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

In this work we develop a Bayesian setting to infer unknown parameters in initial-boundary value problems related to linear parabolic partial dierential 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 eld 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 diusivity is the unknown parameter. We assume that the thermal diusivity parameter can be modeled a priori through a lognormal random variable or by means of a space- dependent stationary lognormal random eld. Synthetic data are used to test the inference. We exploit the behavior of the non-normalized log posterior distribution of the thermal diusivity. Then, we use the Laplace method to obtain an approx- imated 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 dierent experimental setups.