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November  2015, 9(4): 1051-1067. doi: 10.3934/ipi.2015.9.1051

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, China

3. 

Department of Mathematics, University of Iowa, Iowa City, IA 52242, United States

Received  October 2014 Revised  May 2015 Published  October 2015

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.
Citation: Xiaoliang Cheng, Rongfang Gong, Weimin Han. A new Kohn-Vogelius type formulation for inverse source problems. Inverse Problems and Imaging, 2015, 9 (4) : 1051-1067. doi: 10.3934/ipi.2015.9.1051
References:
[1]

L. Afraites, M. Dambrine and D. Kateb, Conformal mappings and shape derivatives for the transmission problem with a single measurement, Numer. Func. Anal. Opt., 28 (2007), 519-551. doi: 10.1080/01630560701381005.

[2]

K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, $3^{rd}$ edition, Springer-Verlag, New York, 2009. doi: 10.1007/978-1-4419-0458-4.

[3]

D. P. Bertsekas and E. M. Gafni, Projection method for variational inequalities with applications to the traffic assignment problem, Math. Prog. Study, 17 (1982), 139-159.

[4]

R. W. Cottle, F. Giannessi and J. L. Lions, Variational Inequalities and Complementarity Problems: Theory and Applications, John Wiley and Sons, New York, 1980.

[5]

L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998.

[6]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.

[7]

W. Han, W. X. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography, Inverse Probl., 22 (2006), 1659-1675. doi: 10.1088/0266-5611/22/5/008.

[8]

W. Han, K. Kazmi, W. X. Cong and G. Wang, Bioluminescence tomography with optimized optical parameters, Inverse Probl., 23 (2007), 1215-1228. doi: 10.1088/0266-5611/23/3/022.

[9]

V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York, 1998. doi: 10.1007/978-1-4899-0030-2.

[10]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.

[11]

C. Qin, J. Feng, S. Zhu, X. Ma, J. Zhong, P. Wu, Z. Jin and J. Tian, Recent advances in bioluminescence tomography: Methodology and system as well as application, Laser Photon. Rev., 8 (2014), 94-114. doi: 10.1002/lpor.201280011.

[12]

S. J. Song and J. G. Huang, Solving an inverse problem from bioluminescence tomography by minimizing an energy-like functional, J. Comput. Anal. Appl., 14 (2012), 544-558.

[13]

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Wiley, New York, 1977.

show all references

References:
[1]

L. Afraites, M. Dambrine and D. Kateb, Conformal mappings and shape derivatives for the transmission problem with a single measurement, Numer. Func. Anal. Opt., 28 (2007), 519-551. doi: 10.1080/01630560701381005.

[2]

K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, $3^{rd}$ edition, Springer-Verlag, New York, 2009. doi: 10.1007/978-1-4419-0458-4.

[3]

D. P. Bertsekas and E. M. Gafni, Projection method for variational inequalities with applications to the traffic assignment problem, Math. Prog. Study, 17 (1982), 139-159.

[4]

R. W. Cottle, F. Giannessi and J. L. Lions, Variational Inequalities and Complementarity Problems: Theory and Applications, John Wiley and Sons, New York, 1980.

[5]

L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998.

[6]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.

[7]

W. Han, W. X. Cong and G. Wang, Mathematical theory and numerical analysis of bioluminescence tomography, Inverse Probl., 22 (2006), 1659-1675. doi: 10.1088/0266-5611/22/5/008.

[8]

W. Han, K. Kazmi, W. X. Cong and G. Wang, Bioluminescence tomography with optimized optical parameters, Inverse Probl., 23 (2007), 1215-1228. doi: 10.1088/0266-5611/23/3/022.

[9]

V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York, 1998. doi: 10.1007/978-1-4899-0030-2.

[10]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.

[11]

C. Qin, J. Feng, S. Zhu, X. Ma, J. Zhong, P. Wu, Z. Jin and J. Tian, Recent advances in bioluminescence tomography: Methodology and system as well as application, Laser Photon. Rev., 8 (2014), 94-114. doi: 10.1002/lpor.201280011.

[12]

S. J. Song and J. G. Huang, Solving an inverse problem from bioluminescence tomography by minimizing an energy-like functional, J. Comput. Anal. Appl., 14 (2012), 544-558.

[13]

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Wiley, New York, 1977.

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