A globally convergent numerical method for a 1-d inverse medium problem with experimental data

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November 2016, 10(4): 1057-1085. doi: 10.3934/ipi.2016032

A globally convergent numerical method for a 1-d inverse medium problem with experimental data

Michael V. Klibanov ^1, , Loc H. Nguyen ^1, , Anders Sullivan ^2, and Lam Nguyen ^2,

1.	Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28213, United States, United States
2.	US Army Research Laboratory, 2800 Powder Mill Road, Adelphy, MD 20783-1197, United States, United States

Received February 2016 Revised April 2016 Published October 2016

Abstract
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In this paper, a reconstruction method for the spatially distributed dielectric constant of a medium from the back scattering wave field in the frequency domain is considered. Our approach is to propose a globally convergent algorithm, which does not require any knowledge of a small neighborhood of the solution of the inverse problem in advance. The Quasi-Reversibility Method (QRM) is used in the algorithm. The convergence of the QRM is proved via a Carleman estimate. The method is tested on both computationally simulated and experimental data.

Keywords: electromagnetic waves., Coefficient inverse scattering problem, globally convergent algorithm, dielectric constant.

Mathematics Subject Classification: 34L25, 35P25, 78A4.

Citation: Michael V. Klibanov, Loc H. Nguyen, Anders Sullivan, Lam Nguyen. A globally convergent numerical method for a 1-d inverse medium problem with experimental data. Inverse Problems & Imaging, 2016, 10 (4) : 1057-1085. doi: 10.3934/ipi.2016032

References:

[1]	L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems,, Springer, (2012). doi: 10.1007/978-1-4419-7805-9.
[2]	L. Beilina and M. V. Klibanov, A new approximate mathematical model for global convergence for a coefficient inverse problem with backscattering data,, J. Inverse and Ill-Posed Problems, 20 (2012), 512. doi: 10.1515/jip-2012-0063.
[3]	L. Beilina, Energy estimates and numerical verification of the stabilized domain decomposition finite element/finite difference approach for the Maxwell's system in time domain,, Central European Journal of Mathematics, 11 (2013), 702. doi: 10.2478/s11533-013-0202-3.
[4]	L. Bourgeois and J. Dardé, About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains,, Applicable Analalysis, 89 (2010), 1745. doi: 10.1080/00036810903393809.
[5]	L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/9/095016.
[6]	L. Bourgeois and J. Dardé, The "exterior approach" to solve the inverse obstacle problem for the Stokes system,, Inverse Problems and Imaging, 8 (2014), 23. doi: 10.3934/ipi.2014.8.23.
[7]	H. T. Chuah, K. Y. Lee and T. W. Lau, Dielectric constants of rubber and oil palm leaf samples at X-band,, IEEE Trans. on Geoscience and Remote Sensing, 33 (1995), 221. doi: 10.1109/36.368205.
[8]	S. I. Kabanikhin, On linear regularization of multidimensional inverse problems for hyperbolic equations,, Soviet Mathematics Doklady, 40 (1990), 579.
[9]	S. I. Kabanikhin, A. D. Satybaev and M. A. Shishlenin, Direct Methods of Solving Inverse Hyperbolic Problems,, VSP, (2005).
[10]	S. I. Kabanikhin, K. K. Sabelfeld, N. S. Novikov and M. A. Shishlenin, Numerical solution of the multidimensional Gelfand-Levitan equation,, J. Inverse and Ill-Posed Problems, 23 (2015), 439. doi: 10.1515/jiip-2014-0018.
[11]	A. L. Karchevsky, M. V. Klibanov, L. Nguyen, N. Pantong and A. Sullivan, The Krein method and the globally convergent method for experimental data,, Applied Numerical Mathematics, 74 (2013), 111. doi: 10.1016/j.apnum.2013.09.003.
