# American Institute of Mathematical Sciences

• Previous Article
Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem
• IPI Home
• This Issue
• Next Article
A parallel space-time domain decomposition method for unsteady source inversion problems
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
 [1] Fabien Caubet, Marc Dambrine, Djalil Kateb, Chahnaz Zakia Timimoun. A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid. Inverse Problems & Imaging, 2013, 7 (1) : 123-157. doi: 10.3934/ipi.2013.7.123 [2] Martin Hanke, William Rundell. On rational approximation methods for inverse source problems. Inverse Problems & Imaging, 2011, 5 (1) : 185-202. doi: 10.3934/ipi.2011.5.185 [3] Thorsten Hohage, Mihaela Pricop. Nonlinear Tikhonov regularization in Hilbert scales for inverse boundary value problems with random noise. Inverse Problems & Imaging, 2008, 2 (2) : 271-290. doi: 10.3934/ipi.2008.2.271 [4] Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010 [5] Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems & Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059 [6] Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042 [7] Armin Lechleiter, Marcel Rennoch. Non-linear Tikhonov regularization in Banach spaces for inverse scattering from anisotropic penetrable media. Inverse Problems & Imaging, 2017, 11 (1) : 151-176. doi: 10.3934/ipi.2017008 [8] Kenrick Bingham, Yaroslav Kurylev, Matti Lassas, Samuli Siltanen. Iterative time-reversal control for inverse problems. Inverse Problems & Imaging, 2008, 2 (1) : 63-81. doi: 10.3934/ipi.2008.2.63 [9] Stefan Kindermann, Antonio Leitão. Convergence rates for Kaczmarz-type regularization methods. Inverse Problems & Imaging, 2014, 8 (1) : 149-172. doi: 10.3934/ipi.2014.8.149 [10] Vinicius Albani, Adriano De Cezaro. A connection between uniqueness of minimizers in Tikhonov-type regularization and Morozov-like discrepancy principles. Inverse Problems & Imaging, 2019, 13 (1) : 211-229. doi: 10.3934/ipi.2019012 [11] Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511 [12] Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control & Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013 [13] Victor Isakov, Shuai Lu. Inverse source problems without (pseudo) convexity assumptions. Inverse Problems & Imaging, 2018, 12 (4) : 955-970. doi: 10.3934/ipi.2018040 [14] Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641 [15] Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems & Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709 [16] Stefan Kindermann, Andreas Neubauer. On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization. Inverse Problems & Imaging, 2008, 2 (2) : 291-299. doi: 10.3934/ipi.2008.2.291 [17] Daijun Jiang, Hui Feng, Jun Zou. Overlapping domain decomposition methods for linear inverse problems. Inverse Problems & Imaging, 2015, 9 (1) : 163-188. doi: 10.3934/ipi.2015.9.163 [18] Konstantinos Chrysafinos. Error estimates for time-discretizations for the velocity tracking problem for Navier-Stokes flows by penalty methods. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1077-1096. doi: 10.3934/dcdsb.2006.6.1077 [19] Toshiyuki Suzuki. Energy methods for Hartree type equations with inverse-square potentials. Evolution Equations & Control Theory, 2013, 2 (3) : 531-542. doi: 10.3934/eect.2013.2.531 [20] Vinicius Albani, Adriano De Cezaro, Jorge P. Zubelli. On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy. Inverse Problems & Imaging, 2016, 10 (1) : 1-25. doi: 10.3934/ipi.2016.10.1

2018 Impact Factor: 1.469