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Method for recovering boundary data in a two-dimensional Poisson equation on annular domain. (English) Zbl 1457.65165

Summary: 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.

MSC:

65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
80A23 Inverse problems in thermodynamics and heat transfer
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B65 Smoothness and regularity of solutions to PDEs
35R30 Inverse problems for PDEs
35B25 Singular perturbations in context of PDEs
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