Why linear sampling really seems to work

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August 2008, 2(3): 373-395. doi: 10.3934/ipi.2008.2.373

Why linear sampling really seems to work

Martin Hanke ^1,

1.	Institute of Mathematics, Johannes Gutenberg-Universität, 55099 Mainz, Germany

Received June 2008 Revised June 2008 Published July 2008

Abstract
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We reconsider the Linear Sampling Method by Colton and Kirsch, and provide an analysis which may serve as a justification of the method for problems where the Factorization Method is known to work. As a by-product, however, we obtain convincing arguments that one popular implementation of the Linear Sampling Method may not be as robust as is commonly believed. Our approach stems from the theory of regularization methods for linear ill-posed operator equations. More precisely, we derive a novel asymptotic analysis of the Tikhonov method if the exact right-hand side is inconsistent, i.e., does not belong to the (dense) range of the corresponding operator. It appears possible that our results can be a starting point to derive a calibration of standard implementations of the Linear Sampling Method, in order to obtain reconstructions of the scattering obstacles that go beyond an approximate localization of their respective positions.

Keywords: factorization method., Inverse scattering, sampling method.

Mathematics Subject Classification: Primary: 35R30, 65N2.

Citation: Martin Hanke. Why linear sampling really seems to work. Inverse Problems and Imaging, 2008, 2 (3) : 373-395. doi: 10.3934/ipi.2008.2.373

References:

[1]	T. Arens, Why linear sampling works, Inverse Problems, 20 (2004), 163-173. doi: 10.1088/0266-5611/20/1/010.
[2]	T. Arens and A. Lechleiter, "The Linear Sampling Method Revisited," manuscript, 2007.
[3]	F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory: An Introduction," Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006.
[4]	A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for the transmission problem in three-dimensional linear elasticity, Inverse Problems, 18 (2002), 547-558. doi: 10.1088/0266-5611/18/3/303.
[5]	D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003.
[6]	D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," 2nd Ed., Applied Mathematical Sciences, 93, Springer, Berlin, 1998.
[7]	D. Colton and P. Monk, A linear sampling method for the detection of leukemia using microwaves II, SIAM J. Appl. Math., 60 (1999), 241-255. doi: 10.1137/S003613999834426X.
[8]	D. Colton, M. Piani and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems, Inverse Problems, 13 (1997), 1477-1493. doi: 10.1088/0266-5611/13/6/005.
[9]	H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.
[10]	B. Gebauer, M. Hanke, A. Kirsch, W. Muniz and C. Schneider, A sampling method for detecting buried objects using electromagnetic scattering, Inverse Problems, 21 (2005), 2035-2050. doi: 10.1088/0266-5611/21/6/015.
[11]	I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series and Products," 7th Ed., Academic Press, New York, 2007.
[12]	C. W. Groetsch, "The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind," Research Notes in Mathematics, 105, Pitman (Advanced Publishing Program), Boston, MA, 1984.
[13]	H. Haddar and P. Monk, The linear sampling method for solving the electromagnetic inverse medium problem, Inverse Problems, 18 (2002), 891-906. doi: 10.1088/0266-5611/18/3/323.
[14]	A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009.
[15]	A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford University Press, Oxford, 2008.
[16]	P. Monk, "Finite Element Methods for Maxwell's Equations," Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2003.
[17]	V. A. Morozov, On the solution of functional equations by the method of regularization, Dokl. Akad. Nauk SSSR, 167, 510-512 (Russian), translated as Soviet Math. Dokl., 7 (1966), 414-417.
[18]	V. A. Morozov, "Methods for Solving Incorrectly Posed Problems," Translated from the Russian by A. B. Aries. Translation edited by Z. Nashed. Springer-Verlag, New York, 1984.
[19]	S. Nintcheu Fata and B. B. Guzina, A linear sampling method for near-field inverse problems in elastodynamics, Inverse Problems, 20 (2004), 713-736. doi: 10.1088/0266-5611/20/3/005.
[20]	A. Tacchino, J. Coyle and M. Piana, Numerical validation of the linear sampling method, Inverse Problems, 18 (2002), 511-527. doi: 10.1088/0266-5611/18/3/301.
[21]	G. M. Vainikko, The discrepancy principle for a class of regularization methods, USSR Comp. Math. Math. Phys., 22 (1982), 1-19. doi: 10.1016/0041-5553(82)90120-3.
[22]	G. M. Vainikko, The critical level of discrepancy in regularization methods, USSR Comp. Math. Math. Phys., 23 (1983), 1-19. doi: 10.1016/S0041-5553(83)80068-8.

