Jonas Šukys

Multilevel Monte Carlo Finite Volume Methods for Shallow Water Equations with Uncertain Topography in Multi-dimensions

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 multilevel 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.

Static load balancing for multi-level monte carlo finite volume solvers

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.

Multi-level Monte Carlo finite volume methods for uncertainty quantification of acoustic wave propagation in random heterogeneous layered medium

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 recently developed 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.

Adaptive Load Balancing for Massively Parallel Multi-Level Monte Carlo Solvers

The Multi-Level Monte Carlo algorithm was shown to be a robust solver for uncertainty quantification in the solutions of multi-dimensional systems of stochastic conservation laws. For random fluxes or random initial data with large variances, the time step of the explicit time stepping scheme becomes random due to the random CFL stability restriction. Such sample path dependent complexity of the underlying deterministic solver renders our static load balancing of the MLMC algorithm very inefficient. We introduce an adaptive load balancing procedure based on two key ingredients: (1) pre-computation of the time step size for each draw of random inputs (realization), (2) distribution of the samples using the greedy algorithm to “workers” with heterogeneous speeds of execution. Numerical experiments showing strong scaling are presented.

Multi-level Monte Carlo Finite Difference and Finite Volume Methods for Stochastic Linear Hyperbolic Systems

We consider stochastic linear hyperbolic systems of conservation laws in several space dimensions. We prove existence and uniqueness of a random weak solution and provide estimates for the space-time as well as statistical regularity of the solution in terms of the corresponding estimates for the random input data. Multi-Level Monte Carlo Finite Difference and Finite Volume algorithms are used to approximate such statistical moments in an efficient manner. We present novel probabilistic computational complexity analysis which takes into account the sample path dependent complexity of the underlying FDM/FVM solver, due to the random CFL-restricted time step size on account of the wave speed in a random medium. Error bounds for mean square error vs. expected computational work are obtained. We present numerical experiments with uncertain uniformly as well as log-normally distributed wave speeds that illustrate the theoretical results.