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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 & 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. doi: 10.1080/01630560701381005. Google Scholar

[2]

K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework,, $3^{rd}$ edition, (2009). doi: 10.1007/978-1-4419-0458-4. Google Scholar

[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. Google Scholar

[4]

R. W. Cottle, F. Giannessi and J. L. Lions, Variational Inequalities and Complementarity Problems: Theory and Applications,, John Wiley and Sons, (1980). Google Scholar

[5]

L. C. Evans, Partial Differential Equations,, American Mathematical Society, (1998). Google Scholar

[6]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman, (1985). Google Scholar

[7]

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

[8]

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

[9]

V. Isakov, Inverse Problems for Partial Differential Equations,, Springer, (1998). doi: 10.1007/978-1-4899-0030-2. Google Scholar

[10]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971). Google Scholar

[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. doi: 10.1002/lpor.201280011. Google Scholar

[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. Google Scholar

[13]

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems,, Wiley, (1977). Google Scholar

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. doi: 10.1080/01630560701381005. Google Scholar

[2]

K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework,, $3^{rd}$ edition, (2009). doi: 10.1007/978-1-4419-0458-4. Google Scholar

[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. Google Scholar

[4]

R. W. Cottle, F. Giannessi and J. L. Lions, Variational Inequalities and Complementarity Problems: Theory and Applications,, John Wiley and Sons, (1980). Google Scholar

[5]

L. C. Evans, Partial Differential Equations,, American Mathematical Society, (1998). Google Scholar

[6]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman, (1985). Google Scholar

[7]

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

[8]

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

[9]

V. Isakov, Inverse Problems for Partial Differential Equations,, Springer, (1998). doi: 10.1007/978-1-4899-0030-2. Google Scholar

[10]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971). Google Scholar

[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. doi: 10.1002/lpor.201280011. Google Scholar

[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. Google Scholar

[13]

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems,, Wiley, (1977). Google Scholar

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