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

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074

[2]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[3]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[4]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[5]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[6]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072

[7]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[8]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

[9]

Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044

[10]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[11]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[12]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[13]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[14]

Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020, 3 (4) : 219-227. doi: 10.3934/mfc.2020013

[15]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108

[16]

Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 201-215. doi: 10.3934/dcds.2020263

[17]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[18]

Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026

[19]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[20]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268

2019 Impact Factor: 1.373

Metrics

  • PDF downloads (74)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]