A hierarchical Bayesian setting for an inverse problem in linear parabolic PDEs with noisy boundary conditions

TL;DR

This paper develops a hierarchical Bayesian framework to infer unknown parameters in linear parabolic PDEs with noisy boundary data, demonstrated through heat equation examples, using Laplace approximation to efficiently estimate posteriors.

Contribution

It introduces an analytical marginalization of the likelihood in a Bayesian setting for inverse PDE problems with noisy boundary conditions, applied to heat equation parameter inference.

Findings

Effective inference of thermal diffusivity using synthetic data.

Laplace approximation provides accurate posterior estimates without MCMC.

Quantitative analysis of information gain and predictive densities.

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.