Multi-level Monte Carlo Finite Volume Research Articles

Multilevel Monte Carlo finite volume methods for random conservation laws with discontinuous flux

We consider conservation laws with discontinuous flux where the initial datum, the flux function, and the discontinuous spatial dependency coefficient are subject to randomness. We establish a notion of random adapted entropy solutions to these equations and prove well-posedness provided that the spatial dependency coefficient is piecewise constant with finitely many discontinuities. In particular, the setting under consideration allows the flux to change across finitely many points in space whose positions are uncertain. We propose a single- and multilevel Monte Carlo method based on a finite volume approximation for each sample. Our analysis includes convergence rate estimates of the resulting Monte Carlo and multilevel Monte Carlo finite volume methods as well as error versus work rates showing that the multilevel variant outperforms the single-level method in terms of efficiency. We present numerical experiments motivated by two-phase reservoir simulations for reservoirs with varying geological properties.

A multilevel Monte Carlo finite difference method for random scalar degenerate convection–diffusion equations

This paper proposes a finite difference multilevel Monte Carlo algorithm for degenerate parabolic convection–diffusion equations where the convective and diffusive fluxes are allowed to be random. We establish a notion of stochastic entropy solutions to these equations. Our chief goal is to efficiently compute approximations to statistical moments of these stochastic entropy solutions. To this end, we design a multilevel Monte Carlo method based on a finite volume scheme for each sample. We present a novel convergence rate analysis of the combined multilevel Monte Carlo finite volume method, allowing in particular for low [Formula: see text]-integrability of the random solution with [Formula: see text], and low deterministic convergence rates (here, the theoretical rate is [Formula: see text]). We analyze the design and error versus work of the multilevel estimators. We obtain that the maximal rate (based on optimizing possibly the pessimistic upper bounds on the discretization error) is obtained for [Formula: see text], for finite volume convergence rate of [Formula: see text]. We conclude with numerical experiments.

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.

Numerical Solution of Scalar Conservation Laws with Random Flux Functions

We consider scalar hyperbolic conservation laws in several space dimensions, with a class of random (and parametric) flux functions. We propose a Karhunen--Loève expansion on the state space of the random flux. For random flux functions which are continuously differentiable with respect to the state variable $u$, we prove the existence of a unique random entropy solution. Using a Karhunen--Loève spectral decomposition of the random flux into principal components with respect to the state variables, we introduce a family of parametric, deterministic entropy solutions on high-dimensional parameter spaces. We prove bounds on the sensitivity of the parametric and of the random entropy solutions on the Karhunen--Loève parameters. We also outline the convergence analysis for two classes of discretization schemes, the multilevel Monte Carlo finite volume method (MLMCFVM), developed in [S. Mishra and C. Schwab, Math. Comp., 81 (2012), pp. 1979--2018], [S. Mishra, C. Schwab, and J. Šukys, J. Comput. Phys., 231 (2012), pp. 3365--3388], and [S. Mishra, C. Schwab, and J. Šukys, Multi-level Monte Carlo finite volume methods for uncertainty quantification in nonlinear systems of balance laws, in Uncertainty Quantification in Computational Fluid Dynamics, Lecture Notes in Comput. Sci. Eng. 92, Springer, Heidelberg, 2013, pp. 225--294], and the stochastic collocation finite volume method (SCFVM) of [S. Tokareva, Stochastic Finite Volume Methods for Computational Uncertainty Quantification in Hyperbolic Conservation Laws, Ph.D. dissertation, ETH Diss. Nr. 21498, ETH Zürich, Zürich, Switzerland]. (A corrected version is attached.)

Multi-level Monte Carlo finite volume method for shallow water equations with uncertain parameters applied to landslides-generated tsunamis

Two layer Savage–Hutter type shallow water PDEs model flows such as tsunamis generated by rockslides. On account of heterogeneities in the composition of the granular matter, these models contain uncertain parameters like the ratio of densities of layers, Coulomb and interlayer friction. These parameters are modeled statistically and quantifying the resulting solution uncertainty (UQ) is a crucial task in geophysics. We propose a novel paradigm for UQ that combines the recently developed IFCP spatial discretizations with the recently developed Multi-level Monte Carlo (MLMC) statistical sampling method and provides a fast, accurate and computationally efficient framework to compute statistical quantities of interest. Numerical experiments, including realistic simulations of the Lituya Bay mega tsunami, are presented to illustrate the robustness of the proposed UQ algorithm.

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.

Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions

We extend the multi-level Monte Carlo (MLMC) 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 is described and applied to simulate uncertain solutions of the Euler equations and ideal magnetohydrodynamics (MHD) equations. Numerical experiments showing the robustness, efficiency and scalability of the proposed algorithm are presented.

Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data

We consider scalar hyperbolic conservation laws in spatial dimension d ≥ 1 d\geq 1 with stochastic initial data. We prove existence and uniqueness of a random-entropy solution and give sufficient conditions on the initial data that ensure the existence of statistical moments of any order k k of this random entropy solution. We present a class of numerical schemes of multi-level Monte Carlo Finite Volume (MLMC-FVM) type for the approximation of the ensemble average of the random entropy solutions as well as of their k k -point space-time correlation functions. These schemes are shown to obey the same accuracy vs. work estimate as a single application of the finite volume solver for the corresponding deterministic problem. Numerical experiments demonstrating the efficiency of these schemes are presented. In certain cases, statistical moments of discontinuous solutions are found to be more regular than pathwise solutions.

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.