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Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem
Why linear sampling really seems to work
1.  Institute of Mathematics, Johannes GutenbergUniversität, 55099 Mainz, Germany 
References:
[1] 
T. Arens, Why linear sampling works, Inverse Problems, 20 (2004), 163173. doi: 10.1088/02665611/20/1/010. 
[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, SpringerVerlag, Berlin, 2006. 
[4] 
A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for the transmission problem in threedimensional linear elasticity, Inverse Problems, 18 (2002), 547558. doi: 10.1088/02665611/18/3/303. 
[5] 
D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383393. doi: 10.1088/02665611/12/4/003. 
[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), 241255. 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), 14771493. doi: 10.1088/02665611/13/6/005. 
[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), 20352050. doi: 10.1088/02665611/21/6/015. 
[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), 891906. doi: 10.1088/02665611/18/3/323. 
[14] 
A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 14891512. doi: 10.1088/02665611/14/6/009. 
[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, 510512 (Russian), translated as Soviet Math. Dokl., 7 (1966), 414417. 
[18] 
V. A. Morozov, "Methods for Solving Incorrectly Posed Problems," Translated from the Russian by A. B. Aries. Translation edited by Z. Nashed. SpringerVerlag, New York, 1984. 
[19] 
S. Nintcheu Fata and B. B. Guzina, A linear sampling method for nearfield inverse problems in elastodynamics, Inverse Problems, 20 (2004), 713736. doi: 10.1088/02665611/20/3/005. 
[20] 
A. Tacchino, J. Coyle and M. Piana, Numerical validation of the linear sampling method, Inverse Problems, 18 (2002), 511527. doi: 10.1088/02665611/18/3/301. 
[21] 
G. M. Vainikko, The discrepancy principle for a class of regularization methods, USSR Comp. Math. Math. Phys., 22 (1982), 119. doi: 10.1016/00415553(82)901203. 
[22] 
G. M. Vainikko, The critical level of discrepancy in regularization methods, USSR Comp. Math. Math. Phys., 23 (1983), 119. doi: 10.1016/S00415553(83)800688. 
show all references
References:
[1] 
T. Arens, Why linear sampling works, Inverse Problems, 20 (2004), 163173. doi: 10.1088/02665611/20/1/010. 
[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, SpringerVerlag, Berlin, 2006. 
[4] 
A. Charalambopoulos, D. Gintides and K. Kiriaki, The linear sampling method for the transmission problem in threedimensional linear elasticity, Inverse Problems, 18 (2002), 547558. doi: 10.1088/02665611/18/3/303. 
[5] 
D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383393. doi: 10.1088/02665611/12/4/003. 
[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), 241255. 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), 14771493. doi: 10.1088/02665611/13/6/005. 
[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), 20352050. doi: 10.1088/02665611/21/6/015. 
[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), 891906. doi: 10.1088/02665611/18/3/323. 
[14] 
A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 14891512. doi: 10.1088/02665611/14/6/009. 
[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, 510512 (Russian), translated as Soviet Math. Dokl., 7 (1966), 414417. 
[18] 
V. A. Morozov, "Methods for Solving Incorrectly Posed Problems," Translated from the Russian by A. B. Aries. Translation edited by Z. Nashed. SpringerVerlag, New York, 1984. 
[19] 
S. Nintcheu Fata and B. B. Guzina, A linear sampling method for nearfield inverse problems in elastodynamics, Inverse Problems, 20 (2004), 713736. doi: 10.1088/02665611/20/3/005. 
[20] 
A. Tacchino, J. Coyle and M. Piana, Numerical validation of the linear sampling method, Inverse Problems, 18 (2002), 511527. doi: 10.1088/02665611/18/3/301. 
[21] 
G. M. Vainikko, The discrepancy principle for a class of regularization methods, USSR Comp. Math. Math. Phys., 22 (1982), 119. doi: 10.1016/00415553(82)901203. 
[22] 
G. M. Vainikko, The critical level of discrepancy in regularization methods, USSR Comp. Math. Math. Phys., 23 (1983), 119. doi: 10.1016/S00415553(83)800688. 
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