Polynomial Chaos Surrogate Construction for Random Fields with Parametric Uncertainty

Mueller, Joy N.; Sargsyan, Khachik; Daniels, Craig J.; Najm, Habib N.

doi:10.1137/23m1613505

Polynomial Chaos Surrogate Construction for Random Fields with Parametric Uncertainty

Journal Article · 02 January 2025 · SIAM/ASA Journal on Uncertainty Quantification

DOI:https://doi.org/10.1137/23m1613505· OSTI ID:2502157

^[1]; ^[1]; Daniels, Craig J. ^[1]; ^[1]

Sandia National Lab. (SNL-CA), Livermore, CA (United States)

Engineering and applied science rely on computational experiments to rigorously study physical systems. The mathematical models used to probe these systems are highly complex, and sampling-intensive studies often require prohibitively many simulations for acceptable accuracy. Surrogate models provide a means of circumventing the high computational expense of sampling such complex models. In particular, polynomial chaos expansions (PCEs) have been successfully used for uncertainty quantification studies of deterministic models where the dominant source of uncertainty is parametric. We discuss an extension to conventional PCE surrogate modeling to enable surrogate construction for stochastic computational models that have intrinsic noise in addition to parametric uncertainty. We develop a PCE surrogate on a joint space of intrinsic and parametric uncertainty, enabled by Rosenblatt transformations, which are evaluated via kernel density estimation of the associated conditional cumulative distributions. Furthermore, we extend the construction to random field data via the Karhunen–Loève expansion. We then take advantage of closed-form solutions for computing PCE Sobol indices to perform a global sensitivity analysis of the model which quantifies the intrinsic noise contribution to the overall model output variance. Additionally, the resulting joint PCE is generative in the sense that it allows generating random realizations at any input parameter setting that are statistically approximately equivalent to realizations from the underlying stochastic model. The method is demonstrated on a chemical catalysis example model and a synthetic example controlled by a parameter that enables a switch from unimodal to bimodal response distributions.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-CA), Livermore, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES). Chemical Sciences, Geosciences & Biosciences Division (CSGB)

Grant/Contract Number:: NA0003525

OSTI ID:: 2502157

Report Number(s):: SAND2025--00499J

Journal Information:: SIAM/ASA Journal on Uncertainty Quantification, Journal Name: SIAM/ASA Journal on Uncertainty Quantification Journal Issue: 1 Vol. 13; ISSN 2166-2525

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

References (55)

