Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data

Mishra, S.; Schwab, Ch.

doi:10.1090/S0025-5718-2012-02574-9

Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data
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by S. Mishra and Ch. Schwab PDF

Math. Comp. 81 (2012), 1979-2018 Request permission

Abstract:

We consider scalar hyperbolic conservation laws in spatial dimension $d\geq 1$ with stochastic initial data. We prove existence and uniqueness of a random-entropy solution and give sufficient conditions on the initial data that ensure the existence of statistical moments of any order $k$ of this random entropy solution. We present a class of numerical schemes of multi-level Monte Carlo Finite Volume (MLMC-FVM) type for the approximation of the ensemble average of the random entropy solutions as well as of their $k$-point space-time correlation functions. These schemes are shown to obey the same accuracy vs. work estimate as a single application of the finite volume solver for the corresponding deterministic problem. Numerical experiments demonstrating the efficiency of these schemes are presented. In certain cases, statistical moments of discontinuous solutions are found to be more regular than pathwise solutions.

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