  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.
