A Hierarchical Bayesian Setting for an Inverse Problem in Linear Parabolic PDEs with Noisy Boundary Conditions
Type
ArticleKAUST Department
Applied Mathematics and Computational Science ProgramComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Date
2016-05-12Online Publication Date
2016-05-12Print Publication Date
2017-06Permanent link to this record
http://hdl.handle.net/10754/624030
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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.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.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.Publisher
Institute of Mathematical StatisticsJournal
Bayesian AnalysisarXiv
1501.04739Additional Links
http://projecteuclid.org/euclid.ba/1463078272ae974a485f413a2113503eed53cd6c53
10.1214/16-BA1007
Scopus Count
Except where otherwise noted, this item's license is described as Archived with thanks to Bayesian Analysis.Creative Commons Attribution 4.0 International License.