Coordinate transformation and Polynomial Chaos for the Bayesian inference of a Gaussian process with parametrized prior covariance function

Handle URI:
http://hdl.handle.net/10754/581497
Title:
Coordinate transformation and Polynomial Chaos for the Bayesian inference of a Gaussian process with parametrized prior covariance function
Authors:
Sraj, Ihab ( 0000-0002-6158-472X ) ; Le Maître, Olivier P.; Knio, Omar; Hoteit, Ibrahim ( 0000-0002-3751-4393 )
Abstract:
This paper addresses model dimensionality reduction for Bayesian inference based on prior Gaussian fields with uncertainty in the covariance function hyper-parameters. The dimensionality reduction is traditionally achieved using the Karhunen-Loève expansion of a prior Gaussian process assuming covariance function with fixed hyper-parameters, despite the fact that these are uncertain in nature. The posterior distribution of the Karhunen-Loève coordinates is then inferred using available observations. The resulting inferred field is therefore dependent on the assumed hyper-parameters. Here, we seek to efficiently estimate both the field and covariance hyper-parameters using Bayesian inference. To this end, a generalized Karhunen-Loève expansion is derived using a coordinate transformation to account for the dependence with respect to the covariance hyper-parameters. Polynomial Chaos expansions are employed for the acceleration of the Bayesian inference using similar coordinate transformations, enabling us to avoid expanding explicitly the solution dependence on the uncertain hyper-parameters. We demonstrate the feasibility of the proposed method on a transient diffusion equation by inferring spatially-varying log-diffusivity fields from noisy data. The inferred profiles were found closer to the true profiles when including the hyper-parameters’ uncertainty in the inference formulation.
KAUST Department:
Physical Sciences and Engineering (PSE) Division; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Coordinate transformation and Polynomial Chaos for the Bayesian inference of a Gaussian process with parametrized prior covariance function 2015 Computer Methods in Applied Mechanics and Engineering
Publisher:
Elsevier BV
Journal:
Computer Methods in Applied Mechanics and Engineering
Issue Date:
22-Oct-2015
DOI:
10.1016/j.cma.2015.10.002
Type:
Article
ISSN:
00457825
Additional Links:
http://linkinghub.elsevier.com/retrieve/pii/S0045782515003217
Appears in Collections:
Articles; Physical Sciences and Engineering (PSE) Division; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorSraj, Ihaben
dc.contributor.authorLe Maître, Olivier P.en
dc.contributor.authorKnio, Omaren
dc.contributor.authorHoteit, Ibrahimen
dc.date.accessioned2015-11-01T10:50:06Zen
dc.date.available2015-11-01T10:50:06Zen
dc.date.issued2015-10-22en
dc.identifier.citationCoordinate transformation and Polynomial Chaos for the Bayesian inference of a Gaussian process with parametrized prior covariance function 2015 Computer Methods in Applied Mechanics and Engineeringen
dc.identifier.issn00457825en
dc.identifier.doi10.1016/j.cma.2015.10.002en
dc.identifier.urihttp://hdl.handle.net/10754/581497en
dc.description.abstractThis paper addresses model dimensionality reduction for Bayesian inference based on prior Gaussian fields with uncertainty in the covariance function hyper-parameters. The dimensionality reduction is traditionally achieved using the Karhunen-Loève expansion of a prior Gaussian process assuming covariance function with fixed hyper-parameters, despite the fact that these are uncertain in nature. The posterior distribution of the Karhunen-Loève coordinates is then inferred using available observations. The resulting inferred field is therefore dependent on the assumed hyper-parameters. Here, we seek to efficiently estimate both the field and covariance hyper-parameters using Bayesian inference. To this end, a generalized Karhunen-Loève expansion is derived using a coordinate transformation to account for the dependence with respect to the covariance hyper-parameters. Polynomial Chaos expansions are employed for the acceleration of the Bayesian inference using similar coordinate transformations, enabling us to avoid expanding explicitly the solution dependence on the uncertain hyper-parameters. We demonstrate the feasibility of the proposed method on a transient diffusion equation by inferring spatially-varying log-diffusivity fields from noisy data. The inferred profiles were found closer to the true profiles when including the hyper-parameters’ uncertainty in the inference formulation.en
dc.language.isoenen
dc.publisherElsevier BVen
dc.relation.urlhttp://linkinghub.elsevier.com/retrieve/pii/S0045782515003217en
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering, 22 October 2015. DOI: 10.1016/j.cma.2015.10.002en
dc.subjectKarhunen-Loève expansionen
dc.subjectDimensionality reductionen
dc.subjectMarkov Chain Monte Carloen
dc.subjectPolynomial chaosen
dc.subjectBayesian inferenceen
dc.titleCoordinate transformation and Polynomial Chaos for the Bayesian inference of a Gaussian process with parametrized prior covariance functionen
dc.typeArticleen
dc.contributor.departmentPhysical Sciences and Engineering (PSE) Divisionen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalComputer Methods in Applied Mechanics and Engineeringen
dc.eprint.versionPost-printen
dc.contributor.institutionLIMSI-CNRS, BP 133, Bt 508, 91403 Orsay Cedex, Franceen
dc.contributor.institutionDepartment of Mechanical Engineering and Materials Science, Duke University, 144 Hudson Hall, Durham, North Carolina 27708, USAen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorSraj, Ihaben
kaust.authorKnio, Omaren
kaust.authorHoteit, Ibrahimen
kaust.authorHoteit, Ibrahimen
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