A Hierarchical Bayesian Setting for an Inverse Problem in Linear Parabolic PDEs with Noisy Boundary Conditions

Handle URI:
http://hdl.handle.net/10754/624030
Title:
A Hierarchical Bayesian Setting for an Inverse Problem in Linear Parabolic PDEs with Noisy Boundary Conditions
Authors:
Ruggeri, Fabrizio; Sawlan, Zaid A; Scavino, Marco ( 0000-0001-5114-853X ) ; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
In this work we develop a Bayesian setting to infer unknown parameters in initial-boundary value problems related to linear parabolic partial differential equations. We realistically assume that the boundary data are noisy, for a given prescribed initial condition. We show how to derive the joint likelihood function for the forward problem, given some measurements of the solution field subject to Gaussian noise. Given Gaussian priors for the time-dependent Dirichlet boundary values, we analytically marginalize the joint likelihood using the linearity of the equation. Our hierarchical Bayesian approach is fully implemented in an example that involves the heat equation. In this example, the thermal diffusivity is the unknown parameter. We assume that the thermal diffusivity parameter can be modeled a priori through a lognormal random variable or by means of a space-dependent stationary lognormal random field. Synthetic data are used to test the inference. We exploit the behavior of the non-normalized log posterior distribution of the thermal diffusivity. Then, we use the Laplace method to obtain an approximated Gaussian posterior and therefore avoid costly Markov Chain Monte Carlo computations. Expected information gains and predictive posterior densities for observable quantities are numerically estimated using Laplace approximation for different experimental setups.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Ruggeri F, Sawlan Z, Scavino M, Tempone R (2017) A Hierarchical Bayesian Setting for an Inverse Problem in Linear Parabolic PDEs with Noisy Boundary Conditions. Bayesian Analysis 12: 407–433. Available: http://dx.doi.org/10.1214/16-BA1007.
Publisher:
Institute of Mathematical Statistics
Journal:
Bayesian Analysis
Issue Date:
12-May-2016
DOI:
10.1214/16-BA1007
Type:
Article
ISSN:
1936-0975
Sponsors:
Part of this work was carried out while F. Ruggeri and M. Scavino were Visiting Professors at KAUST. Z. Sawlan, M. Scavino and R. Tempone are members of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.
Additional Links:
http://projecteuclid.org/euclid.ba/1463078272
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorRuggeri, Fabrizioen
dc.contributor.authorSawlan, Zaid Aen
dc.contributor.authorScavino, Marcoen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2017-06-05T06:02:23Z-
dc.date.available2017-06-05T06:02:23Z-
dc.date.issued2016-05-12en
dc.identifier.citationRuggeri F, Sawlan Z, Scavino M, Tempone R (2017) A Hierarchical Bayesian Setting for an Inverse Problem in Linear Parabolic PDEs with Noisy Boundary Conditions. Bayesian Analysis 12: 407–433. Available: http://dx.doi.org/10.1214/16-BA1007.en
dc.identifier.issn1936-0975en
dc.identifier.doi10.1214/16-BA1007en
dc.identifier.urihttp://hdl.handle.net/10754/624030-
dc.description.abstractIn this work we develop a Bayesian setting to infer unknown parameters in initial-boundary value problems related to linear parabolic partial differential equations. We realistically assume that the boundary data are noisy, for a given prescribed initial condition. We show how to derive the joint likelihood function for the forward problem, given some measurements of the solution field subject to Gaussian noise. Given Gaussian priors for the time-dependent Dirichlet boundary values, we analytically marginalize the joint likelihood using the linearity of the equation. Our hierarchical Bayesian approach is fully implemented in an example that involves the heat equation. In this example, the thermal diffusivity is the unknown parameter. We assume that the thermal diffusivity parameter can be modeled a priori through a lognormal random variable or by means of a space-dependent stationary lognormal random field. Synthetic data are used to test the inference. We exploit the behavior of the non-normalized log posterior distribution of the thermal diffusivity. Then, we use the Laplace method to obtain an approximated Gaussian posterior and therefore avoid costly Markov Chain Monte Carlo computations. Expected information gains and predictive posterior densities for observable quantities are numerically estimated using Laplace approximation for different experimental setups.en
dc.description.sponsorshipPart of this work was carried out while F. Ruggeri and M. Scavino were Visiting Professors at KAUST. Z. Sawlan, M. Scavino and R. Tempone are members of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.en
dc.publisherInstitute of Mathematical Statisticsen
dc.relation.urlhttp://projecteuclid.org/euclid.ba/1463078272en
dc.rightsArchived with thanks to Bayesian Analysis.Creative Commons Attribution 4.0 International License.en
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/en
dc.subjectBayesian inferenceen
dc.subjectHeat equationen
dc.subjectLinear parabolic PDEsen
dc.subjectNoisy boundary parametersen
dc.subjectThermal diffusivityen
dc.titleA Hierarchical Bayesian Setting for an Inverse Problem in Linear Parabolic PDEs with Noisy Boundary Conditionsen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalBayesian Analysisen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionCNR - IMATI, Consiglio Nazionale delle Ricerche, Milano, , Italyen
dc.contributor.institutionInstituto de Estadística (IESTA), Universidad de la República, Montevideo, , Uruguayen
kaust.authorSawlan, Zaid Aen
kaust.authorScavino, Marcoen
kaust.authorTempone, Raulen
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