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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), 163. 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, (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), 547. 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), 383. doi: 10.1088/02665611/12/4/003. 
[6] 
D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," 2nd Ed.,, Applied Mathematical Sciences, (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. 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. 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 (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. 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, (2007). 
[12] 
C. W. Groetsch, "The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind,", Research Notes in Mathematics, 105 (1984). 
[13] 
H. Haddar and P. Monk, The linear sampling method for solving the electromagnetic inverse medium problem,, Inverse Problems, 18 (2002), 891. 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), 1489. doi: 10.1088/02665611/14/6/009. 
[15] 
A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems,", Oxford University Press, (2008). 
[16] 
P. Monk, "Finite Element Methods for Maxwell's Equations,", Numerical Mathematics and Scientific Computation. Oxford University Press, (2003). 
[17] 
V. A. Morozov, On the solution of functional equations by the method of regularization,, Dokl. Akad. Nauk SSSR, 7 (1966), 510. 
[18] 
V. A. Morozov, "Methods for Solving Incorrectly Posed Problems,", Translated from the Russian by A. B. Aries. Translation edited by Z. Nashed. SpringerVerlag, (1984). 
[19] 
S. Nintcheu Fata and B. B. Guzina, A linear sampling method for nearfield inverse problems in elastodynamics,, Inverse Problems, 20 (2004), 713. 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), 511. 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), 1. 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), 1. doi: 10.1016/S00415553(83)800688. 
show all references
References:
[1] 
T. Arens, Why linear sampling works,, Inverse Problems, 20 (2004), 163. 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, (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), 547. 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), 383. doi: 10.1088/02665611/12/4/003. 
[6] 
D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," 2nd Ed.,, Applied Mathematical Sciences, (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. 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. 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 (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. 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, (2007). 
[12] 
C. W. Groetsch, "The Theory of Tikhonov Regularization for Fredholm Integral Equations of the First Kind,", Research Notes in Mathematics, 105 (1984). 
[13] 
H. Haddar and P. Monk, The linear sampling method for solving the electromagnetic inverse medium problem,, Inverse Problems, 18 (2002), 891. 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), 1489. doi: 10.1088/02665611/14/6/009. 
[15] 
A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems,", Oxford University Press, (2008). 
[16] 
P. Monk, "Finite Element Methods for Maxwell's Equations,", Numerical Mathematics and Scientific Computation. Oxford University Press, (2003). 
[17] 
V. A. Morozov, On the solution of functional equations by the method of regularization,, Dokl. Akad. Nauk SSSR, 7 (1966), 510. 
[18] 
V. A. Morozov, "Methods for Solving Incorrectly Posed Problems,", Translated from the Russian by A. B. Aries. Translation edited by Z. Nashed. SpringerVerlag, (1984). 
[19] 
S. Nintcheu Fata and B. B. Guzina, A linear sampling method for nearfield inverse problems in elastodynamics,, Inverse Problems, 20 (2004), 713. 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), 511. 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), 1. 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), 1. doi: 10.1016/S00415553(83)800688. 
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