Monte Carlo and multi-level Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws

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 twoand 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. Date: April 3, 2012. 1991 Mathematics Subject Classification. 65M12, 65M22, 65M08, 65M55, 65C05, 35L65.