Data-driven polynomial chaos expansion for machine learning regression Torre, Emiliano; Marelli, Stefano; Embrechts, Paul Journal of Computational Physics, Vol. 388 https://doi.org/10.1016/j.jcp.2019.03.039	journal	July 2019
Stochastic spectral methods for efficient Bayesian solution of inverse problems Marzouk, Youssef M.; Najm, Habib N.; Rahn, Larry A. Journal of Computational Physics, Vol. 224, Issue 2 https://doi.org/10.1016/j.jcp.2006.10.010	journal	June 2007
Handbook of Uncertainty Quantification Ghanem, Roger; Higdon, David; Owhadi, Houman https://doi.org/10.1007/978-3-319-12385-1	reference-book	January 2017
A spectral surrogate model for stochastic simulators computed from trajectory samples Lüthen, Nora; Marelli, Stefano; Sudret, Bruno Computer Methods in Applied Mechanics and Engineering, Vol. 406 https://doi.org/10.1016/j.cma.2022.115875	journal	March 2023
Bayesian Compressive Sensing Using Laplace Priors Babacan, S. D.; Molina, R.; Katsaggelos, A. K. IEEE Transactions on Image Processing, Vol. 19, Issue 1 https://doi.org/10.1109/TIP.2009.2032894	journal	January 2010
Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems Marzouk, Youssef M.; Najm, Habib N. Journal of Computational Physics, Vol. 228, Issue 6 https://doi.org/10.1016/j.jcp.2008.11.024	journal	April 2009
A Radial Basis Function Method for Global Optimization Gutmann, H. -M. Journal of Global Optimization, Vol. 19, Issue 3, p. 201-227 https://doi.org/10.1023/A:1011255519438	journal	March 2001
A Stochastic Projection Method for Fluid Flow Le Maı̂ tre, Olivier P.; Knio, Omar M.; Najm, Habib N. Journal of Computational Physics, Vol. 173, Issue 2 https://doi.org/10.1006/jcph.2001.6889	journal	November 2001
Sampling via Measure Transport: An Introduction Marzouk, Youssef; Moselhy, Tarek; Parno, Matthew Handbook of Uncertainty Quantification https://doi.org/10.1007/978-3-319-12385-1_23	book	June 2017
Emulation of Stochastic Simulators Using Generalized Lambda Models Zhu, Xujia; Sudret, Bruno SIAM/ASA Journal on Uncertainty Quantification, Vol. 9, Issue 4 https://doi.org/10.1137/20M1337302	journal	January 2021
Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach Ghanem, Roger; Red-Horse, John Physica D: Nonlinear Phenomena, Vol. 133, Issue 1-4 https://doi.org/10.1016/S0167-2789(99)00102-5	journal	September 1999
Building Accurate Emulators for Stochastic Simulations via Quantile Kriging Plumlee, Matthew; Tuo, Rui Technometrics, Vol. 56, Issue 4 https://doi.org/10.1080/00401706.2013.860919	journal	October 2014
Stochastic Finite Elements: A Spectral Approach Ghanem, Roger G.; Spanos, Pol D. Springer New York, NY https://doi.org/10.1007/978-1-4612-3094-6	book	January 1991
Dimensionality Reduction for Complex Models via Bayesian Compressive Sensing Sargsyan, Khachik; Safta, Cosmin; Najm, Habib N. International Journal for Uncertainty Quantification, Vol. 4, Issue 1 https://doi.org/10.1615/Int.J.UncertaintyQuantification.2013006821	journal	January 2014
Stochastic Spectral Embedding Marelli, Stefano; Wagner, P. -R.; Lataniotis, Christos International Journal for Uncertainty Quantification, Vol. 11, Issue 2 https://doi.org/10.1615/Int.J.UncertaintyQuantification.2020034395	journal	January 2021
Design and Analysis of Computer Experiments Sacks, Jerome; Welch, William J.; Mitchell, Toby J. Statistical Science, Vol. 4, Issue 4 https://doi.org/10.1214/ss/1177012413	journal	November 1989
Numerical methods for the discretization of random fields by means of the Karhunen–Loève expansion Betz, Wolfgang; Papaioannou, Iason; Straub, Daniel Computer Methods in Applied Mechanics and Engineering, Vol. 271 https://doi.org/10.1016/j.cma.2013.12.010	journal	April 2014
Surrogate Models for Uncertainty Propagation and Sensitivity Analysis Sargsyan, Khachik Handbook of Uncertainty Quantification https://doi.org/10.1007/978-3-319-12385-1_22	book	January 2017
A multigrid solver for two-dimensional stochastic diffusion equations Le Maı̂tre, O. P.; Knio, O. M.; Debusschere, B. J. Computer Methods in Applied Mechanics and Engineering, Vol. 192, Issue 41-42 https://doi.org/10.1016/S0045-7825(03)00457-2	journal	October 2003
