Method for recovering boundary data in a two-dimensional Poisson equation on annular domain

Abstract

In this paper, we consider a two-dimensional inverse boundary value problem for Poisson equation on annular domain, consisting of recovering boundary data on the inner boundary from temperature data on the outer circle. This problem is ill-posed in the sense of Hadamard, i.e., small errors in the data can lead to arbitrarily large perturbations in the solution. We establish an infinite singular value expansion for sought boundary data assuming noise-free temperature measurements. In the case of corrupted data, the truncated series is used as a regularized solution; i.e. ill-posedness is dealt with by filtering away high frequencies in the solution. The truncation parameter is determined by Morozov’s discrepancy principle and an error estimate to quantify the accuracy of the computed approximate solution is derived. The proposed regularization method is illustrated by numerical simulations using synthetic data.