Abstract
A mathematical formulation of conservation and of balance laws with random input data, specifically with random initial conditions, random source terms and random flux functions, is reviewed. The concept of random entropy solution is specified. For scalar conservation laws in multi-dimensions, recent results on the existence and on the uniqueness of random entropy solutions with finite variances are presented. The combination of Monte Carlo sampling with Finite Volume Method discretization in space and time for the numerical approximation of the statistics of random entropy solutions is proposed. The finite variance of random entropy solutions is used to prove asymptotic error estimates for combined Monte Carlo Finite Volume Method discretizations of scalar conservation laws with random inputs. A Multi-Level extension of combined Monte Carlo Finite Volume Method (MC-FVM) discretizations is proposed and asymptotic error bounds are presented in the case of scalar, nonlinear hyperbolic conservation laws. Sparse tensor constructions for the computation of compressed approximations of two- and k-point space-time correlation functions of random entropy solutions are introduced.Asymptotic error versus work estimates indicate superiority of Multi-Level versions of MC-FVM over the plain MC-FVM, under comparable assumptions on the random input data. In particular, it is shown that these compressed sparse tensor approximations converge essentially at the same rate as the MLMC-FVM estimators for the mean solutions.Extensions of the proposed algorithms to nonlinear, hyperbolic systems of balance laws are outlined. Multiresolution discretizations of random source terms which are exactly bias-free are indicated.Implementational aspects of these Multi-Level Monte Carlo Finite Volume methods, in particular results on large scale random number generation, scalability and resilience on emerging massively parallel computing platforms, are discussed.
Supported by ERC StG No. 306279 SPARCCLE.
Supported by ERC AdG No. 247277 STAHDPDE.
Supported in part by ETH CHIRP1-03 10-1 and CSCS production project ID S366.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
R. Abgrall. A simple, flexible and generic deterministic approach to uncertainty quantification in non-linear problems. Rapport de Recherche, INRIA, 2007.
ALSVID. Available from http://folk.uio.no/mcmurry/amhd.
ALSVID-UQ. Available from http://www.sam.math.ethz.ch/alsvid-uq.
K. Aziz and A. Settari. Fundamentals of petroleum reservoir simulation. Applied Science Publishers, London, 1979.
A. Barth, Ch. Schwab and N. Zollinger. Multilevel MC Method for Elliptic PDEs with Stochastic Coefficients. Numerische Mathematik, Volume 119(1), pp. 123–161, 2011.
T. J. Barth. Numerical methods for gas-dynamics systems on unstructured meshes. An Introduction to Recent Developments in Theory and Numerics of Conservation Laws, pp. 195–285. Lecture Notes in Computational Science and Engineering volume 5, Springer, Berlin. Eds: D. Kroner, M. Ohlberger, and Rohde, C., 1999.
P. D. Bates, S. N. Lane and R. I. Ferguson. Parametrization, Validation and Uncertainty analysis of CFD models of fluvial and flood hydraulics in natural environments. Computational Fluid Dynamics: Applications in environmental hydraulics, John Wiley and sons, pp. 193–212, 2005.
Q. Y. Chen, D. Gottlieb and J. S. Hesthaven. Uncertainty analysis for steady flow in a dual throat nozzle. J. Comput. Phys, 204, pp. 378–398, 2005.
B. Cockburn and C-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput., 52, pp. 411–435, 1989.
Constantine M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics (2nd Ed.). Springer Verlag, 2005.
Josef Dick, Franes Y. Kuo and Ian H. Sloan. High dimensional integration: the Quasi Monte-Carlo way. Acta Numerica, to appear, 2013.
R. Eymard, T. Gallouët, and R. Herbin. Finite volume methods, in Handbook of numerical analysis, Vol. VII, pp. 713–1020, North-Holland, Amsterdam, 2000.
P. F. Fisher and N. J. Tate. Causes and consequences of error in digital elevation models. Prog. in Phy. Geography, 30(4), pp. 467–489, 2006.
