Uncertainty quantification in kinetic and hyperbolic problems

Multilevel Monte-Carlo FV and FT Methods for hyperbolic PDEs with random input data

Christoph Schwab



We consider random scalar, nonlinear hyperbolic conservation laws in spatial dimension d ? 1 with bounded random flux functions. There exists a unique random entropy solution (i.e., a strongly measurable mapping from a probability space into C([0, T];L1(Rd))) with finite second moments. We present a convergence analysis of a Multi-Level Monte-Carlo Front-Tracking (MLMCFT) algorithm. It is based on “pathwise" application of the Front-Tracking Method for deterministic SCLs. We compare the MLMCFT algorithms to Multi-Level Monte-Carlo Finite- Volume methods. Due to the absence of a CFL time step restriction in the pathwise front tracking scheme, we can prove favourable complexity estimates: in spatial dimension d ? 2, the mean field of the random entropy solution can be approximated numerically with (up to logarithmic terms) the same complexity as the solution of one instance of the deterministic problem, on the same mesh. We then present results on large scale simulations of MLMC for linear acoustic wave propagation in heterogeneous media with log-gaussian random coefficients. Here, conventional explicit timestepping schemes encounter the CFL constraint which, due to the lognormal gaussian constitutive parameter, is random. A probabilistic complexity analysis is presented. Implementation with a novel adaptive load balancing algorithm achieves near linear strong scaling. Joint work with S. Mishra, N. Risebro, J. Sukys and F. Weber.