Spectral Representation and Reduced Order Modeling of the Dynamics of Stochastic Reaction Networks via Adaptive Data Partitioning Sargsyan, Khachik; Debusschere, Bert; Najm, Habib SIAM Journal on Scientific Computing, Vol. 31, Issue 6 https://doi.org/10.1137/090747932	journal	January 2010
A review on quantile regression for stochastic computer experiments Torossian, Léonard; Picheny, Victor; Faivre, Robert Reliability Engineering & System Safety, Vol. 201 https://doi.org/10.1016/j.ress.2020.106858	journal	September 2020
Uncertainty Quantification of Detonation Through Adapted Polynomial Chaos Liang, Xiao; Wang, R.; Ghanem, Roger International Journal for Uncertainty Quantification, Vol. 10, Issue 1 https://doi.org/10.1615/Int.J.UncertaintyQuantification.2020030630	journal	January 2020
Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion Oladyshkin, S.; Nowak, W. Reliability Engineering & System Safety, Vol. 106 https://doi.org/10.1016/j.ress.2012.05.002	journal	October 2012
Remarks on a Multivariate Transformation Rosenblatt, Murray The Annals of Mathematical Statistics, Vol. 23, Issue 3 https://doi.org/10.1214/aoms/1177729394	journal	September 1952
Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes Debusschere, Bert J.; Najm, Habib N.; Pébay, Philippe P. SIAM Journal on Scientific Computing, Vol. 26, Issue 2 https://doi.org/10.1137/S1064827503427741	journal	January 2004
Stochastic Polynomial Chaos Expansions to Emulate Stochastic Simulators Zhu, Xujia; Sudret, Bruno International Journal for Uncertainty Quantification, Vol. 13, Issue 2 https://doi.org/10.1615/Int.J.UncertaintyQuantification.2022042912	journal	January 2023
Polynomial chaos representation of spatio-temporal random fields from experimental measurements Das, Sonjoy; Ghanem, Roger; Finette, Steven Journal of Computational Physics, Vol. 228, Issue 23 https://doi.org/10.1016/j.jcp.2009.08.025	journal	December 2009
Cross-Validation and the Estimation of Conditional Probability Densities Hall, Peter; Racine, Jeff; Li, Qi Journal of the American Statistical Association, Vol. 99, Issue 468 https://doi.org/10.1198/016214504000000548	journal	December 2004
Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark Lüthen, Nora; Marelli, Stefano; Sudret, Bruno SIAM/ASA Journal on Uncertainty Quantification, Vol. 9, Issue 2 https://doi.org/10.1137/20M1315774	journal	January 2021
Analyzing Stochastic Computer Models: A Review with Opportunities Baker, Evan; Barbillon, Pierre; Fadikar, Arindam Statistical Science, Vol. 37, Issue 1 https://doi.org/10.1214/21-STS822	journal	February 2022
Ground Heat Flux Reconstruction Using Bayesian Uncertainty Quantification Machinery and Surrogate Modeling Zhou, Wenbo; Zhang, Liujing; Sheshukov, Aleksey Earth and Space Science, Vol. 11, Issue 3 https://doi.org/10.1029/2023EA003435	journal	March 2024
Replication-Based Emulation of the Response Distribution of Stochastic Simulators Using Generalized Lambda Distributions Zhu, Xujia; Sudret, Bruno International Journal for Uncertainty Quantification, Vol. 10, Issue 3 https://doi.org/10.1615/Int.J.UncertaintyQuantification.2020033029	journal	January 2020
Modeling uncertainty in flow simulations via generalized polynomial chaos Xiu, Dongbin; Karniadakis, George Em Journal of Computational Physics, Vol. 187, Issue 1 https://doi.org/10.1016/S0021-9991(03)00092-5	journal	May 2003
Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics Le Maître, O. P.; Knio, Omar M. Scientific Computation https://doi.org/10.1007/978-90-481-3520-2	book	January 2010
Surrogate Modeling of Stochastic Functions-Application to Computational Electromagnetic Dosimetry Azzi, Soumaya; Huang, Yuanyuan; Sudret, Bruno International Journal for Uncertainty Quantification, Vol. 9, Issue 4 https://doi.org/10.1615/Int.J.UncertaintyQuantification.2019029103	journal	January 2019
Emulators for stochastic simulation codes Moutoussamy, Vincent; Nanty, Simon; Pauwels, Benoît ESAIM: Proceedings and Surveys, Vol. 48 https://doi.org/10.1051/proc/201448005	journal	January 2015
Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes Huang, S. P.; Quek, S. T.; Phoon, K. K. International Journal for Numerical Methods in Engineering, Vol. 52, Issue 9 https://doi.org/10.1002/nme.255	journal	November 2001
