Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

Abstract. A formal mean square error expansion (MSE) is derived for Euler-Maruyama numerical solutions of stochastic differential equations (SDE). The error expansion is used to construct a pathwise, a posteriori, adaptive time-stepping Euler-Maruyama algorithm for numerical solutions of SDE, and the resulting algorithm is incorporated into a multilevel Monte Carlo (MLMC) algorithm for weak approximations of SDE. This gives an efficient MSE adaptive MLMC algorithm for handling a number of low-regularity approximation problems. In low-regularity numerical example problems, the developed adaptive MLMC algorithm is shown to outperform the uniform time-stepping MLMC algorithm by orders of magnitude, producing output whose error with high probability is bounded by TOL ą 0 at the near-optimal MLMC cost rate O`TOL´2 logpTOLq 4˘.

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“…Presently, MLMC is a vibrant and growing research topic, (cf. [10,9,3,4,34,26,13], and references therein).…”

Section: Uniform and Adaptive Time-stepping Mlmcmentioning

confidence: 99%

Construction of a Mean Square Error Adaptive Euler–Maruyama Method With Applications in Multilevel Monte Carlo

Hoel

Häppölä

Tempone

2016

Springer Proceedings in Mathematics &Amp; Statistics

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“…. , α k ∈ R are coefficients that weight the differences y [15,46]. Both correct an estimate of high-fidelity quantities with a sum of estimates of differences of lower-fidelity quantities.…”

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confidence: 99%

Optimal Model Management for Multifidelity Monte Carlo Estimation

Peherstorfer¹,

Willcox²,

Gunzburger³

2016

SIAM J. Sci. Comput.

205

233

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Abstract. This work presents an optimal model management strategy that exploits multifidelity surrogate models to accelerate the estimation of statistics of outputs of computationally expensive high-fidelity models. Existing acceleration methods typically exploit a multilevel hierarchy of surrogate models that follow a known rate of error decay and computational costs; however, a general collection of surrogate models, which may include projection-based reduced models, data-fit models, support vector machines, and simplified-physics models, does not necessarily give rise to such a hierarchy. Our multifidelity approach provides a framework to combine an arbitrary number of surrogate models of any type. Instead of relying on error and cost rates, an optimization problem balances the number of model evaluations across the high-fidelity and surrogate models with respect to error and costs. We show that a unique analytic solution of the model management optimization problem exists under mild conditions on the models. Our multifidelity method makes occasional recourse to the high-fidelity model; in doing so it provides an unbiased estimator of the statistics of the high-fidelity model, even in the absence of error bounds and error estimators for the surrogate models. Numerical experiments with linear and nonlinear examples show that speedups by orders of magnitude are obtained compared to Monte Carlo estimation that invokes a single model only. 1. Introduction. Multilevel techniques have a long and successful history in computational science and engineering, e.g., multigrid for solving systems of equations [8,25,9], multilevel discretizations for representing functions [50,18,10], and multilevel Monte Carlo and multilevel stochastic collocation for estimating mean solutions of partial differential equations (PDEs) with stochastic parameters [27,22,45]. These multilevel techniques typically start with a fine-grid discretization-a high-fidelity model-of the underlying PDE or function. The fine-grid discretization is chosen to guarantee an approximation of the output of interest with the accuracy required by the current problem at hand. Additionally, a hierarchy of coarser discretizations-lowerfidelity surrogate models-is constructed, where a parameter (e.g., mesh width) controls the trade-off between error and computational costs. Changing this parameter gives rise to a multilevel hierarchy of discretizations with known error and cost rates. Multilevel techniques use these error and cost rates to distribute the computational work among the discretizations in the hierarchy, shifting most of the work onto the

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“…In contrast, the variance reduction in MLMC depends on the convergence rate of the FE approximation to the system response, see e.g. [8,15,22,63].…”

Section: Remark 32mentioning

confidence: 99%

Multilevel Estimation of Rare Events

Ullmann¹,

Papaioannou²

2015

SIAM/ASA J. Uncertainty Quantification

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This work is motivated by the need to estimate the probability of rare events in engineering systems with random inputs. We introduce a multilevel estimator which is based on and generalises the idea of subset simulation. The novel estimator employs a hierarchy of approximations to the system response computed with different resolutions. This leads to reduced computational costs compared to subset simulation. We study the statistical properties and implementation details of the proposed estimator. Markov chain Monte Carlo runs are required within the estimator and we demonstrate that the nestedness of the associated multilevel failure domains enables a perfect MCMC simulation without burn-in. We show that nestedness follows from certain simple one-dimensional failure domains. In high dimensions we propose a modification of the multilevel estimator which uses level-dependent stochastic input dimensions. We report on numerical experiments in 1D and 2D physical space; in particular, we estimate rare events arising from a Darcy flow problem with random permeability.

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Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

Cited by 183 publications

References 32 publications

Construction of a Mean Square Error Adaptive Euler–Maruyama Method With Applications in Multilevel Monte Carlo

Construction of a Mean Square Error Adaptive Euler–Maruyama Method With Applications in Multilevel Monte Carlo

Optimal Model Management for Multifidelity Monte Carlo Estimation

Multilevel Estimation of Rare Events

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