Joint Estimation of Robin Coefficient and Domain Boundary for the Poisson Problem

TL;DR

This paper develops a Bayesian framework for jointly estimating the Robin boundary coefficient and the domain boundary shape in a Poisson problem, using linearised uncertainty quantification and MCMC sampling, with applications in corrosion and thermal analysis.

Contribution

It introduces a method that exploits invariance properties to avoid re-meshing and combines linearised analysis with full Bayesian sampling for joint shape and coefficient inference.

Findings

Efficient computation of MAP estimates using Gauss-Newton.

Successful joint inference demonstrated through numerical experiments.

Gradient and derivative calculations enable effective MCMC exploration.

Abstract

We consider the problem of simultaneously inferring the heterogeneous coefficient field for a Robin boundary condition on an inaccessible part of the boundary along with the shape of the boundary for the Poisson problem. Such a problem arises in, for example, corrosion detection, and thermal parameter estimation. We carry out both linearised uncertainty quantification, based on a local Gaussian approximation, and full exploration of the joint posterior using Markov chain Monte Carlo (MCMC) sampling. By exploiting a known invariance property of the Poisson problem, we are able to circumvent the need to re-mesh as the shape of the boundary changes. The linearised uncertainty analysis presented here relies on a local linearisation of the parameter-to-observable map, with respect to both the Robin coefficient and the boundary shape, evaluated at the maximum a posteriori (MAP) estimates. Computation of the MAP estimate is carried out using the Gauss-Newton method. On the other hand, to explore the full joint posterior we use the Metropolis-adjusted Langevin algorithm (MALA), which requires the gradient of the log-posterior. We thus derive both the Fr\'{e}chet derivative of the solution to the Poisson problem with respect to the Robin coefficient and the boundary shape, and the gradient of the log-posterior, which is efficiently computed using the so-called adjoint approach. The performance of the approach is demonstrated via several numerical experiments with simulated data.