American Institute of Mathematical Sciences

Advanced
February  2011, 5(1): 185-202. doi: 10.3934/ipi.2011.5.185

On rational approximation methods for inverse source problems

Martin Hanke 1, and  William Rundell 2,

1. 

Institute of Mathematics, Johannes Gutenberg-Universität, 55099 Mainz
2. 

Department of Mathematics, Texas A&M University, College Station, TX 77843-3368

Received  March 2010 Revised  July 2010 Published  February 2011

The basis of most imaging methods is to detect hidden obstacles or inclusions within a body when one can only make measurements on an exterior surface. Such is the ubiquity of these problems, the underlying model can lead to a partial differential equation of any of the major types, but here we focus on the case of steady-state electrostatic or thermal imaging and consider boundary value problems for Laplace's equation. Our inclusions are interior forces with compact support and our data consists of a single measurement of (say) voltage/current or temperature/heat flux on the external boundary. We propose an algorithm that under certain assumptions allows for the determination of the support set of these forces by solving a simpler "equivalent point source" problem, and which uses a Newton scheme to improve the corresponding initial approximation.
Keywords: rational approximation., Inverse source problem.
Mathematics Subject Classification: Primary: 35R2.
Citation: Martin Hanke, William Rundell. On rational approximation methods for inverse source problems. Inverse Problems and Imaging, 2011, 5 (1) : 185-202. doi: 10.3934/ipi.2011.5.185
References:
[1]

L. V. Ahlfors, "Complex Analysis," McGraw-Hill, New York, third edition, 1979.

[2]

S. Andrieux and A. Ben Abda, Identification of planar cracks by complete overdetermined data: inversion formulae, Inverse Problems, 12 (1996), 553-563. doi: 10.1088/0266-5611/12/5/002.

[3]

G. A. Baker and P. Graves-Morris, "Padé Approximants," Cambridge University Press, Cambridge, second edition, 1996.

[4]

L. Baratchart, A. Ben Abda, F. Ben Hassen and J. Leblond, Recovery of pointwise sources or small inclusions in 2D domains and rational approximation, Inverse Problems, 21 (2005), 51-74. doi: 10.1088/0266-5611/21/1/005.

[5]

L. Baratchart, J. Leblond, F. Mandréa and E. B. Saff, How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian? Inverse Problems, 15 (1999), 79-90. doi: 10.1088/0266-5611/15/1/012.

[6]

D. J. Cedio-Fengya, S. Moskow and M. S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems, 14 (1998), 553-595. doi: 10.1088/0266-5611/14/3/011.

[7]

Y.-S. Chung and S.-Y. Chung, Identification of the combination of monopolar and dipolar sources for elliptic equations, Inverse Problems, 25 (2009), 085006. doi: 10.1088/0266-5611/25/8/085006.

[8]

A. El Badia and T. Ha-Duong, An inverse source problem in potential analysis, Inverse Problems, 16 (2000), 651-663. doi: 10.1088/0266-5611/16/3/308.

[9]

H. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Kluwer Academic Publishers, Dordrecht, 1996.

[10]

W. B. Gragg and G. D. Johnson, The Laurent-Padé table, In "Information Processing 74, Proceedings IFIP Congress," pages 632-637. North-Holland, Amsterdam, 1974.

[11]

M. Hanke, On real-time algorithms for the location search of discontinuous conductivities with one measurement, Inverse Problems, 24 (2008), 045005. doi: 10.1088/0266-5611/24/4/045005.

[12]

M. Hanke, N. Hyvönen, M. Lehn and S. Reusswig, Source supports in electrostatics, BIT, 48 (2008), 245-264. doi: 10.1007/s10543-008-0172-1.

[13]

F. Hettlich and W. Rundell, Iterative methods for the reconstruction of an inverse potential problem, Inverse Problems, 12 (1996), 251-266. doi: 10.1088/0266-5611/12/3/006.

[14]

V. Isakov, "Inverse Problems for Partial Differential Equations," Springer-Verlag, New York, second edition, 2005.

[15]

H. Kang and H. Lee, Identification of simple poles via boundary measurements and an application of EIT, Inverse Problems, 20 (2004), 1853-1863. doi: 10.1088/0266-5611/20/6/010.

[16]

O. Kwon, J. K. Seo and J. R. Yoon, A real time algorithm for the location search of discontinuous conductivities with one measurement, Comm. Pure Appl. Math., 55 (2002), 1-29. doi: 10.1002/cpa.3009.

show all references

References:
[1]

L. V. Ahlfors, "Complex Analysis," McGraw-Hill, New York, third edition, 1979.

[2]

S. Andrieux and A. Ben Abda, Identification of planar cracks by complete overdetermined data: inversion formulae, Inverse Problems, 12 (1996), 553-563. doi: 10.1088/0266-5611/12/5/002.

[3]

G. A. Baker and P. Graves-Morris, "Padé Approximants," Cambridge University Press, Cambridge, second edition, 1996.

[4]

L. Baratchart, A. Ben Abda, F. Ben Hassen and J. Leblond, Recovery of pointwise sources or small inclusions in 2D domains and rational approximation, Inverse Problems, 21 (2005), 51-74. doi: 10.1088/0266-5611/21/1/005.

[5]

L. Baratchart, J. Leblond, F. Mandréa and E. B. Saff, How can the meromorphic approximation help to solve some 2D inverse problems for the Laplacian? Inverse Problems, 15 (1999), 79-90. doi: 10.1088/0266-5611/15/1/012.

