A hierarchical Bayesian setting for an inverse problem in linear parabolic PDEs with noisy boundary conditions
Abstract
In this work we develop a Bayesian setting to infer unknown parameters in initialboundary 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 timedependent 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 spacedependent stationary lognormal random field. Synthetic data are used to test the inference. We exploit the behavior of the nonnormalized 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2015
 arXiv:
 arXiv:1501.04739
 Bibcode:
 2015arXiv150104739R
 Keywords:

 Statistics  Methodology
 EPrint:
 30 pages, submitted, January 2015