Multi-level Monte Carlo Finite Difference and Finite Volume Methods for Stochastic Linear Hyperbolic Systems

Abstract

We consider stochastic linear hyperbolic systems of conservation laws in several space dimensions. We prove existence and uniqueness of a random weak solution and provide estimates for the space-time as well as statistical regularity of the solution in terms of the corresponding estimates for the random input data. Multi-Level Monte Carlo Finite Difference and Finite Volume algorithms are used to approximate such statistical moments in an efficient manner. We present novel probabilistic computational complexity analysis which takes into account the sample path dependent complexity of the underlying FDM/FVM solver, due to the random CFL-restricted time step size on account of the wave speed in a random medium. Error bounds for mean square error vs. expected computational work are obtained. We present numerical experiments with uncertain uniformly as well as log-normally distributed wave speeds that illustrate the theoretical results © Springer-Verlag Berlin Heidelberg 2013.