A continuation multilevel Monte Carlo algorithm

Collier, Nathan; Haji-Ali, Abdul-Lateef; Nobile, Fabio; Schwerin, Erik von; Tempone, Raúl

doi:10.1007/s10543-014-0511-3

Cited by 110 publications

(152 citation statements)

References 29 publications

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“…Some way of balancing the error sources is necessary to make the error asymptotically vanish. One might want the errors to converge with different rates, for example to make the multilevel Monte Carlo estimator asymptotically a normal distribution, which requires the stochastic error to decrease with any strictly greater rate than the numerical error (see [8,Lemma 7.1]). However, in practice, an even balance between the errors is justified since it prevents one source of error to dominate the other.…”

Section: Monte Carlo Methods For Estimating Failure Probabilitymentioning

confidence: 99%

“…The sample size N L for the deepest level L in the MLMC estimator is typically too small for estimating Var [Y L ] reliably using sample variance. See [8,12] for elaborate discussions about this. The approach used here is to estimate the variance for the lower levels and extrapolate it to the deeper levels.…”

Section: Multilevel Monte Carlo Methodsmentioning

confidence: 99%

“…An improvement of the MC estimator for application on numerical simulations of controllable numerical accuracy is the multilevel Monte Carlo (MLMC) estimator [20] first introduced in the context of differential equations in [15]. It has since then been applied to elliptic PDEs in [3,7,25,29] and has been further analyzed and extended in [8,18,19]. It exploits the convergence of numerical solutions with respect to some discretization parameter h (typically mesh size) and uses a series of corrector estimators of increasing cost but decreasing variance to allow for redistribution of the variance reduction effort to cheap low-accuracy problems.…”

Section: Introductionmentioning

confidence: 99%

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Multilevel Monte Carlo methods for computing failure probability of porous media flow systems

Fagerlund

Hellman

Målqvist

et al. 2016

Advances in Water Resources

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Multilevel Monte Carlo methods for computing failure probability of porous media flow systems. Advances in Water Resources AbstractWe study improvements of the standard and multilevel Monte Carlo method for point evaluation of the cumulative distribution function (failure probability) applied to porous media two-phase flow simulations with uncertain permeability. To illustrate the methods, we study an injection scenario where we consider sweep efficiency of the injected phase as quantity of interest and seek the probability that this quantity of interest is smaller than a critical value. In the sampling procedure, we use computable error bounds on the sweep efficiency functional to identify small subsets of realizations to solve highest accuracy by means of what we call selective refinement. We quantify the performance gains possible by using selective refinement in combination with both the standard and multilevel Monte Carlo method. We also identify issues in the process of practical implementation of the methods. We conclude that significant savings in computational cost are possible for failure probability estimation in a realistic setting using the selective refinement technique, both in combination with standard and multilevel Monte Carlo.

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Section: Monte Carlo Methods For Estimating Failure Probabilitymentioning

confidence: 99%

Section: Multilevel Monte Carlo Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multilevel Monte Carlo methods for computing failure probability of porous media flow systems

Fagerlund

Hellman

Målqvist

et al. 2016

Advances in Water Resources

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“…Equation (14) shows that the total mass of solute in the domain increases through solute injection across the upper boundary and decreases through the first order reaction. We also note that in this particular problem, the more CO 2 is absorbed through the upper boundary, the more the process is considered to be e cient.…”

Section: Mathematical Model: Surface Flux In C-ed Processes In Porousmentioning

confidence: 99%

“…For modelling the correlation of Z, i.e., expression (21), we use the following exponential covariance function [29,54,14,13]:…”

Section: Generation Of Random Permeability Fieldsmentioning

confidence: 99%

Gaussian process modelling for uncertainty quantification in convectively-enhanced dissolution processes in porous media

Crevillén-García

Wilkinson

Shah

et al. 2017

Advances in Water Resources

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(2017) Gaussian process modelling for uncertainty quantification in convectively-enhanced dissolution processes in porous media. Advances in Water Resources, 99 . pp. 1-14. Permanent WRAP URL:http://wrap.warwick.ac.uk/85680 Copyright and reuse:The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions. Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription. AbstractNumerical groundwater flow and dissolution models of physico-chemical processes in deep aquifers are usually subject to uncertainty in one or more of the model input parameters. This uncertainty is propagated through the equations and needs to be quantified and characterised in order to rely on the model outputs. In this paper we present a Gaussian process emulation method as a tool for performing uncertainty quantification in mathematical models for convection and dissolution processes in porous media. One of the advantages of this method is its ability to significantly reduce the computational cost of an uncertainty analysis, while yielding accurate results, compared to classical Monte Carlo methods. We apply the methodology to a model of convectively-enhanced dissolution processes occurring during carbon capture and storage. In this model, the Gaussian process methodology fails due to the presence of multiple branches of solutions emanating from a bifurcation point, i.e., two equilibrium states exist rather than one. To overcome this issue we use a classifier as a precursor to the Gaussian process emulation, after which we are able to successfully perform a full uncertainty analysis in the vicinity of the bifurcation point.

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Multilevel double loop Monte Carlo and stochastic collocation methods with importance sampling for Bayesian optimal experimental design

Beck

Dia

Espath

et al. 2020

Numerical Meth Engineering

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An optimal experimental set-up maximizes the value of data for statistical inferences and predictions. An optimal set-up is particularly important for experiments that are time consuming or expensive to perform. In the context of partial differential equations (PDEs), multilevel methods have been proven to dramatically reduce the computational complexity of their single-level counterparts. Here, two multilevel methods, which efficiently compute the expected information gain using a Kullback-Leibler divergence measure in simulation-based Bayesian optimal experimental design, are proposed. The first method is a multilevel double loop Monte Carlo (MLDLMC) with importance sampling that greatly reduces the computational work of the inner loop. The second proposed method is a multilevel double loop stochastic collocation (MLDLSC) with importance sampling, which performs a high-dimensional integration by deterministic quadrature on sparse grids. In both methods, the Laplace approximation is used as an effective means of importance sampling, and the optimal values of the method parameters are determined by minimizing the average computational work, subject to a desired error tolerance. The computational efficiencies of the methods are demonstrated by computing the expected information gain from an electrical impedance tomography experiment where the fiber orientation in composite laminate materials are inferred through Bayesian inversion. MLDLSC performs better than MLDLMC when the regularity of the underlying computational model, with respect to the additive noise and the unknown parameters, can be exploited.

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A continuation multilevel Monte Carlo algorithm

Cited by 110 publications

References 29 publications

Multilevel Monte Carlo methods for computing failure probability of porous media flow systems

Multilevel Monte Carlo methods for computing failure probability of porous media flow systems

Gaussian process modelling for uncertainty quantification in convectively-enhanced dissolution processes in porous media

Multilevel double loop Monte Carlo and stochastic collocation methods with importance sampling for Bayesian optimal experimental design

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