G. Fishman. Monte Carlo. Springer, 1996.
U.S. Fjordholm, S. Mishra, and E. Tadmor. Well-balanced, energy stable schemes for the shallow water equations with varying topology. J. Comput. Phys., 230, pp. 5587–5609, 2011.
F. Fuchs, A. D. McMurry, S. Mishra, N. H. Risebro and K. Waagan. Approximate Riemann solver based high-order finite volume schemes for the Godunov-Powell form of ideal MHD equations in multi-dimensions. Comm. Comput. Phys., 9, pp. 324–362, 2011.
M. Giles. Improved multilevel Monte Carlo convergence using the Milstein scheme. Preprint NA-06/22, Oxford computing lab, Oxford, U.K, 2006.
M. Giles. Multilevel Monte Carlo path simulation. Oper. Res., 56, pp. 607–617, 2008.
E. Godlewski and P.A. Raviart. Hyperbolic Systems of Conservation Laws. Mathematiques et Applications, Ellipses Publ., Paris, 1991.
S. Gottlieb, C. W. Shu and E. Tadmor. High order time discretizations with strong stability property. SIAM. Review, 43, pp. 89–112, 2001.
A. Harten, B. Engquist, S. Osher and S. R. Chakravarty. Uniformly high order accurate essentially non-oscillatory schemes. J. Comput. Phys., pp. 231–303, 1987s.
S. Heinrich. Multilevel Monte Carlo methods. Large-scale scientific computing, Third international conference LSSC 2001, Sozopol, Bulgaria, 2001, Lecture Notes in Computer Science, Vol 2170, pp. 58–67, Springer Verlag, 2001.
P. L’Ecuyer and F. Panneton. Fast Random Number Generators Based on Linear Recurrences Modulo 2: Overview and Comparison. Proceedings of the 2005 Winter Simulation Conference, pp. 110–119, IEEE press, 2005.
P. L’Ecuyer and F. Panneton. Fast Random Number Generators Based on Linear Recurrences Modulo 2. ACM Trans. Math. Software, 32, pp. 1–16, 2006.
R.A. LeVeque. Numerical Solution of Hyperbolic Conservation Laws. Cambridge Univ. Press 2005.
R. LeVeque, D. George and M. Berger. Tsunami modeling with adaptively refined finite volume methods. Acta Numerica, 20, pp. 211–289, 2011.
G. Lin, C.H. Su and G. E. Karniadakis. The stochastic piston problem. PNAS 101, pp. 15840–15845, 2004.
X. Ma and N. Zabaras. An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. J. Comp. Phys, 228, pp. 3084–3113, 2009.
M. Matsumoto and T. Nishimura. Mersenne Twister: a 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Trans. Modeling and Computer Simulation, 8, pp. 3–30, Jan. 1998.
S. Mishra and Ch. Schwab. Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data. Math. Comp. 280(81), pp. 1979–2018, 2012.
S. Mishra, N.H. Risebro, Ch. Schwab and S. Tokareva. Numerical solution of scalar conservation laws with random flux functions. SAM Technical Report No. 2012-35, in review, 2012. Also available from http://www.sam.math.ethz.ch/sam_reports/index.php?id=2012-35.
S. Mishra, Ch. Schwab and J. Šukys. Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions. J. Comp. Phys., 231(8), pp. 3365–3388, 2012.
S. Mishra, Ch. Schwab, and J. Šukys. Multi-level Monte Carlo Finite Volume methods for shallow water equations with uncertain topography in multi-dimensions. SIAM J. Sci. Comput., 34(6), pp. B761–B784, 2012.
S. Pauli and P. Arbenz and Ch. Schwab. Intrinsic fault tolerance of multi level Monte Carlo methods. SAM Technical Report 2012-24, Seminar für Angewandte Mathematik ETH Zürich, 2012. Also available from http://www.sam.math.ethz.ch/sam_reports/index.php?id=2012-24.
