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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 |
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). Google Scholar |
[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. Google Scholar |
[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). Google Scholar |
[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). Google Scholar |
[26] |
J. A. Scales, M. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems,, J. Computational Physics, 103 (1992), 258. Google Scholar |
[27] |
, Table of dielectric constants,, , (). Google Scholar |
[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). Google Scholar |
[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. Google Scholar |
[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). Google Scholar |
[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). Google Scholar |
[26] |
J. A. Scales, M. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems,, J. Computational Physics, 103 (1992), 258. Google Scholar |
[27] |
, Table of dielectric constants,, , (). Google Scholar |
[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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