Multilevel Monte Carlo Finite Volume Methods for Random Conservation Laws with Discontinuous Flux
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We consider a random scalar hyperbolic conservation law in one spatial dimension with bounded random flux functions which are discontinuous in the spatial variable. We show that there exists a unique random entropy solution to the conservation law corresponding to the specific entropy condition used to solve the deterministic case. Assuming the empirical convergence rates of the underlying deterministic problem over a broad range of parameters, we present a convergence analysis of a multilevel Monte Carlo Finite Volume Method (MLMC-FVM). It is based on a pathwise application of the finite volume method for the deterministic conservation laws. We show that the work required to compute the MLMC-FVM solutions is an order lower than the work required to compute the Monte Carlo Finite Volume Method solutions with equal accuracy.
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