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 & Imaging, 2011, 5 (1) : 185-202. doi: 10.3934/ipi.2011.5.185
References:
[1]

L. V. Ahlfors, "Complex Analysis,", McGraw-Hill, (1979).   Google Scholar

[2]

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

[3]

G. A. Baker and P. Graves-Morris, "Padé Approximants,", Cambridge University Press, (1996).   Google Scholar

[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.  doi: 10.1088/0266-5611/21/1/005.  Google Scholar

[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.  doi: 10.1088/0266-5611/15/1/012.  Google Scholar

[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.  doi: 10.1088/0266-5611/14/3/011.  Google Scholar

[7]

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

[8]

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

[9]

H. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,", Kluwer Academic Publishers, (1996).   Google Scholar

[10]

W. B. Gragg and G. D. Johnson, The Laurent-Padé table,, In, (1974), 632.   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Springer-Verlag, (2005).   Google Scholar

[15]

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

[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.  doi: 10.1002/cpa.3009.  Google Scholar

show all references

References:
[1]

L. V. Ahlfors, "Complex Analysis,", McGraw-Hill, (1979).   Google Scholar

[2]

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

[3]

G. A. Baker and P. Graves-Morris, "Padé Approximants,", Cambridge University Press, (1996).   Google Scholar

[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.  doi: 10.1088/0266-5611/21/1/005.  Google Scholar

[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.  doi: 10.1088/0266-5611/15/1/012.  Google Scholar

[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.  doi: 10.1088/0266-5611/14/3/011.  Google Scholar

[7]

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

[8]

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

[9]

H. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,", Kluwer Academic Publishers, (1996).   Google Scholar

[10]

W. B. Gragg and G. D. Johnson, The Laurent-Padé table,, In, (1974), 632.   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

V. Isakov, "Inverse Problems for Partial Differential Equations,", Springer-Verlag, (2005).   Google Scholar

[15]

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

[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.  doi: 10.1002/cpa.3009.  Google Scholar

[1]

Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems & Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035
[2]

Kenichi Sakamoto, Masahiro Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control & Related Fields, 2011, 1 (4) : 509-518. doi: 10.3934/mcrf.2011.1.509
[3]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367
[4]

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

Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems & Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791
[6]

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
[7]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014
[8]

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 & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032
[9]

Loc H. Nguyen, Qitong Li, Michael V. Klibanov. A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media. Inverse Problems & Imaging, 2019, 13 (5) : 1067-1094. doi: 10.3934/ipi.2019048
[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 & Continuous Dynamical Systems - A, 2008, 22 (4) : 955-972. doi: 10.3934/dcds.2008.22.955
[12]

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
[13]

Janne M.J. Huttunen, J. P. Kaipio. Approximation errors in nonstationary inverse problems. Inverse Problems & Imaging, 2007, 1 (1) : 77-93. doi: 10.3934/ipi.2007.1.77
[14]

Victor Isakov, Shuai Lu. Inverse source problems without (pseudo) convexity assumptions. Inverse Problems & Imaging, 2018, 12 (4) : 955-970. doi: 10.3934/ipi.2018040
[15]

Joep H.M. Evers, Sander C. Hille, Adrian Muntean. Modelling with measures: Approximation of a mass-emitting object by a point source. Mathematical Biosciences & Engineering, 2015, 12 (2) : 357-373. doi: 10.3934/mbe.2015.12.357
[16]

Victor Isakov, Joseph Myers. On the inverse doping profile problem. Inverse Problems & Imaging, 2012, 6 (3) : 465-486. doi: 10.3934/ipi.2012.6.465
[17]

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
[18]

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
[19]

Hui-Ling Li, Heng-Ling Wang, Xiao-Liu Wang. A quasilinear parabolic problem with a source term and a nonlocal absorption. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1945-1956. doi: 10.3934/cpaa.2018092
[20]

Xinmin Xiang. The long-time behaviour for nonlinear Schrödinger equation and its rational pseudospectral approximation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 469-488. doi: 10.3934/dcdsb.2005.5.469

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences