On rational approximation methods for inverse source problems

Martin Hanke; William Rundell

doi:10.3934/ipi.2011.5.185

Article Contents

2011, Volume 5, Issue 1: 185-202. Doi: 10.3934/ipi.2011.5.185

This issue Previous Article Correlation priors Next Article Acoustically invisible gateways

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

Abstract / Introduction Related Papers Cited by

Abstract

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:

Inverse source problem,
rational approximation.

Mathematics Subject Classification: Primary: 35R20.

Citation:

Related Papers

Cited by

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.