Bayesian Treed Multivariate Gaussian Process With Adaptive Design: Application to a Carbon Capture Unit

Konomi, Bledar A.; Karagiannis, Georgios; Sarkar, Avik; X, Sun; Lin, Guang

doi:10.1080/00401706.2013.879078

Cited by 23 publications

(33 citation statements)

References 28 publications

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“…In fact, our examples imply that higher-order PC coefficients may be sparser in the spatial domain and adequately approximated by lower-degree polynomial series. The proposed method can be possibly extended to consider discontinuity by using binary tree partitioning (Chipman et al, 1998;Konomi et al, 2014) or capture smaller-scale variations, unexplained by the gPC part, by modeling the total truncation error term as a Gaussian process (O'Hagan, 1978;Bilionis et al, 2013). We believe that the former can lead to simpler gPC expansions while the latter can improve the estimates.…”

Section: Discussionmentioning

confidence: 99%

A Bayesian mixed shrinkage prior procedure for spatial–stochastic basis selection and evaluation of gPC expansions: Applications to elliptic SPDEs

Karagiannis

Konomi

Lin

2015

Journal of Computational Physics

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(2015) 'A Bayesian mixed shrinkage prior procedure for spatialstochastic basis selection and evaluation of gPC expansions : applications to elliptic SPDEs.', Journal of computational physics., 284 . pp. 528-546. Further information on publisher's website: 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. AbstractWe propose a new fully Bayesian method to efficiently obtain the spectral representation of a spatial random field, which can conduct spatial-stochastic basis selection and evaluation of generalized Polynomial Chaos (gPC) expansions when the number of the available basis functions is significantly larger than the size of the training data-set. We develop a fully Bayesian stochastic procedure, called mixed shrinkage prior (MSP), which performs both basis selection and coefficient evaluation simultaneously. MSP involves assigning a prior probability on the gPC structure and assigning conjugate priors on the expansion coefficients that can be thought of as mixtures of Ridge-LASSO shrinkage priors, in augmented form. The method offers a number of advantages over existing compressive sensing methods in gPC literature, such that it recovers possible sparse structures in both stochastic and spatial domains while the resulted expansion can be re-used directly to economically obtain results at any spatial input values. Yet, it inherits all the advantages of Bayesian model uncertainty methods, e.g. accounts for uncertainty about basis significance and provides interval estimation through posterior distributions. A unique highlight of the MSP procedure is that it can address heterogeneous sparsity in the spatial domain for different random dimensions. Furthermore, it yields a compromise between Ridge and LASSO regressions, and hence combines a weak (l 2 -norm) and strong (l 1 -norm) shrinkage, in an adaptive, data-driven manner. We demonstrate the good performance of the proposed method, and compare it against other existing compressive sensing ones on elliptic stochastic partial differential equations.

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Section: Discussionmentioning

confidence: 99%

A Bayesian mixed shrinkage prior procedure for spatial–stochastic basis selection and evaluation of gPC expansions: Applications to elliptic SPDEs

Karagiannis

Konomi

Lin

2015

Journal of Computational Physics

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“…So it is challenging to run a large number of simulations to study the behaviors of the sorbent distribution under different operating conditions. Therefore, the Gaussian process model is used instead as an effective tool for the uncertainty quantification purpose [5]. The training data set was on a 46×101 input-time grid, and we randomly held out 4 input points on the same time grid for evaluating model performances.…”

Section: The Regenerator Of a Carbon Capture Unitmentioning

confidence: 99%

“…For example, [4] proposed a stationary multi-output Gaussian process emulator based on separable cross-covariance. Also based on separable cross-covariance, [5] generalized the work in [6] to a Bayesian Treed Multivariate Gaussian process model, accounting for both the nonstationarity and the multivariate features of the data.…”

Section: Introductionmentioning

confidence: 99%

Full scale multi-output Gaussian process emulator with nonseparable auto-covariance functions

Zhang

Konomi

Sang

et al. 2015

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. AbstractGaussian process emulator with separable covariance function has been utilized extensively in modeling large computer model outputs. The assumption of separability imposes constraints on the emulator and may negatively affect its performance in some applications where separability may not hold. We propose a multi-output Gaussian process emulator with a nonseparable auto-covariance function to avoid limitations of using separable emulators. In addition, to facilitate the computation of nonseparable emulator, we introduce a new computational method, referred to as the Full-Scale approximation method with block modulating function (FSA-Block) approach. The FSA-Block approach is very flexible and can apply to partially separable covariance models. We illustrate the effectiveness of our method through simulation studies and compare it with emulator with separable covariances. We also apply our method to a real computer code of the carbon capture system.

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“…It requires the users to resolve the stochastic problem in each of the random element. A few recent works on the development of efficient stochastic algorithms for handling discontinuities also exist [20][21][22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

An Adaptive WENO Collocation Method for Differential Equations with Random Coefficients

et al. 2016

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Abstract:The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the smoothness requirement; thus a direct application of the method will cause poor performance and slow convergence rate due to the well known Gibbs phenomenon. To address the issue, we propose an adaptive high-order multi-element stochastic collocation scheme by incorporating a WENO (Weighted Essentially non-oscillatory) interpolation procedure and an adaptive mesh refinement (AMR) strategy. The proposed multi-element stochastic collocation scheme requires only repetitive runs of an existing deterministic solver at each interpolation point, similar to the Monte Carlo method. Furthermore, the scheme takes advantage of robustness and the high-order nature of the WENO interpolation procedure, and efficacy and efficiency of the AMR strategy. When the proposed scheme is applied to stochastic problems with non-smooth solutions, the Gibbs phenomenon is mitigated by the WENO methodology in the random space, and the errors around discontinuities in the stochastic space are significantly reduced by the AMR strategy. The numerical experiments for some benchmark stochastic problems, such as the Kraichnan-Orszag problem and Burgers' equation with random initial conditions, demonstrate the reliability, efficiency and efficacy of the proposed scheme.

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Bayesian Treed Multivariate Gaussian Process With Adaptive Design: Application to a Carbon Capture Unit

Cited by 23 publications

References 28 publications

A Bayesian mixed shrinkage prior procedure for spatial–stochastic basis selection and evaluation of gPC expansions: Applications to elliptic SPDEs

A Bayesian mixed shrinkage prior procedure for spatial–stochastic basis selection and evaluation of gPC expansions: Applications to elliptic SPDEs

Full scale multi-output Gaussian process emulator with nonseparable auto-covariance functions

An Adaptive WENO Collocation Method for Differential Equations with Random Coefficients

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