For initial boundary value problems of the linear wave equation whose coefficients depend on countably many random parameters e.g. via a Karhunen-Loeve or a wavelet expansion,we show analyticity of weak solutions with respect to these parameters.The stochastic problem is reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameter space. This problem is discretized by polynomial chaos approximations in the countably many parameters. We present convergence rates for best N-term approximations of polynomial chaos type, for the resulting space-time approximations and compare to Multi-Level Monte Carlo (MLMC) space-time discretizations. We generalize to MLMC discretizations of nonlinear hyperbolic conservation laws with random initial conditions, and report numerical experiments for Compressible Euler and Ideal Magnetohydrodynamics with stochastic data.
Joint work with Siddartha Mishra, Jonas Sukys, David Berhardsgruetter SAM, ETH and with Viet-Ha Hoang, Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371.