Quantifying uncertain system outputs via the multilevel Monte Carlo method — Part I: Central moment estimation

Krumscheid, Sebastian; Nobile, Fabio; Pisaroni, Michele

doi:10.1016/j.jcp.2020.109466

Cited by 27 publications

(38 citation statements)

References 13 publications

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“…In [14] we presented an extension of the classical MLMC concept for the estimation of arbitrary order central statistical moments…”

Section: C-mlmcmentioning

confidence: 99%

“…where h p denotes an appropriate h-statistic of order p. One of the method's key ingredients is the use of such h-statistics [20] as unbiased central moment estimators with minimal variance for the level-wise contributions. In [21,22] we further extended the MLMC concept to accurately approximate the Cumulative distribution function (CDF) of a random system output and robustness measures such as quantiles (also known as Value at Risk, VaR) or coherent risk measures [23] such as the conditional value at risk (CVaR [24]). The α-quantile q α is given by:…”

Section: C-mlmcmentioning

confidence: 99%

“…The idea proposed in [21,22] to approximate simultaneously the CDF, quantiles q α and CV aR α of a certain output QoI Q is to first approximate the function H Q : Θ → R defined as:…”

Section: C-mlmcmentioning

confidence: 99%

“…In a previous work we presented a robust and effective Continuation Multi Level Monte Carlo (C-MLMC) methodology [12], following the ideas of Collier et al [13], which is capable of propagating operational and geometrical uncertainties in compressible inviscid flow problems and allows an effective reduction of the overall computational efforts compared to MC and classical MLMC. In [14] we introduced an extension of the standard MLMC approach to compute arbitrary order central statistical moments.…”

Section: Introductionmentioning

confidence: 99%

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Continuation Multilevel Monte Carlo Evolutionary Algorithm for Robust Aerodynamic Shape Design

Pisaroni

Nobile

Leyland

2019

Journal of Aircraft

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The majority of problems in aircraft production and operation require decisions made in the presence of uncertainty. For this reason aerodynamic designs obtained with traditional deterministic optimization techniques seeking only optimality in a specific set of conditions may have very poor off-design performances or may even be unreliable. In this work, we present a novel approach for robust and reliability-based design optimization of aerodynamic shapes based on the combination of single and multi-objective Evolutionary Algorithms and a Continuation Multi Level Monte Carlo methodology to compute objective functions and constraints that involve statistical moments or statistical quantities such as quantiles, also called Value at risk, (VaR) and Conditional Value at Risk (CVaR) without relying on derivatives and meta-models. Detailed numerical studies are presented for the RAE-2822 transonic airfoil design affected by geometrical and operational uncertainties.

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“…In [14] we presented an extension of the classical MLMC concept for the estimation of arbitrary order central statistical moments…”

Section: C-mlmcmentioning

confidence: 99%

Section: C-mlmcmentioning

confidence: 99%

“…The idea proposed in [21,22] to approximate simultaneously the CDF, quantiles q α and CV aR α of a certain output QoI Q is to first approximate the function H Q : Θ → R defined as:…”

Section: C-mlmcmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Continuation Multilevel Monte Carlo Evolutionary Algorithm for Robust Aerodynamic Shape Design

Pisaroni

Nobile

Leyland

2019

Journal of Aircraft

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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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Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Zhang

2020

WIREs Computational Stats

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Uncertainty quantification (UQ) includes the characterization, integration, and propagation of uncertainties that result from stochastic variations and a lack of knowledge or data in the natural world. Monte Carlo (MC) method is a sampling‐based approach that has widely used for quantification and propagation of uncertainties. However, the standard MC method is often time‐consuming if the simulation‐based model is computationally intensive. This article gives an overview of modern MC methods to address the existing challenges of the standard MC in the context of UQ. Specifically, multilevel Monte Carlo (MLMC) extending the concept of control variates achieves a significant reduction of the computational cost by performing most evaluations with low accuracy and corresponding low cost, and relatively few evaluations at high accuracy and corresponding high cost. Multifidelity Monte Carlo (MFMC) accelerates the convergence of standard Monte Carlo by generalizing the control variates with different models having varying fidelities and varying computational costs. Multimodel Monte Carlo method (MMMC), having a different setting of MLMC and MFMC, aims to address the issue of UQ and propagation when data for characterizing probability distributions are limited. Multimodel inference combined with importance sampling is proposed for quantifying and efficiently propagating the uncertainties resulting from small data sets. All of these three modern MC methods achieve a significant improvement of computational efficiency for probabilistic UQ, particularly uncertainty propagation. An algorithm summary and the corresponding code implementation are provided for each of the modern MC methods. The extension and application of these methods are discussed in detail. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Monte Carlo Methods Statistical and Graphical Methods of Data Analysis > Sampling

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Quantifying uncertain system outputs via the multilevel Monte Carlo method — Part I: Central moment estimation

Cited by 27 publications

References 13 publications

Continuation Multilevel Monte Carlo Evolutionary Algorithm for Robust Aerodynamic Shape Design

Continuation Multilevel Monte Carlo Evolutionary Algorithm for Robust Aerodynamic Shape Design

High-dimensional and higher-order multifidelity Monte Carlo estimators

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

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