[6]

D. J. Cedio-Fengya, S. Moskow and M. S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems, 14 (1998), 553-595. doi: 10.1088/0266-5611/14/3/011.

[7]

Y.-S. Chung and S.-Y. Chung, Identification of the combination of monopolar and dipolar sources for elliptic equations, Inverse Problems, 25 (2009), 085006. doi: 10.1088/0266-5611/25/8/085006.

[8]

A. El Badia and T. Ha-Duong, An inverse source problem in potential analysis, Inverse Problems, 16 (2000), 651-663. doi: 10.1088/0266-5611/16/3/308.

[9]

H. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Kluwer Academic Publishers, Dordrecht, 1996.

[10]

W. B. Gragg and G. D. Johnson, The Laurent-Padé table, In "Information Processing 74, Proceedings IFIP Congress," pages 632-637. North-Holland, Amsterdam, 1974.

[11]

M. Hanke, On real-time algorithms for the location search of discontinuous conductivities with one measurement, Inverse Problems, 24 (2008), 045005. doi: 10.1088/0266-5611/24/4/045005.

[12]

M. Hanke, N. Hyvönen, M. Lehn and S. Reusswig, Source supports in electrostatics, BIT, 48 (2008), 245-264. doi: 10.1007/s10543-008-0172-1.

[13]

F. Hettlich and W. Rundell, Iterative methods for the reconstruction of an inverse potential problem, Inverse Problems, 12 (1996), 251-266. doi: 10.1088/0266-5611/12/3/006.

[14]

V. Isakov, "Inverse Problems for Partial Differential Equations," Springer-Verlag, New York, second edition, 2005.

[15]

H. Kang and H. Lee, Identification of simple poles via boundary measurements and an application of EIT, Inverse Problems, 20 (2004), 1853-1863. doi: 10.1088/0266-5611/20/6/010.

[16]

O. Kwon, J. K. Seo and J. R. Yoon, A real time algorithm for the location search of discontinuous conductivities with one measurement, Comm. Pure Appl. Math., 55 (2002), 1-29. doi: 10.1002/cpa.3009.

[1]

Xiaoli Feng, Meixia Zhao, Peijun Li, Xu Wang. An inverse source problem for the stochastic wave equation. Inverse Problems and Imaging, 2022, 16 (2) : 397-415. doi: 10.3934/ipi.2021055
[2]

Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems and Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035
[3]

Kenichi Sakamoto, Masahiro Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control and Related Fields, 2011, 1 (4) : 509-518. doi: 10.3934/mcrf.2011.1.509
[4]

Yuxuan Gong, Xiang Xu. Inverse random source problem for biharmonic equation in two dimensions. Inverse Problems and Imaging, 2019, 13 (3) : 635-652. doi: 10.3934/ipi.2019029
[5]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367
[6]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 1-21. doi: 10.3934/dcdss.2021006
[7]

Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems and Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791
[8]

Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems and Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059
[9]

Shuli Chen, Zewen Wang, Guolin Chen. Cauchy problem of non-homogenous stochastic heat equation and application to inverse random source problem. Inverse Problems and Imaging, 2021, 15 (4) : 619-639. doi: 10.3934/ipi.2021008
[10]

Weihua Liu, Andrew Klapper. AFSRs synthesis with the extended Euclidean rational approximation algorithm. Advances in Mathematics of Communications, 2017, 11 (1) : 139-150. doi: 10.3934/amc.2017008
[11]

Hassan Emamirad, Arnaud Rougirel. A functional calculus approach for the rational approximation with nonuniform partitions. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 955-972. doi: 10.3934/dcds.2008.22.955
[12]

Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations and Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032
[13]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems and Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014
[14]

Loc H. Nguyen, Qitong Li, Michael V. Klibanov. A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media. Inverse Problems and Imaging, 2019, 13 (5) : 1067-1094. doi: 10.3934/ipi.2019048
[15]

El Mustapha Ait Ben Hassi, Salah-Eddine Chorfi, Lahcen Maniar, Omar Oukdach. Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions. Evolution Equations and Control Theory, 2021, 10 (4) : 837-859. doi: 10.3934/eect.2020094
[16]

Zhiyuan Li, Yikan Liu, Masahiro Yamamoto. Inverse source problem for a one-dimensional time-fractional diffusion equation and unique continuation for weak solutions. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022027
[17]

Justyna Szpond, Grzegorz Malara. The containment problem and a rational simplicial arrangement. Electronic Research Announcements, 2017, 24: 123-128. doi: 10.3934/era.2017.24.013
[18]

Janne M.J. Huttunen, J. P. Kaipio. Approximation errors in nonstationary inverse problems. Inverse Problems and Imaging, 2007, 1 (1) : 77-93. doi: 10.3934/ipi.2007.1.77
[19]

Victor Isakov, Shuai Lu. Inverse source problems without (pseudo) convexity assumptions. Inverse Problems and Imaging, 2018, 12 (4) : 955-970. doi: 10.3934/ipi.2018040
[20]

Victor Isakov, Joseph Myers. On the inverse doping profile problem. Inverse Problems and Imaging, 2012, 6 (3) : 465-486. doi: 10.3934/ipi.2012.6.465

2021 Impact Factor: 1.483

Tools

Metrics

  • PDF downloads (337)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy