Inverse Problems and Imaging (IPI)

A new Kohn-Vogelius type formulation for inverse source problems

Pages: 1051 - 1067, Volume 9, Issue 4, November 2015      doi:10.3934/ipi.2015.9.1051

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Xiaoliang Cheng - Department of Mathematics, Zhejiang University, Hangzhou 310027, China (email)
Rongfang Gong - Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China (email)
Weimin Han - Department of Mathematics, University of Iowa, Iowa City, IA 52242, United States (email)

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.

Keywords:  Inverse source problems, Tikhonov regularization, Kohn-Vogelius type functional, error estimates, iterative methods.
Mathematics Subject Classification:  Primary: 65N21, 65F22; Secondary: 49J40.

Received: October 2014;      Revised: May 2015;      Available Online: October 2015.