Multifidelity Monte Carlo Estimation of Variance and Sensitivity Indices

Qian, Elizabeth; Peherstorfer, Benjamin; O’Malley, Daniel; Vesselinov, Velimir V.; Willcox, Karen

doi:10.1137/17m1151006

Cited by 63 publications

(51 citation statements)

References 48 publications

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“…In recent years, there has been an increased attention in the machine learning community to develop predictive methods that enable the effective fusion of variable fidelity information sources. Some techniques are specially tailored to estimate the uncertainty in the model predictions [41,14,2,27,30] and parameter estimation [17]. To predict the model output, Kennedy and O'Hagan [16] first proposed to use Gaussian process priors to perform multi-fidelity regression.…”

Section: Motivationmentioning

confidence: 99%

Multi-fidelity classification using Gaussian processes: Accelerating the prediction of large-scale computational models

Costabal

Perdikaris

Kuhl

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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Machine learning techniques typically rely on large datasets to create accurate classifiers. However, there are situations when data is scarce and expensive to acquire. This is the case of studies that rely on state-of-the-art computational models which typically take days to run, thus hindering the potential of machine learning tools. In this work, we present a novel classifier that takes advantage of lower fidelity models and inexpensive approximations to predict the binary output of expensive computer simulations. We postulate an autoregressive model between the different levels of fidelity with Gaussian process priors. We adopt a fully Bayesian treatment for the hyper-parameters and use Markov Chain Mont Carlo samplers. We take advantage of the probabilistic nature of the classifier to implement active learning strategies. We also introduce a sparse approximation to enhance the ability of themulti-fidelity classifier to handle large datasets. We test these multi-fidelity classifiers against their single-fidelity counterpart with synthetic data, showing a median computational cost reduction of 23% for a target accuracy of 90%. In an application to cardiac electrophysiology, the multi-fidelity classifier achieves an F1 score, the harmonic mean of precision and recall, of 99.6% compared to 74.1% of a single-fidelity classifier when both are trained with 50 samples. In general, our results show that the multi-fidelity classifiers outperform their single-fidelity counterpart in terms of accuracy in all cases. We envision that this new tool will enable researchers to study classification problems that would otherwise be prohibitively expensive. Source code is available at https://github.com/fsahli/MFclass.

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Section: Motivationmentioning

confidence: 99%

Multi-fidelity classification using Gaussian processes: Accelerating the prediction of large-scale computational models

Costabal

Perdikaris

Kuhl

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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“…The first test is a well-known problem in the context of multifidelity methods [3]. There, we consider the expectation, variance, and Sobol index estimators.…”

Section: Choosing the Computational Budgetmentioning

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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“…However, recent works have proposed different ways to estimate PDFs and shown very promising results . Extensions for sensitivity analysis have also recently been made …”

Section: Frequentist Approachesmentioning

confidence: 99%

“…[21,22] Extensions for sensitivity analysis have also recently been made. [23] Since its introduction for geometric integration, [1] MLMC has been applied to several problems, such as path simulations in finance, [2] subsurface flow, [24][25][26] and inviscid incompressible flow. [27,28] The MLMC method has gained tremendous traction in recent years and several extensions have been proposed quite recently, see for example, Giles, [18] for a detailed review including recent developments.…”

Section: Multilevel Monte Carlomentioning

confidence: 99%

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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Multifidelity Monte Carlo Estimation of Variance and Sensitivity Indices

Cited by 63 publications

References 48 publications

Multi-fidelity classification using Gaussian processes: Accelerating the prediction of large-scale computational models

Multi-fidelity classification using Gaussian processes: Accelerating the prediction of large-scale computational models

High-dimensional and higher-order multifidelity Monte Carlo estimators

Multifidelity approaches for uncertainty quantification

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