Stabilized multilevel Monte Carlo method for stiff stochastic differential equations

A multilevel Monte Carlo (MLMC) method for mean square stable stochastic differential equations with multiple scales is proposed. For such problems, that we call stiff, the performance of MLMC methods based on classical explicit methods deteriorates because of the time step restriction to resolve the fastest scales that prevents to exploit all the levels of the MLMC approach. We show that by switching to explicit stabilized stochastic methods and balancing the stabilization procedure simultaneously with the hierarchical sampling strategy of MLMC methods, the computational cost for stiff systems is significantly reduced, while keeping the computational algorithm fully explicit and easy to implement. Numerical experiments on linear and nonlinear stochastic differential equations and on a stochastic partial differential equation illustrate the performance of the stabilized MLMC method and corroborate our theoretical findings. (C) 2013 Elsevier Inc. All rights reserved

Uncertainty Quantification by MLMC and Local Time-stepping For Wave Propagation

Because of their robustness, efficiency and non-intrusiveness, Monte Carlo methods are probably the most popular approach in uncertainty quantification to computing expected values of quantities of interest (QoIs). Multilevel Monte Carlo (MLMC) methods significantly reduce the computational cost by distributing the sampling across a hierarchy of discretizations and allocating most samples to the coarser grids. For time dependent problems, spatial coarsening typically entails an increased time-step. Geometric constraints, however, may impede uniform coarsening thereby forcing some elements to remain small across all levels. If explicit time-stepping is used, the time-step will then be dictated by the smallest element on each level for numerical stability. Hence, the increasingly stringent CFL condition on the time-step on coarser levels significantly reduces the advantages of the multilevel approach. By adapting the time-step to the locally refined elements on each level, local time-stepping (LTS) methods permit to restore the efficiency of MLMC methods even in the presence of complex geometry without sacrificing the explicitness and inherent parallelism

Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions

The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. This property is discussed for approximations of infinite-dimensional stochastic differential equations and necessary and sufficient conditions ensuring mean-square stability are given. They are applied to typical discretization schemes such as combinations of spectral Galerkin, finite element, Euler-Maruyama, Milstein, Crank-Nicolson, and forward and backward Euler methods. Furthermore, results on the relation to stability properties of corresponding analytical solutions are provided. Simulations of the stochastic heat equation illustrate the theory.Comment: 22 pages, 4 figures; deleted a section; shortened the presentation of results; corrected typo

Uncertainty quantification by multilevel Monte Carlo and local time-stepping

Because of their robustness, efficiency, and non intrusiveness, Monte Carlo methods are probablythe most popular approach in uncertainty quantification for computing expected values of quantitiesof interest. Multilevel Monte Carlo (MLMC) methods significantly reduce the computational costby distributing the sampling across a hierarchy of discretizations and allocating most samples tothe coarser grids. For time dependent problems, spatial coarsening typically entails an increasedtime step. Geometric constraints, however, may impede uniform coarsening thereby forcing someelements to remain small across all levels. If explicit time-stepping is used, the time step will thenbe dictated by the smallest element on each level for numerical stability. Hence, the increasinglystringent CFL condition on the time step on coarser levels significantly reduces the advantages of themultilevel approach. To overcome that bottleneck we propose to combine the multilevel approach ofMLMC with local time-stepping. By adapting the time step to the locally refined elements on eachlevel, the efficiency of MLMC methods is restored even in the presence of complex geometry withoutsacrificing the explicitness and inherent parallelism. In a careful cost comparison, we quantify thereduction in computational cost for local refinement either inside a small fixed region or towards areentrant corner

Rapid covariance-based sampling of linear SPDE approximations in the multilevel Monte Carlo method

The efficient simulation of the mean value of a non-linear functional of the solution to a linear stochastic partial differential equation (SPDE) with additive Gaussian noise is considered. A Galerkin finite element method is employed along with an implicit Euler scheme to arrive at a fully discrete approximation of the mild solution to the equation. A scheme is presented to compute the covariance of this approximation, which allows for rapid sampling in a Monte Carlo method. This is then extended to a multilevel Monte Carlo method, for which a scheme to compute the cross-covariance between the approximations at different levels is presented. In contrast to traditional path-based methods it is not assumed that the Galerkin subspaces at these levels are nested. The computational complexities of the presented schemes are compared to traditional methods and simulations confirm that, under suitable assumptions, the costs of the new schemes are significantly lower.Comment: 18 pages, 5 figures; numerical simulations revised, implementation section added; To appear in Monte Carlo and Quasi-Monte Carlo Methods - MCQMC, Rennes, France, July 201