show all references

References:

[1]	T. Arens, Why linear sampling works, Inverse Problems, 20 (2004), 163-173. doi: 10.1088/0266-5611/20/1/010.
[2]	T. Arens and A. Lechleiter, "The Linear Sampling Method Revisited," manuscript, 2007.
[3]	F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory: An Introduction," Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006.
[4]	A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for the transmission problem in three-dimensional linear elasticity, Inverse Problems, 18 (2002), 547-558. doi: 10.1088/0266-5611/18/3/303.
[5]	D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003.
[6]	D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," 2nd Ed., Applied Mathematical Sciences, 93, Springer, Berlin, 1998.
[7]	D. Colton and P. Monk, A linear sampling method for the detection of leukemia using microwaves II, SIAM J. Appl. Math., 60 (1999), 241-255. doi: 10.1137/S003613999834426X.
[8]	D. Colton, M. Piani and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems, Inverse Problems, 13 (1997), 1477-1493. doi: 10.1088/0266-5611/13/6/005.
[9]	H. W. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems," Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.
[10]	B. Gebauer, M. Hanke, A. Kirsch, W. Muniz and C. Schneider, A sampling method for detecting buried objects using electromagnetic scattering, Inverse Problems, 21 (2005), 2035-2050. doi: 10.1088/0266-5611/21/6/015.
[11]	I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series and Products," 7th Ed., Academic Press, New York, 2007.
[12]	C. W. Groetsch, "The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind," Research Notes in Mathematics, 105, Pitman (Advanced Publishing Program), Boston, MA, 1984.
[13]	H. Haddar and P. Monk, The linear sampling method for solving the electromagnetic inverse medium problem, Inverse Problems, 18 (2002), 891-906. doi: 10.1088/0266-5611/18/3/323.
[14]	A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009.
[15]	A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford University Press, Oxford, 2008.
[16]	P. Monk, "Finite Element Methods for Maxwell's Equations," Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2003.
[17]	V. A. Morozov, On the solution of functional equations by the method of regularization, Dokl. Akad. Nauk SSSR, 167, 510-512 (Russian), translated as Soviet Math. Dokl., 7 (1966), 414-417.
[18]	V. A. Morozov, "Methods for Solving Incorrectly Posed Problems," Translated from the Russian by A. B. Aries. Translation edited by Z. Nashed. Springer-Verlag, New York, 1984.
[19]	S. Nintcheu Fata and B. B. Guzina, A linear sampling method for near-field inverse problems in elastodynamics, Inverse Problems, 20 (2004), 713-736. doi: 10.1088/0266-5611/20/3/005.
[20]	A. Tacchino, J. Coyle and M. Piana, Numerical validation of the linear sampling method, Inverse Problems, 18 (2002), 511-527. doi: 10.1088/0266-5611/18/3/301.
[21]	G. M. Vainikko, The discrepancy principle for a class of regularization methods, USSR Comp. Math. Math. Phys., 22 (1982), 1-19. doi: 10.1016/0041-5553(82)90120-3.
[22]	G. M. Vainikko, The critical level of discrepancy in regularization methods, USSR Comp. Math. Math. Phys., 23 (1983), 1-19. doi: 10.1016/S0041-5553(83)80068-8.

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