Multilevel Monte Carlo methods

Giles, Michael B.

doi:10.1017/s096249291500001x

Cited by 543 publications

(465 citation statements)

References 101 publications

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“…In the multilevel Monte Carlo framework, one usually considers a particular statistic, such as evaluations of the cumulative distribution function (CDF: Elfverson et al , 2014; Giles et al , ; Wilson and Baker, ), probability density function (PDF: Bierig and Chernov, ) or expected values (Giles, ; Cliffe et al , ), selecting the ensemble sizes/finest level of resolution so that the overall multilevel estimator produces an efficient and accurate approximation.…”

Section: Introductionmentioning

confidence: 99%

“…An alternative approach to this could be approximating each verification or scoring measure, such as the calibration or sharpness, individually and directly from the multilevel hierarchy of ensembles. For example, one could use a MLMC approximation for the CDF (Elfverson et al , 2014; Giles et al , ) to help compute a rank histogram to evaluate the forecast calibration. Each different MLMC approximation typically comes with a framework to implement it, such as a smoothing scheme in the former of those two studies.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the calibration of multilevel Monte Carlo ensemble forecasts

Gregory

Cotter

2017

Quart J Royal Meteoro Soc

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The multilevel Monte Carlo method can efficiently compute statistical estimates of discretized random variables for a given error tolerance. Traditionally, only a certain statistic is computed from a particular implementation of multilevel Monte Carlo. This article considers the multilevel case in which one wants to verify and evaluate a single ensemble that forms an empirical approximation to many different statistics, namely an ensemble forecast. We propose a simple algorithm that, in the univariate case, allows one to derive a statistically consistent single ensemble forecast from the hierarchy of ensembles that are formed during an implementation of multilevel Monte Carlo. This ensemble forecast then allows the entire multilevel hierarchy of ensembles to be evaluated using standard ensemble forecast verification techniques. We demonstrate the case of evaluating the calibration of the forecast.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the calibration of multilevel Monte Carlo ensemble forecasts

Gregory

Cotter

2017

Quart J Royal Meteoro Soc

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“…Finally, to show the effectiveness of our approach in approximating challenging quantities of interest, we compute the parametric von Mises stress for the optimal design using the biresolution approach and compare it with MC and multilevel Monte Carlo (MLMC) simulations . Figure shows a realization of high‐ and low‐resolution von Mises stresses associated with 100 × 100 and 10 × 10 meshes on one of 43 quadrature points.…”

Section: Numerical Illustrationmentioning

confidence: 99%

Parametric topology optimization with multiresolution finite element models

Keshavarzzadeh

Kirby

Narayan

2019

Numerical Meth Engineering

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Summary We present a methodical procedure for topology optimization under uncertainty with multiresolution finite element (FE) models. We use our framework in a bifidelity setting where a coarse and a fine mesh corresponding to low‐ and high‐resolution models are available. The inexpensive low‐resolution model is used to explore the parameter space and approximate the parameterized high‐resolution model and its sensitivity, where parameters are considered in both structural load and stiffness. We provide error bounds for bifidelity FE approximations and their sensitivities and conduct numerical studies to verify these theoretical estimates. We demonstrate our approach on benchmark compliance minimization problems, where we show significant reduction in computational cost for expensive problems such as topology optimization under manufacturing variability, reliability‐based topology optimization, and three‐dimensional topology optimization while generating almost identical designs to those obtained with a single‐resolution mesh. We also compute the parametric von Mises stress for the generated designs via our bifidelity FE approximation and compare them with standard Monte Carlo simulations. The implementation of our algorithm, which extends the well‐known 88‐line topology optimization code in MATLAB, is provided.

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“…The multi-level aspect of this method can be seen as a close cousin of the multi-level Monte Carlo methods described in e.g. [8,12,16].…”

Section: Multi-level Collocation Polynomial Chaos Expansion Collocatimentioning

confidence: 99%

“…Multi-level Monte Carlo uses a hierarchy of forward models of increasing computational complexity (e.g. uniform spatial grid refinement) and calculates the expectation as being that of the coarsest level, plus a correction based on the difference in expectation between consecutive levels [16]. Sensitivity derivative methods calculate a correction based on the sensitivity of the forward model with respect to the stochastic parameters about the mean parameters.…”

Section: Introductionmentioning

confidence: 99%

Accelerating Monte Carlo estimation with derivatives of high-level finite element models

Hauseux

Hale

Bordas

2017

Computer Methods in Applied Mechanics and Engineering

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In this paper we demonstrate the ability of a derivative-driven Monte Carlo estimator to accelerate the propagation of uncertainty through two high-level non-linear finite element models. The use of derivative information amounts to a correction to the standard Monte Carlo estimation procedure that reduces the variance under certain conditions. We express the finite element models in variational form using the high-level Unified Form Language (UFL). We derive the tangent linear model automatically from this high-level description and use it to efficiently calculate the required derivative information. To study the effectiveness of the derivative-driven method we consider two stochastic PDEs; a onedimensional Burgers equation with stochastic viscosity and a three-dimensional geometrically non-linear Mooney-Rivlin hyperelastic equation with stochastic density and volumetric material parameter. Our results show that for these problems the first-order derivative-driven Monte Carlo method is around one order of magnitude faster than the standard Monte Carlo method and at the cost of only one extra tangent linear solution per estimation problem. We find similar trends when comparing with a modern non-intrusive multi-level polynomial chaos expansion method. We parallelise the task of the repeated forward model evaluations across a cluster using the ipyparallel and mpi4py software tools. A complete working example showing the solution of the stochastic viscous Burgers equation is included as supplementary material.

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Multilevel Monte Carlo methods

Cited by 543 publications

References 101 publications

On the calibration of multilevel Monte Carlo ensemble forecasts

On the calibration of multilevel Monte Carlo ensemble forecasts

Parametric topology optimization with multiresolution finite element models

Accelerating Monte Carlo estimation with derivatives of high-level finite element models

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