Uncertainty propagation using infinite mixture of Gaussian processes and variational Bayesian inference

Chen, Peng; Zabaras, Nicholas; Bilionis, Ilias

doi:10.1016/j.jcp.2014.12.028

Cited by 55 publications

(39 citation statements)

References 36 publications

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“…The implementation, however, is rather technical. For more details see the publications of Bilionis in the subject, [30,31,32,33,34]. In our numerical examples we use M = 100.…”

Section: Epistemic Uncertainty On the Solution Of A Stochastic Optimimentioning

confidence: 99%

Extending Expected Improvement for High-Dimensional Stochastic Optimization of Expensive Black-Box Functions

Pandita

Bilionis

Panchal

2016

Journal of Mechanical Design

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Design optimization under uncertainty is notoriously difficult when the objective function is expensive to evaluate. State-of-the-art techniques, e.g, stochastic optimization or sampling average approximation, fail to learn exploitable patterns from collected data and require an excessive number of objective function evaluations. There is a need for techniques that alleviate the high cost of information acquisition and select sequential simulations optimally. In the field of deterministic single-objective unconstrained global optimization, the Bayesian global optimization (BGO) approach has been relatively successful in addressing the information acquisition problem. BGO builds a probabilistic surrogate of the expensive objective function and uses it to define an information acquisition function (IAF) whose role is to quantify the merit of making new objective evaluations. Specifically, BGO iterates between making the observations with the largest expected IAF and rebuilding the probabilistic surrogate, until a convergence criterion is met. In this work, we extend the expected improvement (EI) IAF to the case of design optimization under uncertainty wherein the EI policy is reformulated to filter out parametric and measurement uncertainties. To increase the robustness of our approach in the low sample regime, we employ a fully Bayesian interpretation of Gaussian processes by constructing a particle approximation of the posterior of its hyperparameters using adaptive Markov chain Monte Carlo. We verify and validate our approach by solving two synthetic optimization problems under uncertainty and demonstrate it by solving the oil-well-placement problem with uncertainties in the permeability field and the oil price time series.

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“…The implementation, however, is rather technical. For more details see the publications of Bilionis in the subject, [30,31,32,33,34]. In our numerical examples we use M = 100.…”

Section: Epistemic Uncertainty On the Solution Of A Stochastic Optimimentioning

confidence: 99%

Extending Expected Improvement for High-Dimensional Stochastic Optimization of Expensive Black-Box Functions

Pandita

Bilionis

Panchal

2016

Journal of Mechanical Design

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“…Note that in our framework, the components of mixture (2.1) are computer models (simulators), unlike other works [18,19,20] in the literature where the components are different statistical models referring to the same computer model.…”

Section: Basic Formulationmentioning

confidence: 99%

Selection of polynomial chaos bases via Bayesian model uncertainty methods with applications to sparse approximation of PDEs with stochastic inputs

Karagiannis

Lin

2014

Journal of Computational Physics

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Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractFor many real systems, several computer models may exist with different physics and predictive abilities.To achieve more accurate simulations/predictions, it is desirable for these models to be properly combined and calibrated. We propose the Bayesian calibration of computer model mixture method which relies on the idea of representing the real system output as a mixture of the available computer model outputs with unknown input dependent weight functions. The method builds a fully Bayesian predictive model as an emulator for the real system output by combining, weighting, and calibrating the available models in the Bayesian framework. Moreover, it fits a mixture of calibrated computer models that can be used by the domain scientist as a mean to combine the available computer models, in a flexible and principled manner, and perform reliable simulations. It can address realistic cases where one model may be more accurate than the others at different input values because the mixture weights, indicating the contribution of each model, are functions of the input. Inference on the calibration parameters can consider multiple computer models associated with different physics. The method does not require knowledge of the fidelity order of the models.We provide a technique able to mitigate the computational overhead due to the consideration of multiple computer models that is suitable to the mixture model framework. We implement the proposed method in a real world application involving the Weather Research and Forecasting large-scale climate model.

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“…Because the surrogate model can be queried very cheaply, one can use it as a replacement of the original simulator and perform UQ tasks using MC techniques. Popular choices for surrogate models in the literature include, Gaussian processes [11,12,13,14,15], polynomial chaos expansions [16,17,18,19], radial basis functions [20] and relevance vector machines [21]. Despite their success, these methods become intractable for problems in which the number of input stochastic dimensions is large.…”

Section: Introductionmentioning

confidence: 99%

Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification

Tripathy

Bilionis

2018

Journal of Computational Physics

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181

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State-of-the-art computer codes for simulating real physical systems are often characterized by vast number of input parameters. Performing uncertainty quantification (UQ) tasks with Monte Carlo (MC) methods is almost always infeasible because of the need to perform hundreds of thousands or even millions of forward model evaluations in order to obtain convergent statistics. One, thus, tries to construct a cheap-to-evaluate surrogate model to replace the forward model solver. For systems with large numbers of input parameters, one has to deal with the curse of dimensionality -the exponential increase in the volume of the input space, as the number of parameters increases linearly. Suitable dimensionality reduction techniques are used to address the curse of dimensionality. A popular class of dimensionality reduction methods are those that attempt to recover a low dimensional representation of the high dimensional feature space. However, such methods often tend to overestimate the intrinsic dimensionality of the input feature space. In this work, we demonstrate the use of deep neural networks (DNN) to construct surrogate models for numerical simulators.We parameterize the structure of the DNN in a manner that lends the DNN surrogate the interpretation of recovering a low dimensional nonlinear 1

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Uncertainty propagation using infinite mixture of Gaussian processes and variational Bayesian inference

Cited by 55 publications

References 36 publications

Extending Expected Improvement for High-Dimensional Stochastic Optimization of Expensive Black-Box Functions

Extending Expected Improvement for High-Dimensional Stochastic Optimization of Expensive Black-Box Functions

Selection of polynomial chaos bases via Bayesian model uncertainty methods with applications to sparse approximation of PDEs with stochastic inputs

Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification

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