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Title: Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems

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Journal Article: Publisher's Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 282; Journal Issue: C; Related Information: CHORUS Timestamp: 2017-07-04 09:23:05; Journal ID: ISSN 0021-9991
Country of Publication:
United States

Citation Formats

Lu, Fei, Morzfeld, Matthias, Tu, Xuemin, and Chorin, Alexandre J. Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems. United States: N. p., 2015. Web. doi:10.1016/
Lu, Fei, Morzfeld, Matthias, Tu, Xuemin, & Chorin, Alexandre J. Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems. United States. doi:10.1016/
Lu, Fei, Morzfeld, Matthias, Tu, Xuemin, and Chorin, Alexandre J. 2015. "Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems". United States. doi:10.1016/
title = {Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems},
author = {Lu, Fei and Morzfeld, Matthias and Tu, Xuemin and Chorin, Alexandre J.},
abstractNote = {},
doi = {10.1016/},
journal = {Journal of Computational Physics},
number = C,
volume = 282,
place = {United States},
year = 2015,
month = 2

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Publisher's Version of Record at 10.1016/

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Cited by: 5works
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  • Polynomial chaos expansions are used to reduce the computational cost in the Bayesian solutions of inverse problems by creating a surrogate posterior that can be evaluated inexpensively. We show, by analysis and example, that when the data contain significant information beyond what is assumed in the prior, the surrogate posterior can be very different from the posterior, and the resulting estimates become inaccurate. One can improve the accuracy by adaptively increasing the order of the polynomial chaos, but the cost may increase too fast for this to be cost effective compared to Monte Carlo sampling without a surrogate posterior.
  • Abstract not provided.
  • We consider a Bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field, endowed with a hierarchical Gaussian process prior. Computational challenges in this construction arise from the need for repeated evaluations of the forward model (e.g., in the context of Markov chain Monte Carlo) and are compounded by high dimensionality of the posterior. We address these challenges by introducing truncated Karhunen-Loeve expansions, based on the prior distribution, to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the supportmore » of the prior. We seek a solution of this problem using Galerkin projection on a polynomial chaos basis, and use the solution to construct a reduced-dimensionality surrogate posterior density that is inexpensive to evaluate. We demonstrate the formulation on a transient diffusion equation with prescribed source terms, inferring the spatially-varying diffusivity of the medium from limited and noisy data.« less
  • The growing need for uncertainty analysis of complex computational models has led to an expanding use of meta-models across engineering and sciences. The efficiency of meta-modeling techniques relies on their ability to provide statistically-equivalent analytical representations based on relatively few evaluations of the original model. Polynomial chaos expansions (PCE) have proven a powerful tool for developing meta-models in a wide range of applications; the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials. The classical PCE approach nevertheless faces the “curse of dimensionality”, namely themore » exponential increase of the basis size with increasing input dimension. To address this limitation, the sparse PCE technique has been proposed, in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm. An alternative for developing meta-models with polynomial functions in high-dimensional problems is offered by the newly emerged low-rank approximations (LRA) approach. By exploiting the tensor–product structure of the multivariate basis, LRA can provide polynomial representations in highly compressed formats. Through extensive numerical investigations, we herein first shed light on issues relating to the construction of canonical LRA with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions. Specifically, we examine the selection of optimal rank, stopping criteria in the updating of the polynomial coefficients and error estimation. In the sequel, we confront canonical LRA to sparse PCE in structural-mechanics and heat-conduction applications based on finite-element solutions. Canonical LRA exhibit smaller errors than sparse PCE in cases when the number of available model evaluations is small with respect to the input dimension, a situation that is often encountered in real-life problems. By introducing the conditional generalization error, we further demonstrate that canonical LRA tend to outperform sparse PCE in the prediction of extreme model responses, which is critical in reliability analysis.« less