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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, New York, third edition, 1979. Google Scholar

[2]

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

[3]

G. A. Baker and P. Graves-Morris, "Padé Approximants," Cambridge University Press, Cambridge, second edition, 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), 51-74. 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), 79-90. 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), 553-595. 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), 085006. 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), 651-663. 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, Dordrecht, 1996. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

V. Isakov, "Inverse Problems for Partial Differential Equations," Springer-Verlag, New York, second edition, 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), 1853-1863. 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), 1-29. doi: 10.1002/cpa.3009.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

G. A. Baker and P. Graves-Morris, "Padé Approximants," Cambridge University Press, Cambridge, second edition, 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), 51-74. 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), 79-90. 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), 553-595. 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), 085006. 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), 651-663. 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, Dordrecht, 1996. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

V. Isakov, "Inverse Problems for Partial Differential Equations," Springer-Verlag, New York, second edition, 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), 1853-1863. 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), 1-29. doi: 10.1002/cpa.3009.  Google Scholar

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