits a faster rate of convergence compared to the Monte Carlo techniques. We show theoretical and numerical evidence that certain quadratic quantities of interest are indeed smooth in stochastic space, and exhibit fast convergence.