Global sensitivity analysis for stochastic simulators based on generalized lambda surrogate models Zhu, Xujia; Sudret, Bruno Reliability Engineering & System Safety, Vol. 214 https://doi.org/10.1016/j.ress.2021.107815	journal	October 2021
Principal component analysis and sparse polynomial chaos expansions for global sensitivity analysis and model calibration: Application to urban drainage simulation Nagel, Joseph B.; Rieckermann, Jörg; Sudret, Bruno Reliability Engineering & System Safety, Vol. 195 https://doi.org/10.1016/j.ress.2019.106737	journal	March 2020
Polynomial chaos expansion for sensitivity analysis Crestaux, Thierry; Le Maıˆtre, Olivier; Martinez, Jean-Marc Reliability Engineering & System Safety, Vol. 94, Issue 7 https://doi.org/10.1016/j.ress.2008.10.008	journal	July 2009
Why so many published sensitivity analyses are false: A systematic review of sensitivity analysis practices Saltelli, Andrea; Aleksankina, Ksenia; Becker, William Environmental Modelling & Software, Vol. 114 https://doi.org/10.1016/j.envsoft.2019.01.012	journal	April 2019
A tutorial on support vector regression Smola, Alex J.; Schölkopf, Bernhard Statistics and Computing, Vol. 14, Issue 3 https://doi.org/10.1023/B:STCO.0000035301.49549.88	journal	August 2004
Bayesian calibration with summary statistics for the prediction of xenon diffusion in UO2 nuclear fuel Robbe, Pieterjan; Andersson, David; Bonnet, Luc Computational Materials Science, Vol. 225 https://doi.org/10.1016/j.commatsci.2023.112184	journal	June 2023
Efficient Computation of Sobol' Indices for Stochastic Models Hart, J. L.; Alexanderian, A.; Gremaud, P. A. SIAM Journal on Scientific Computing, Vol. 39, Issue 4 https://doi.org/10.1137/16M106193X	journal	January 2017
Global sensitivity analysis of stochastic computer models with joint metamodels Marrel, Amandine; Iooss, Bertrand; Da Veiga, Sébastien Statistics and Computing, Vol. 22, Issue 3 https://doi.org/10.1007/s11222-011-9274-8	journal	November 2011
Sensitivity analysis practices: Strategies for model-based inference Saltelli, Andrea; Ratto, Marco; Tarantola, Stefano Reliability Engineering & System Safety, Vol. 91, Issue 10-11 https://doi.org/10.1016/j.ress.2005.11.014	journal	October 2006
Global sensitivity analysis using polynomial chaos expansions Sudret, Bruno Reliability Engineering & System Safety, Vol. 93, Issue 7 https://doi.org/10.1016/j.ress.2007.04.002	journal	July 2008
Regression Quantiles Koenker, Roger; Bassett, Gilbert Econometrica, Vol. 46, Issue 1 https://doi.org/10.2307/1913643	journal	January 1978
A Practical Guide to Surface Kinetic Monte Carlo Simulations Andersen, Mie; Panosetti, Chiara; Reuter, Karsten Frontiers in Chemistry, Vol. 7 https://doi.org/10.3389/fchem.2019.00202	journal	April 2019
Practical Heteroscedastic Gaussian Process Modeling for Large Simulation Experiments Binois, Mickaël; Gramacy, Robert B.; Ludkovski, Mike Journal of Computational and Graphical Statistics, Vol. 27, Issue 4 https://doi.org/10.1080/10618600.2018.1458625	journal	July 2018
Global sensitivity analysis of computer models with functional inputs Iooss, Bertrand; Ribatet, Mathieu Reliability Engineering & System Safety, Vol. 94, Issue 7 https://doi.org/10.1016/j.ress.2008.09.010	journal	July 2009
Error propagation in first-principles kinetic Monte Carlo simulation Döpking, Sandra; Matera, Sebastian Chemical Physics Letters, Vol. 674 https://doi.org/10.1016/j.cplett.2017.02.043	journal	April 2017
Kinetic Monte Carlo simulations for heterogeneous catalysis: Fundamentals, current status, and challenges Pineda, M.; Stamatakis, M. The Journal of Chemical Physics, Vol. 156, Issue 12 https://doi.org/10.1063/5.0083251	journal	March 2022
Nonintrusive Polynomial Chaos Expansions for Sensitivity Analysis in Stochastic Differential Equations Jimenez, M. Navarro; Le Maître, O. P.; Knio, O. M. SIAM/ASA Journal on Uncertainty Quantification, Vol. 5, Issue 1 https://doi.org/10.1137/16M1061989	journal	January 2017
Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics Najm, Habib N. Annual Review of Fluid Mechanics, Vol. 41, Issue 1 https://doi.org/10.1146/annurev.fluid.010908.165248	journal	January 2009

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