• Previous Article
    On the regularization of the inverse conductivity problem with discontinuous conductivities
  • IPI Home
  • This Issue
  • Next Article
    Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem
August  2008, 2(3): 373-395. doi: 10.3934/ipi.2008.2.373

Why linear sampling really seems to work

1. 

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

Received  June 2008 Revised  June 2008 Published  July 2008

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.
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.

[1]

Qinghua Wu, Guozheng Yan. The factorization method for a partially coated cavity in inverse scattering. Inverse Problems and Imaging, 2016, 10 (1) : 263-279. doi: 10.3934/ipi.2016.10.263

[2]

Fang Zeng. Extended sampling method for interior inverse scattering problems. Inverse Problems and Imaging, 2020, 14 (4) : 719-731. doi: 10.3934/ipi.2020033

[3]

Andreas Kirsch, Albert Ruiz. The Factorization Method for an inverse fluid-solid interaction scattering problem. Inverse Problems and Imaging, 2012, 6 (4) : 681-695. doi: 10.3934/ipi.2012.6.681

[4]

Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems and Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757

[5]

Jianliang Li, Jiaqing Yang, Bo Zhang. A linear sampling method for inverse acoustic scattering by a locally rough interface. Inverse Problems and Imaging, 2021, 15 (5) : 1247-1267. doi: 10.3934/ipi.2021036

[6]

Jun Guo, Qinghua Wu, Guozheng Yan. The factorization method for cracks in elastic scattering. Inverse Problems and Imaging, 2018, 12 (2) : 349-371. doi: 10.3934/ipi.2018016

[7]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems and Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[8]

Guanqiu Ma, Guanghui Hu. Factorization method for inverse time-harmonic elastic scattering with a single plane wave. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022050

[9]

Tielei Zhu, Jiaqing Yang. A non-iterative sampling method for inverse elastic wave scattering by rough surfaces. Inverse Problems and Imaging, 2022, 16 (4) : 997-1017. doi: 10.3934/ipi.2022009

[10]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems and Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[11]

Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems and Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577

[12]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems and Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291

[13]

Armin Lechleiter. The factorization method is independent of transmission eigenvalues. Inverse Problems and Imaging, 2009, 3 (1) : 123-138. doi: 10.3934/ipi.2009.3.123

[14]

Guanghui Hu, Andreas Kirsch, Tao Yin. Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves. Inverse Problems and Imaging, 2016, 10 (1) : 103-129. doi: 10.3934/ipi.2016.10.103

[15]

Weishi Yin, Jiawei Ge, Pinchao Meng, Fuheng Qu. A neural network method for the inverse scattering problem of impenetrable cavities. Electronic Research Archive, 2020, 28 (2) : 1123-1142. doi: 10.3934/era.2020062

[16]

Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems and Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029

[17]

Deyue Zhang, Yue Wu, Yinglin Wang, Yukun Guo. A direct imaging method for the exterior and interior inverse scattering problems. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022025

[18]

Shixu Meng. A sampling type method in an electromagnetic waveguide. Inverse Problems and Imaging, 2021, 15 (4) : 745-762. doi: 10.3934/ipi.2021012

[19]

Yosra Boukari, Houssem Haddar. The factorization method applied to cracks with impedance boundary conditions. Inverse Problems and Imaging, 2013, 7 (4) : 1123-1138. doi: 10.3934/ipi.2013.7.1123

[20]

Xiaodong Liu. The factorization method for scatterers with different physical properties. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 563-577. doi: 10.3934/dcdss.2015.8.563

2021 Impact Factor: 1.483

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]