Robust multi-level Monte Carlo finite volume methods for systems of hyperbolic conservation laws with random input data

A mathematical formulation of conservation and of balance laws with random input data is reviewed, specifically with random initial conditions, source terms, flux functions and coefficients. The concept of random entropy solution is specified and its statistical moments such as the mean and the variance are presented. For scalar conservation laws in multi-dimensions, recent results by Mishra and Schwab on the existence and uniqueness of random entropy solutions and their statistical moments are presented; these results are then extended to linear hyperbolic systems of conservation laws. 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 discussed. Asymptotic mean square error estimates for combined Monte Carlo Finite Volume Method (MC-FVM) are obtained for scalar and linear hyperbolic systems of conservation laws with random inputs. Following work of Mishra and Schwab, an improved multi-level version of the Monte Carlo Finite Volume Method (MLMC-FVM) is proposed and asymptotic mean square error bounds are presented. Asymptotic mean square error versus expected work estimates, derived using novel probabilistic computational complexity analysis taking into account the sample path dependent complexity of the underlying FVM solve, indicate superiority of MLMC-FVM over plain MC-FVM, under comparable assumptions on the random input data. In particular, it is shown that approximations of statistical moments converge essentially at the same rate as a single solve of a deterministic problem using FVM. Extensions of the proposed algorithms to nonlinear hyperbolic systems of balance laws are outlined. Implementation aspects of MLMC-FVM are discussed, including large scale random number generation, numerically stable and computationally efficient statistical estimators, bias-free multi-resolution discretizations of random source terms and coefficients, and novel static/adaptive load balancing techniques for scalability of MLMC-FVM on emerging massively parallel computing platforms. A new code ALSVID-UQ implementing the MLMC-FVM is described and applied to simulate uncertain solutions of the Euler equations of gas dynamics and ideal magnetohydrodynamics equations of plasma physics with uncertain initial data and equation of state, shallow water equations of oceanography with uncertain bottom topography, acoustic wave equation with log-normally distributed heterogeneous coefficients, and Buckley-Leverett equations of two phase flows with uncertain fluxes. Numerical experiments in one, two and three dimensions with a very large number of uncertainty sources