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

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

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.
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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