Multifidelity importance sampling

Peherstorfer, Benjamin; Cui, Tiangang; Marzouk, Youssef; Willcox, Karen

doi:10.1016/j.cma.2015.12.002

Cited by 118 publications

(90 citation statements)

References 41 publications

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“…The surrogate models are constructed with supervised learning techniques. Another body of work combines the high-fidelity model with a surrogate model in the context of the Monte Carlo method with importance sampling [31,30,13,41,38].…”

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confidence: 99%

Optimal Model Management for Multifidelity Monte Carlo Estimation

Peherstorfer¹,

Willcox²,

Gunzburger³

2016

SIAM J. Sci. Comput.

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218

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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mentioning

confidence: 99%

Optimal Model Management for Multifidelity Monte Carlo Estimation

Peherstorfer¹,

Willcox²,

Gunzburger³

2016

SIAM J. Sci. Comput.

Self Cite

218

233

View full text Add to dashboard Cite

show abstract

“…The MLIS method uses a low‐fidelity model to come up with a suitable IS distribution. Effectively, Peherstorfer et al propose to sample the low‐fidelity model first and then to construct the IS distribution by fitting a Gaussian mixture model to the samples of the low‐fidelity model that fall into the failure domain.…”

Section: Frequentist Approachesmentioning

confidence: 99%

“…Effectively, Peherstorfer et al propose to sample the low‐fidelity model first and then to construct the IS distribution by fitting a Gaussian mixture model to the samples of the low‐fidelity model that fall into the failure domain. While Peherstorfer et al use a simple Monte Carlo scheme to generate low‐fidelity samples in the failure region, there is no reason why more elaborate approaches, such as subset simulation, should not work as well, if not better. For the sake of completeness, we also would like to mention here that the MLIS approach has been extended to make use of multiple low‐fidelity models as well …”

Section: Frequentist Approachesmentioning

confidence: 99%

“…It is commonly known that the standard Monte Carlo estimator suffers from slow convergence, especially when the objective is to compute rare event probabilities as in (3). In simple terms, one can explain this shortcoming by the small number of samples reaching the rare event region.…”

Section: Multilevel Importance Samplingmentioning

confidence: 99%

“…All methods in this category are essentially variance reduction techniques which use low‐fidelity models to reduce the variance of a statistical estimator. Examples are multilevel Monte Carlo (MLMC) and multilevel importance sampling (MLIS). In contrast, methods that belong to the Bayesian category require assumptions about either the functional relationship between model input and output or the relationship between the model output of the different levels or both.…”

Section: Introductionmentioning

confidence: 99%

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Multifidelity approaches for uncertainty quantification

et al. 2019

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The aim of this paper is to give an overview of different multifidelity uncertainty quantification (UQ) schemes. Therefore, different views on multifidelity UQ approaches from a frequentist, Bayesian, and possibilistic perspective are provided and recent developments are discussed. Differences as well as similarities between the methods are highlighted and strategies to construct low‐fidelity models are explained. In addition, two state‐of‐the‐art examples to showcase the capabilities of these methods and the tremendous reduction of computational costs that can be achieved when using these approaches are provided.

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Reduced order modeling of random linear dynamical systems based on a new a posteriori error bound

Hossain

Ghosh

2018

Numerical Meth Engineering

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Reduced order models (ROMs) are becoming increasingly useful for saving computational cost in response prediction of vibrating systems. In a number of applications such as uncertainty quantification, ROMs require robustness over a wide variation of parameters. Accordingly, often they are classified as local and global, based on their performance in the parametric domain. Availability of an error bound of a ROM helps in achieving this robustness, mainly by allowing adaptivity. In this work, for a linear random dynamical system, first, an a posteriori error bound is developed based on the residual in the governing differential equation. Next, based on this error bound, two adaptive methods are proposed for building robust ROMs, that is, one for local, and another for global. While both methods are based on a greedy search approach, a modification is proposed in the training stage of the global ROM for accelerated convergence. These methods are then applied to an uncertainty quantification problem in a statistical simulation framework, and accordingly, two algorithms are developed. A detailed numerical study on vibration of a bladed disk assembly is conducted to study the accuracy and efficiency of the proposed error bound and adaptive ROMs. It is found that these adaptive ROMs provide a considerable speed-up in estimating the probability of failure. KEYWORDS greedy search, probabilistic mechanics, proper orthogonal decomposition, reduced order model, stochastic structural dynamics Abbreviations: HDM, higher dimensional model; POD, proper orthogonal decomposition; ROM, reduced order model; UQ, uncertainty quantification. Int J Numer Methods Eng. 2018;116:741-758.wileyonlinelibrary.com/journal/nme

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Multifidelity importance sampling

Cited by 118 publications

References 41 publications

Optimal Model Management for Multifidelity Monte Carlo Estimation

Optimal Model Management for Multifidelity Monte Carlo Estimation

Multifidelity approaches for uncertainty quantification

Reduced order modeling of random linear dynamical systems based on a new a posteriori error bound

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