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A NEW KOHN-VOGELIUS TYPE FORMULATION FOR INVERSE SOURCE PROBLEMS
Journal article   Open access

A NEW KOHN-VOGELIUS TYPE FORMULATION FOR INVERSE SOURCE PROBLEMS

Xiaoliang Cheng, Rongfang Gong and Weimin Han
Inverse problems and imaging (Springfield, Mo.), Vol.9(4), pp.1051-1067
11/01/2015
DOI: 10.3934/ipi.2015.9.1051
url
https://doi.org/10.3934/ipi.2015.9.1051View
Published (Version of record) Open Access

Abstract

In this paper we propose a Kohn-Vogelius type formulation for an inverse source problem of partial differential equations. The unknown source term is to be determined from both Dirichlet and Neumann boundary conditions. We introduce two different boundary value problems, which depend on two different positive real numbers a and beta, and both of them incorporate the Dirichlet and Neumann data into a single Robin boundary condition. This allows noise in both boundary data. By using the Kohn-Vogelius type Tikhonov regularization, data to be fitted is transferred from boundary into the whole domain, making the problem resolution more robust. More importantly, with the formulation proposed here, satisfactory reconstruction could be achieved for rather small regularization parameter through choosing properly the values of a and beta. This is a desirable property to have since a smaller regularization parameter implies a more accurate approximation of the regularized problem to the original one. The proposed method is studied theoretically. Two numerical examples are provided to show the usefulness of the proposed method.
 Mathematics Physical Sciences Physics Mathematics, Applied Physics, Mathematical Science & Technology

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