[12]	M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation,, SIAM J. Appl. Math, 51 (1991), 1653. doi: 10.1137/0151085.
[13]	M. V. Klibanov and J. Malinsky, Newton-Kantorovich method for 3-dimensional potential inverse scattering problem and stability for the hyperbolic Cauchy problem with time dependent data,, Inverse Problems, 7 (1991), 577. doi: 10.1088/0266-5611/7/4/007.
[14]	M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications,, VSP, (2004). doi: 10.1515/9783110915549.
[15]	M. V. Klibanov and N. T. Thành, Recovering dielectric constants of explosives via a globally strictly convex cost functional,, SIAM J. Appl. Math, 75 (2015), 518. doi: 10.1137/140981198.
[16]	M. V. Klibanov, Carleman estimates for the regularization of ill-posed Cauchy problems,, Applied Numerical Mathematics, 94 (2015), 46. doi: 10.1016/j.apnum.2015.02.003.
[17]	M. G. Krein, On a method of effective solution of an inverse boundary problem,, Dokl. Akad. Nauk SSSR, 94 (1954), 987.
[18]	A. V. Kuzhuget, L. Beilina, M. V. Klibanov, A. Sullivan, L. Nguyen and M. A. Fiddy, Blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/9/095007.
[19]	A. V. Kuzhuget, L. Beilina, M. V. Klibanov, A. Sullivan, L. Nguyen and M. A. Fiddy, Quantitative image recovery from measured blind backscattered data using a globally convergent inverse method,, IEEE Transaction for Geoscience and Remote Sensing, 51 (2013), 2937. doi: 10.1109/TGRS.2012.2211885.
[20]	R. Lattes and J.-L. Lions, The Method of Quasireversibility: Applications to Partial Differential Equations,, Elsevier, (1969).
[21]	D. Lesnic, G. Wakefield, B. D. Sleeman and J. R. Okendon, Determination of the index of refraction of ant-reflection coatings,, Mathematics-in-Industry Case Studies Journal, 2 (2010), 155.
[22]	B. M. Levitan, Inverse Sturm-Liouville Problems,, VSP, (1987).
[23]	N. Nguyen, D. Wong, M. Ressler, F. Koenig, B. Stanton, G. Smith, J. Sichina and K. Kappra, Obstacle avolidance and concealed target detection using the Army Research Lab ultra-wideband synchronous impulse Reconstruction (UWB SIRE) forward imaging radar,, Proc. SPIE 6553 (2007), 6553 (2007).
[24]	V. G. Romanov, Inverse Problems of Mathematical Physics,, VNU Science Press, (1987).
[25]	C. Sanderson, Armadillo: An Open Source C++ Linear Algebra Library for Fast Prototyping and Computationally Intensive Experiments,, Technical Report, (2010).
[26]	J. A. Scales, M. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems,, J. Computational Physics, 103 (1992), 258.
[27]	, Table of dielectric constants,, , ().
[28]	N. T. Thành, L. Beilina, M. V. Klibanov and M. A. Fiddy, Reconstruction of the refractive index from experimental backscattering data using a globally convergent inverse method,, SIAM Journal on Scientific Computing, 36 (2014). doi: 10.1137/130924962.
[29]	N. T. Thành, L. Beilina, M. V. Klibanov and M. A. Fiddy, Imaging of buried objects from experimental backscattering time dependent measurements using a globally convergent inverse algorithm,, SIAM J. Imaging Sciences, 8 (2015), 757. doi: 10.1137/140972469.
[30]	A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems,, Kluwer, (1995). doi: 10.1007/978-94-015-8480-7.
[31]	B. R. Vainberg, Principles of radiation, limiting absorption and limiting amplitude in the general theory of partial differential equations,, Russian Math. Surveys, 21 (1966), 115.
[32]	B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics,, Gordon and Breach Science Publishers, (1989).
[33]	V. S. Vladimirov, Equations of Mathematical Physics,, M. Dekker, (1971).

