Convergence analysis of multilevel Monte Carlo variance estimators and application for random obstacle problems

Bierig, Claudio; Chernov, Alexey

doi:10.1007/s00211-014-0676-3

Cited by 48 publications

(55 citation statements)

References 24 publications

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“…A direct treatment of stochastic obstacles is contained in [10]. The solution u of the stochastic obstacle problem (1.4) not only depends on x ∈ D, but also on the "stochastic parameter" ω ∈ Ω.…”

Section: Introductionmentioning

confidence: 99%

Multilevel Monte Carlo Finite Element Methods for Stochastic Elliptic Variational Inequalities

Kornhuber

Schwab²,

Wolf³

2014

SIAM J. Numer. Anal.

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Abstract. Multi-Level Monte-Carlo Finite Element (MLMC-FE) methods for the solution of stochastic elliptic variational inequalities are introduced, analyzed, and numerically investigated. Under suitable assumptions on the random diffusion coefficient, the random forcing function, and the deterministic obstacle, we prove existence and uniqueness of solutions of "mean-square" and "pathwise" formulations. Suitable regularity results for deterministic, elliptic obstacle problems lead to uniform pathwise error bounds, providing optimal-order error estimates of the statistical error and upper bounds for the corresponding computational cost for classical Monte-Carlo and novel MLMC-FE methods. Utilizing suitable multigrid solvers for the occurring sample problems, in two space dimensions MLMC-FE methods then provide numerical approximations of the expectation of the random solution with the same order of efficiency as for a corresponding deterministic problem, up to logarithmic terms. Our theoretical findings are illustrated by numerical experiments.

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Section: Introductionmentioning

confidence: 99%

Multilevel Monte Carlo Finite Element Methods for Stochastic Elliptic Variational Inequalities

Kornhuber

Schwab²,

Wolf³

2014

SIAM J. Numer. Anal.

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“…• The most appropriate way to compute the variance, as shown by Bierig and Chernov [36] is instead to compute, at each level, the sample variance and compute the differences between the two estimators at the different levels. Compared to the previous method, instead of using a single mean (given by the MLMC estimator), a mean at each level is computed and subtracted to each result at that level.…”

Section: Acknowledgmentsmentioning

confidence: 99%

On the predictivity of pore-scale simulations: Estimating uncertainties with multilevel Monte Carlo

Icardi

Boccardo

Tempone

2016

Advances in Water Resources

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A fast method with tunable accuracy is proposed to estimate errors and uncertainties in porescale and Digital Rock Physics (DRP) problems. The overall predictivity of these studies can be, in fact, hindered by many factors including sample heterogeneity, computational and imaging limitations, model inadequacy and not perfectly known physical parameters. The typical objective of pore-scale studies is the estimation of macroscopic effective parameters such as permeability, effective diffusivity and hydrodynamic dispersion. However, these are often non-deterministic quantities (i.e., results obtained for specific pore-scale sample and setup are not totally reproducible by another "equivalent" sample and setup). The stochastic nature can arise due to the multi-scale heterogeneity, the computational and experimental limitations in considering large samples, and the complexity of the physical models. These approximations, in fact, introduce an error that, being dependent on a large number of complex factors, can be modeled as random. We propose a general simulation tool, based on multilevel Monte Carlo, that can reduce drastically the computational cost needed for computing accurate statistics of effective parameters and other quantities of interest, under any of these random errors. This is, to our knowledge, the first attempt to include Uncertainty Quantification (UQ) in pore-scale physics and simulation. The method can also provide estimates of the discretization error and it is tested on three-dimensional transport problems in heterogeneous materials, where the sampling procedure is done by generation algorithms able to reproduce realistic consolidated and unconsolidated random sphere and ellipsoid packings and arrangements. A totally automatic workflow is developed in an opensource code [1], that include rigid body physics and random packing algorithms, unstructured mesh discretization, finite volume solvers, extrapolation and post-processing techniques. The proposed method can be efficiently used in many porous media applications for problems such as stochastic homogenization/upscaling, propagation of uncertainty from microscopic fluid and rock properties to macro-scale parameters, robust estimation of Representative Elementary Volume size for arbitrary physics.

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“…This is for example the case in the context of cardiac modeling [9]. Moreover, the estimation of high-order moments requires special care [10,11].…”

Section: Introductionmentioning

confidence: 99%

High-dimensional and higher-order multifidelity Monte Carlo estimators

Quaglino

Pezzuto

Krause

2019

Journal of Computational Physics

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Multifidelity Monte Carlo methods rely on a hierarchy of possibly less accurate but statistically correlated simplified or reduced models, in order to accelerate the estimation of statistics of high-fidelity models without compromising the accuracy of the estimates. This approach has recently gained widespread attention in uncertainty quantification [1]. This is partly due to the availability of optimal strategies for the estimation of the expectation of scalar quantities-of-interest [2]. In practice, the optimal strategy for the expectation is also used for the estimation of variance and sensitivity indices [3]. However, a general strategy is still lacking for vector-valued problems, nonlinearly statistically-dependent models, and estimators for which a closed-form expression of the error is unavailable. The focus of the present work is to generalize the standard multifidelity estimators to the above cases. The proposed generalized estimators lead to an optimization problem that can be solved analytically and whose coefficients can be estimated numerically with few runs of the highand low-fidelity models. We analyze the performance of the proposed approach on a selected number of experiments, with a particular focus on cardiac electrophysiology, where a hierarchy of physics-based low-fidelity models is readily available.

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Convergence analysis of multilevel Monte Carlo variance estimators and application for random obstacle problems

Cited by 48 publications

References 24 publications

Multilevel Monte Carlo Finite Element Methods for Stochastic Elliptic Variational Inequalities

Multilevel Monte Carlo Finite Element Methods for Stochastic Elliptic Variational Inequalities

On the predictivity of pore-scale simulations: Estimating uncertainties with multilevel Monte Carlo

High-dimensional and higher-order multifidelity Monte Carlo estimators

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