G. Poette, B. Després and D. Lucor. Uncertainty quantification for systems of conservation laws. J. Comput. Phys. 228, pp. 2443–2467, 2009.
G.D. Prato and J. Zabcyk, Stochastic Equations in infinite dimensions, Cambridge Univ. Press, 1991.
G. Schmidlin. Fast solution algorithms for integral equations in \({\mathbb{R}}^{3}\) . PhD dissertation ETH Zürich No. 15016, 2003.
G. Schmidlin and Ch. Schwab. Wavelet Galerkin BEM on unstructured meshes by aggregation. LNCSE 20, pp. 369–278, Springer Lecture Notes in CSE, Springer Verlag, Berlin Heidelberg New York, 2002.
Ch. Schwab and S. Tokareva. High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data. ESAIM: Mathematical Modelling and Numerical Analysis, ESAIM: M2AN 47, 807–835 (2013) DOI: 10.1051/m2an/2012060, www.esaim-m2an.org.
C. W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory schemes - II. J. Comput. Phys., 83, pp. 32–78, 1989.
C. W. Shu. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. ICASE Technical report, NASA, 1997.
J. Šukys, S. Mishra, and Ch. Schwab. Static load balancing for multi-level Monte Carlo finite volume solvers. PPAM 2011, Part I, LNCS 7203, pp. 245–254. Springer, Heidelberg, 2012.
J. Tryoen, O. Le Maitre, M. Ndjinga and A. Ern. Intrusive projection methods with upwinding for uncertain non-linear hyperbolic systems. Preprint, 2010.
T. von Petersdorff and Ch. Schwab. Sparse Finite Element Methods for Operator Equations with Stochastic Data, Applications of Mathematics 51(2), pp. 145–180, 2006.
X. Wan and G. E. Karniadakis. Long-term behaviour of polynomial chaos in stochastic flow simulations. Comput. Meth. Appl. Mech. Engg. 195, pp. 5582–5596, 2006.
B. P. Welford. Note on a Method for Calculating Corrected Sums of Squares and Products. Technometrics, 4, pp. 419–420, 1962.
J. A. S. Witteveen, A. Loeven, H. Bijl An adaptive stochastic finite element approach based on Newton-Cotes quadrature in simplex elements. Comput. Fluids, 38, pp. 1270–1288, 2009.
D. Xiu and J. S. Hesthaven. High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput., 27, pp. 1118–1139, 2005.
Brutus, ETH Zürich, de.wikipedia.org/wiki/Brutus_(Cluster).
Cray XE6, Swiss National Supercomputing Center (CSCS), Lugano, www.cscs.ch.
MPI: A Message-Passing Interface Standard. Version 2.2, 2009, available from: http://www.mpi-forum.org/docs/mpi-2.2/mpi22-report.pdf.
Open MPI: Open Source High Performance Computing. Available from http://www.open-mpi.org/.
Acknowledgements
The authors wish to express their gratitude to Mr. Luc Grosheintz, a student in the ETH Zürich MSc Applied Mathematics curriculum for performing the numerics for the sparse two-point correlation computations reported in Sect. 6.8. The authors thank the systems support at ETH Zürich parallel Compute Cluster BRUTUS [49] for their support in the production runs for the present paper, and the staff at the Swiss National Supercomputing center (CSCS) [50] at Lugano for their assistance in the large scale Euler and MHD simulations.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Mishra, S., Schwab, C., Šukys, J. (2013). Multi-level Monte Carlo Finite Volume Methods for Uncertainty Quantification in Nonlinear Systems of Balance Laws. In: Bijl, H., Lucor, D., Mishra, S., Schwab, C. (eds) Uncertainty Quantification in Computational Fluid Dynamics. Lecture Notes in Computational Science and Engineering, vol 92. Springer, Cham. https://doi.org/10.1007/978-3-319-00885-1_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-00885-1_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-00884-4
Online ISBN: 978-3-319-00885-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)