show all references

References:

[1]	L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems,, Springer, (2012). doi: 10.1007/978-1-4419-7805-9.
[2]	L. Beilina and M. V. Klibanov, A new approximate mathematical model for global convergence for a coefficient inverse problem with backscattering data,, J. Inverse and Ill-Posed Problems, 20 (2012), 512. doi: 10.1515/jip-2012-0063.
[3]	L. Beilina, Energy estimates and numerical verification of the stabilized domain decomposition finite element/finite difference approach for the Maxwell's system in time domain,, Central European Journal of Mathematics, 11 (2013), 702. doi: 10.2478/s11533-013-0202-3.
[4]	L. Bourgeois and J. Dardé, About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains,, Applicable Analalysis, 89 (2010), 1745. doi: 10.1080/00036810903393809.
[5]	L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/9/095016.
[6]	L. Bourgeois and J. Dardé, The "exterior approach" to solve the inverse obstacle problem for the Stokes system,, Inverse Problems and Imaging, 8 (2014), 23. doi: 10.3934/ipi.2014.8.23.
[7]	H. T. Chuah, K. Y. Lee and T. W. Lau, Dielectric constants of rubber and oil palm leaf samples at X-band,, IEEE Trans. on Geoscience and Remote Sensing, 33 (1995), 221. doi: 10.1109/36.368205.
[8]	S. I. Kabanikhin, On linear regularization of multidimensional inverse problems for hyperbolic equations,, Soviet Mathematics Doklady, 40 (1990), 579.
[9]	S. I. Kabanikhin, A. D. Satybaev and M. A. Shishlenin, Direct Methods of Solving Inverse Hyperbolic Problems,, VSP, (2005).
[10]	S. I. Kabanikhin, K. K. Sabelfeld, N. S. Novikov and M. A. Shishlenin, Numerical solution of the multidimensional Gelfand-Levitan equation,, J. Inverse and Ill-Posed Problems, 23 (2015), 439. doi: 10.1515/jiip-2014-0018.
[11]	A. L. Karchevsky, M. V. Klibanov, L. Nguyen, N. Pantong and A. Sullivan, The Krein method and the globally convergent method for experimental data,, Applied Numerical Mathematics, 74 (2013), 111. doi: 10.1016/j.apnum.2013.09.003.
[12]	M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation,, SIAM J. Appl. Math, 51 (1991), 1653. doi: 10.1137/0151085.
[13]	M. V. Klibanov and J. Malinsky, Newton-Kantorovich method for 3-dimensional potential inverse scattering problem and stability for the hyperbolic Cauchy problem with time dependent data,, Inverse Problems, 7 (1991), 577. doi: 10.1088/0266-5611/7/4/007.
[14]	M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications,, VSP, (2004). doi: 10.1515/9783110915549.
[15]	M. V. Klibanov and N. T. Thành, Recovering dielectric constants of explosives via a globally strictly convex cost functional,, SIAM J. Appl. Math, 75 (2015), 518. doi: 10.1137/140981198.
[16]	M. V. Klibanov, Carleman estimates for the regularization of ill-posed Cauchy problems,, Applied Numerical Mathematics, 94 (2015), 46. doi: 10.1016/j.apnum.2015.02.003.
[17]	M. G. Krein, On a method of effective solution of an inverse boundary problem,, Dokl. Akad. Nauk SSSR, 94 (1954), 987.
[18]	A. V. Kuzhuget, L. Beilina, M. V. Klibanov, A. Sullivan, L. Nguyen and M. A. Fiddy, Blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/9/095007.
[19]	A. V. Kuzhuget, L. Beilina, M. V. Klibanov, A. Sullivan, L. Nguyen and M. A. Fiddy, Quantitative image recovery from measured blind backscattered data using a globally convergent inverse method,, IEEE Transaction for Geoscience and Remote Sensing, 51 (2013), 2937. doi: 10.1109/TGRS.2012.2211885.
[20]	R. Lattes and J.-L. Lions, The Method of Quasireversibility: Applications to Partial Differential Equations,, Elsevier, (1969).
[21]	D. Lesnic, G. Wakefield, B. D. Sleeman and J. R. Okendon, Determination of the index of refraction of ant-reflection coatings,, Mathematics-in-Industry Case Studies Journal, 2 (2010), 155.
[22]	B. M. Levitan, Inverse Sturm-Liouville Problems,, VSP, (1987).
[23]	N. Nguyen, D. Wong, M. Ressler, F. Koenig, B. Stanton, G. Smith, J. Sichina and K. Kappra, Obstacle avolidance and concealed target detection using the Army Research Lab ultra-wideband synchronous impulse Reconstruction (UWB SIRE) forward imaging radar,, Proc. SPIE 6553 (2007), 6553 (2007).
[24]	V. G. Romanov, Inverse Problems of Mathematical Physics,, VNU Science Press, (1987).
[25]	C. Sanderson, Armadillo: An Open Source C++ Linear Algebra Library for Fast Prototyping and Computationally Intensive Experiments,, Technical Report, (2010).
[26]	J. A. Scales, M. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems,, J. Computational Physics, 103 (1992), 258.
[27]	, Table of dielectric constants,, , ().
[28]	N. T. Thành, L. Beilina, M. V. Klibanov and M. A. Fiddy, Reconstruction of the refractive index from experimental backscattering data using a globally convergent inverse method,, SIAM Journal on Scientific Computing, 36 (2014). doi: 10.1137/130924962.
[29]	N. T. Thành, L. Beilina, M. V. Klibanov and M. A. Fiddy, Imaging of buried objects from experimental backscattering time dependent measurements using a globally convergent inverse algorithm,, SIAM J. Imaging Sciences, 8 (2015), 757. doi: 10.1137/140972469.
[30]	A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems,, Kluwer, (1995). doi: 10.1007/978-94-015-8480-7.
[31]	B. R. Vainberg, Principles of radiation, limiting absorption and limiting amplitude in the general theory of partial differential equations,, Russian Math. Surveys, 21 (1966), 115.
[32]	B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics,, Gordon and Breach Science Publishers, (1989).
[33]	V. S. Vladimirov, Equations of Mathematical Physics,, M. Dekker, (1971).

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