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
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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.References
- R. Abgrall. A simple, flexible and generic deterministic approach to uncertainty quantification in non-linear problems. Rapport de Recherche, INRIA, 2007.
- Andrea Barth, Christoph Schwab, and Nathaniel Zollinger, Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients, Numer. Math. 119 (2011), no. 1, 123–161. MR 2824857, DOI 10.1007/s00211-011-0377-0
- Qian-Yong Chen, David Gottlieb, and Jan S. Hesthaven, Uncertainty analysis for the steady-state flows in a dual throat nozzle, J. Comput. Phys. 204 (2005), no. 1, 378–398. MR 2121912, DOI 10.1016/j.jcp.2004.10.019
- Giuseppe Da Prato and Jerzy Zabcyk, Stochastic Equations in infinite dimensions, Cambridge Univ. Press (1991).
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2005. MR 2169977, DOI 10.1007/3-540-29089-3
- Weinan E, K. Khanin, A. Mazel, and Ya. Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. of Math. (2) 151 (2000), no. 3, 877–960. MR 1779561, DOI 10.2307/121126
- Mike Giles, Improved multilevel Monte Carlo convergence using the Milstein scheme, Monte Carlo and quasi-Monte Carlo methods 2006, Springer, Berlin, 2008, pp. 343–358. MR 2479233, DOI 10.1007/978-3-540-74496-2_{2}0
- Michael B. Giles, Multilevel Monte Carlo path simulation, Oper. Res. 56 (2008), no. 3, 607–617. MR 2436856, DOI 10.1287/opre.1070.0496
- Edwige Godlewski and Pierre-Arnaud Raviart, Hyperbolic systems of conservation laws, Mathématiques & Applications (Paris) [Mathematics and Applications], vol. 3/4, Ellipses, Paris, 1991. MR 1304494
- E. Godlewski and P. A. Raviart, The numerical solution of multidimensional Hyperbolic Systems of Conservation Laws, Springer-Verlag, Berlin, Heidelberg, New York (1995).
- Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713–1020. MR 1804748, DOI 10.1086/phos.67.4.188705
- S. Heinrich. Multilevel Monte Carlo methods in Large-scale scientific computing, Third international conference LSSC 2001, Sozopol, Bulgaria, 2001, Lecture Notes in Computer Science, Vol. 2170, Springer-Verlag (2001), pp. 58-67.
- Helge Holden and Nils Henrik Risebro, Front tracking for hyperbolic conservation laws, Applied Mathematical Sciences, vol. 152, Springer-Verlag, New York, 2002. MR 1912206, DOI 10.1007/978-3-642-56139-9
- H. Holden and N. H. Risebro, Conservation laws with a random source, Appl. Math. Optim. 36 (1997), no. 2, 229–241. MR 1455435, DOI 10.1007/s002459900061
- H. Holden, T. Lindstrøm, B. Øksendal, J. Ubøe, and T.-S. Zhang, The Burgers equation with a noisy force and the stochastic heat equation, Comm. Partial Differential Equations 19 (1994), no. 1-2, 119–141. MR 1257000, DOI 10.1080/03605309408821011
- Dietmar Kröner, Numerical schemes for conservation laws, Wiley-Teubner Series Advances in Numerical Mathematics, John Wiley & Sons, Ltd., Chichester; B. G. Teubner, Stuttgart, 1997. MR 1437144
- R.A. LeVeque, Numerical Solution of Hyperbolic Conservation Laws, Cambridge Univ. Press, 2005.
- W. A. Light and E. W. Cheney, Approximation theory in tensor product spaces, Lecture Notes in Mathematics, vol. 1169, Springer-Verlag, Berlin, 1985. MR 817984, DOI 10.1007/BFb0075391
- G. Lin, C. H. Su, and G. E. Karniadakis, The stochastic piston problem, Proc. Natl. Acad. Sci. USA 101 (2004), no. 45, 15840–15845. MR 2099197, DOI 10.1073/pnas.0405889101
- S. Mishra and Ch. Schwab, Entropy Stability of stochastic Galerkin and inifinite dimensional hyperbolic systems obtained from general moment closures, Working Paper, SAM, 2011.
- S. Mishra, Ch. Schwab and J. Šukys, Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions, Report 2011-53, SAM, February 2011 (in review).
- Gaël Poëtte, Bruno Després, and Didier Lucor, Uncertainty quantification for systems of conservation laws, J. Comput. Phys. 228 (2009), no. 7, 2443–2467. MR 2501693, DOI 10.1016/j.jcp.2008.12.018
- Tobias von Petersdorff and Christoph Schwab, Sparse finite element methods for operator equations with stochastic data, Appl. Math. 51 (2006), no. 2, 145–180. MR 2212311, DOI 10.1007/s10492-006-0010-1
- Ch. Schwab and S. Tokareva, High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data, Report SAM 2011-53.
- Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779, DOI 10.1007/978-1-4612-0873-0
- Radu-Alexandru Todor, A new approach to energy-based sparse finite-element spaces, IMA J. Numer. Anal. 29 (2009), no. 1, 72–85. MR 2470940, DOI 10.1093/imanum/drm041
- J. Tryoen, O. Le Maître, M. Ndjinga, and A. Ern, Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems, J. Comput. Phys. 229 (2010), no. 18, 6485–6511. MR 2660316, DOI 10.1016/j.jcp.2010.05.007
- Xiaoliang Wan and George Em Karniadakis, Long-term behavior of polynomial chaos in stochastic flow simulations, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 41-43, 5582–5596. MR 2243318, DOI 10.1016/j.cma.2005.10.016
- William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3
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
- MR Author ID: 305221
- Email: smishra@sam.math.ethz.ch
- 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.
- © Copyright 2012 American Mathematical Society
- Journal: Math. Comp. 81 (2012), 1979-2018
- MSC (2010): Primary 65N30, 65M06, 35L65
- DOI: https://doi.org/10.1090/S0025-5718-2012-02574-9
- MathSciNet review: 2945145