We extend the Multi-Level Monte Carlo (MLMC) algorithm of [19] in order to quantify uncertainty in the solutions of multi-dimensional hyperbolic systems of conservation laws with uncertain initial data. The algorithm is presented and several issues arising in the massively parallel numerical implementation are addressed. In particular, we present a novel load balancing procedure that ensures scalability of the MLMC algorithm on massively parallel hardware. A new code ALSVID-UQ is described and applied to simulate uncertain solutions of the Euler equations and ideal MHD equations. Numerical experiments showing the robustness, efficiency and scalability of the proposed algorithm are presented.

The initial data and bottom topography, used as inputs in shallow water models, are prone to uncertainty due to measurement errors. We model this uncertainty statistically in terms of random shallow water equations. We extend the Multi-Level Monte Carlo (MLMC) algorithm to numerically approximate the random shallow water equations efficiently. The MLMC algorithm is suitably modified to deal with uncertain (and possibly uncorrelated) data on each node of the underlying topography grid by the use of a hierarchical topography representation. Numerical experiments in one and two space dimensions are presented to demonstrate the efficiency of the MLMC algorithm.

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We consider the very challenging problem of efficient uncertainty quantification for acoustic wave propagation in a highly heterogeneous, possibly layered, random medium, characterized by possibly anisotropic, piecewise log-exponentially distributed Gaussian random fields. A multi-level Monte Carlo finite volume method is proposed, along with a novel, bias-free upscaling technique that allows to represent the input random fields, generated using spectral FFT methods, efficiently. Combined together with a novel, dynamic load balancing algorithm that scales to massively parallel computing architectures, the proposed method is able to robustly compute uncertainty for highly realistic random subsurface formations that can contain a very high number (millions) of sources of uncertainty. Numerical experiments, in both two and three space dimensions, illustrating the efficiency of the method are presented. 1 2 SIDDHARTHA MISHRA, CHRISTOPH SCHWAB, AND JONASŠUKYS Ω X(ω) E dP(ω) < ∞.By L 1 (Ω, E) = L 1 ((Ω, F, P), E) we denote the set of all (equivalence classes of) Bochner integrable, E-valued random fields X, equipped with the norm X L 1 (Ω,E) = Ω X(ω) E dP(ω) = E( X E ).

The Multi-Level Monte Carlo finite volumes (MLMC-FVM) algorithm was shown to be a robust and fast solver for uncertainty quantification in the solutions of multi-dimensional systems of stochastic conservation laws. A novel load balancing procedure is used to ensure scalability of the MLMC algorithm on massively parallel hardware. We describe this procedure together with other arising challenges in great detail. Finally, numerical experiments in multi-dimensions showing strong and weak scaling of our implementation are presented.