Joint estimation of Robin coefficient and domain boundary for the Poisson problem. (8th December 2021)

• Record Type: Journal Article
• Authors: Nicholson, Ruanui; Niskanen, Matti
• Is Part Of: Inverse problems. Volume 38:Number 1(2022)
• Publication Date: 2021-12-08
• Subjects: joint estimation -- Metropolis-adjusted Langevin algorithm -- Robin coefficient -- boundary shape -- diffeomorphism -- adjoint approach -- Laplace approximation; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ac3c17
• Languages: English
• ISSNs: 0266-5611

Abstract

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 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, which requires the gradient of the log-posterior. We thus derive both the Fréchet 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.