A new Kohn-Vogelius type formulation for inverse source problems

Xiaoliang Cheng; Rongfang  Gong; Weimin  Han

doi:10.3934/ipi.2015.9.1051

Article Contents

2015, Volume 9, Issue 4: 1051-1067. Doi: 10.3934/ipi.2015.9.1051

This issue Previous Article Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem Next Article A parallel space-time domain decomposition method for unsteady source inversion problems

A new Kohn-Vogelius type formulation for inverse source problems

1.
Department of Mathematics, Zhejiang University, Hangzhou 310027
2.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016
3.
Department of Mathematics, University of Iowa, Iowa City, IA 52242

Received: September 30, 2014

Revised: April 30, 2015

Published: October 2015

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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 $\alpha$ 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 $\alpha$ 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.

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Mathematics Subject Classification: Primary: 65N21, 65F22; Secondary: 49J40.

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