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


Authors: S. Mishra and Ch. Schwab
Journal: Math. Comp. 81 (2012), 1979-2018
MSC (2010): Primary 65N30, 65M06, 35L65
DOI: https://doi.org/10.1090/S0025-5718-2012-02574-9
Published electronically: April 5, 2012
MathSciNet review: 2945145
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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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Additional Information

S. Mishra
Affiliation: Seminar for Applied Mathematics, ETH, HG G. 57.2, Rämistrasse 101, Zürich, Switzerland
Email: smishra@sam.math.ethz.ch

Ch. Schwab
Affiliation: Seminar for Applied Mathematics, ETH, HG G. 57.1, Rämistrasse 101, Zürich, Switzerland
Email: smishra@sam.math.ethz.ch

DOI: https://doi.org/10.1090/S0025-5718-2012-02574-9
Received by editor(s): August 31, 2010
Received by editor(s) in revised form: May 25, 2011
Published electronically: April 5, 2012
Additional Notes: The work of Ch. Schwab was supported in part by ERC grant no. 247277. Ch. Schwab and S. Mishra acknowledge also partial support from ETH grant no. CH1-03 10-1. S. Mishra wishes to thank Claude J. Gittelson for useful discussions.
Article copyright: © Copyright 2012 American Mathematical Society

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