Multi-level Monte Carlo Finite Volume Methods for Uncertainty Quantification in Nonlinear Systems of Balance Laws

  • Siddhartha MishraEmail author
  • Christoph Schwab
  • Jonas Šukys
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 92)


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.


Monte Carlo Finite Volume Method Shallow Water Equation Polynomial Chaos Expansion Stochastic Collocation Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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.


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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Siddhartha Mishra
    • 1
    Email author
  • Christoph Schwab
    • 2
  • Jonas Šukys
    • 3
  1. 1.SAM, ETH ZürichZürichSwitzerland
  2. 2.SAM, ETH ZürichZürichSwitzerland
  3. 3.SAM, ETH ZürichZürichSwitzerland

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