Multi-level Monte Carlo finite difference and finite volume methods for stochastic linear hyperbolic systems

Abstract

We consider stochastic multi-dimensional linear hyperbolic systems of conservation laws. We prove existence and uniqueness of a random weak solution, provide estimates for the regularity of the solution in terms of regularities of input data, and show existence of statistical moments. Bounds for mean square error vs. expected work are proved for the Multi-Level Monte Carlo Finite Volume algorithm which is used to approximate the moments of the solution. Using our implementation called ALSVID-UQ, numerical experiments for acoustic wave equation with uncertain uniformly and log-normally distributed coefficients are conducted.