-
Previous Article
Model-based reconstruction for magnetic particle imaging in 2D and 3D
- IPI Home
- This Issue
-
Next Article
A coupled total variation model with curvature driven for image colorization
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, New York, 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), 512-565.
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), 702-733.
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), 1745-1768.
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), 095016, 21pp.
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), 23-51.
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), 221-223.
doi: 10.1109/36.368205. |
[8] |
S. I. Kabanikhin, On linear regularization of multidimensional inverse problems for hyperbolic equations, Soviet Mathematics Doklady, 40 (1990), 579-583. |
[9] |
S. I. Kabanikhin, A. D. Satybaev and M. A. Shishlenin, Direct Methods of Solving Inverse Hyperbolic Problems, VSP, Utrecht, 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), 439-450.
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), 111-127.
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), 1653-1675.
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), 577-596.
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, Utrecht, The Netherlands, 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), 518-537.
doi: 10.1137/140981198. |
[16] |
M. V. Klibanov, Carleman estimates for the regularization of ill-posed Cauchy problems, Applied Numerical Mathematics, 94 (2015), 46-74.
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), 987-990 (in Russian). |
[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), 095007.
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), 2937-2948.
doi: 10.1109/TGRS.2012.2211885. |
[20] |
R. Lattes and J.-L. Lions, The Method of Quasireversibility: Applications to Partial Differential Equations, Elsevier, New York, 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), 155-173. |
[22] |
B. M. Levitan, Inverse Sturm-Liouville Problems, VSP, Utrecht, 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), 65530H, 8pp. |
[24] |
V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press, b.v., Utrecht, 1987. |
[25] |
C. Sanderson, Armadillo: An Open Source C++ Linear Algebra Library for Fast Prototyping and Computationally Intensive Experiments, Technical Report, NICTA, 2010. |
[26] |
J. A. Scales, M. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems, J. Computational Physics, 103 (1992), 258-268. |
[27] |
, Table of dielectric constants, https://goo.gl/kAxtzB. |
[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), B273-B293.
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), 757-786.
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, London, 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), 115-194. |
[32] |
B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach Science Publishers, New York, 1989. |
[33] |
V. S. Vladimirov, Equations of Mathematical Physics, M. Dekker, New York, 1971. |
show all references
References:
[1] |
L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 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), 512-565.
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), 702-733.
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), 1745-1768.
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), 095016, 21pp.
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), 23-51.
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), 221-223.
doi: 10.1109/36.368205. |
[8] |
S. I. Kabanikhin, On linear regularization of multidimensional inverse problems for hyperbolic equations, Soviet Mathematics Doklady, 40 (1990), 579-583. |
[9] |
S. I. Kabanikhin, A. D. Satybaev and M. A. Shishlenin, Direct Methods of Solving Inverse Hyperbolic Problems, VSP, Utrecht, 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), 439-450.
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), 111-127.
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), 1653-1675.
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), 577-596.
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, Utrecht, The Netherlands, 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), 518-537.
doi: 10.1137/140981198. |
[16] |
M. V. Klibanov, Carleman estimates for the regularization of ill-posed Cauchy problems, Applied Numerical Mathematics, 94 (2015), 46-74.
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), 987-990 (in Russian). |
[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), 095007.
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), 2937-2948.
doi: 10.1109/TGRS.2012.2211885. |
[20] |
R. Lattes and J.-L. Lions, The Method of Quasireversibility: Applications to Partial Differential Equations, Elsevier, New York, 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), 155-173. |
[22] |
B. M. Levitan, Inverse Sturm-Liouville Problems, VSP, Utrecht, 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), 65530H, 8pp. |
[24] |
V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press, b.v., Utrecht, 1987. |
[25] |
C. Sanderson, Armadillo: An Open Source C++ Linear Algebra Library for Fast Prototyping and Computationally Intensive Experiments, Technical Report, NICTA, 2010. |
[26] |
J. A. Scales, M. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems, J. Computational Physics, 103 (1992), 258-268. |
[27] |
, Table of dielectric constants, https://goo.gl/kAxtzB. |
[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), B273-B293.
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), 757-786.
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, London, 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), 115-194. |
[32] |
B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach Science Publishers, New York, 1989. |
[33] |
V. S. Vladimirov, Equations of Mathematical Physics, M. Dekker, New York, 1971. |
[1] |
Michael V. Klibanov, Dinh-Liem Nguyen, Loc H. Nguyen, Hui Liu. A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data. Inverse Problems and Imaging, 2018, 12 (2) : 493-523. doi: 10.3934/ipi.2018021 |
[2] |
Michael V. Klibanov, Thuy T. Le, Loc H. Nguyen, Anders Sullivan, Lam Nguyen. Convexification-based globally convergent numerical method for a 1D coefficient inverse problem with experimental data. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2021068 |
[3] |
Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems and Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139 |
[4] |
Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems and Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951 |
[5] |
Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems and Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010 |
[6] |
Lei Guo, Gui-Hua Lin. Globally convergent algorithm for solving stationary points for mathematical programs with complementarity constraints via nonsmooth reformulations. Journal of Industrial and Management Optimization, 2013, 9 (2) : 305-322. doi: 10.3934/jimo.2013.9.305 |
[7] |
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 |
[8] |
Peijun Li, Xiaokai Yuan. Inverse obstacle scattering for elastic waves in three dimensions. Inverse Problems and Imaging, 2019, 13 (3) : 545-573. doi: 10.3934/ipi.2019026 |
[9] |
Lu Zhao, Heping Dong, Fuming Ma. Inverse obstacle scattering for acoustic waves in the time domain. Inverse Problems and Imaging, 2021, 15 (5) : 1269-1286. doi: 10.3934/ipi.2021037 |
[10] |
Krunal B. Kachhia, Abdon Atangana. Electromagnetic waves described by a fractional derivative of variable and constant order with non singular kernel. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2357-2371. doi: 10.3934/dcdss.2020172 |
[11] |
Xiaoxu Xu, Bo Zhang, Haiwen Zhang. Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency. Inverse Problems and Imaging, 2020, 14 (3) : 489-510. doi: 10.3934/ipi.2020023 |
[12] |
Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems and Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131 |
[13] |
Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems and Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551 |
[14] |
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 |
[15] |
Weijun Zhou. A globally convergent BFGS method for symmetric nonlinear equations. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1295-1303. doi: 10.3934/jimo.2021020 |
[16] |
Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Ⅱ. Bore propagation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5543-5569. doi: 10.3934/dcds.2019244 |
[17] |
Loc H. Nguyen, Qitong Li, Michael V. Klibanov. A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media. Inverse Problems and Imaging, 2019, 13 (5) : 1067-1094. doi: 10.3934/ipi.2019048 |
[18] |
Brian Sleeman. The inverse acoustic obstacle scattering problem and its interior dual. Inverse Problems and Imaging, 2009, 3 (2) : 211-229. doi: 10.3934/ipi.2009.3.211 |
[19] |
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 |
[20] |
Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems and Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035 |
2021 Impact Factor: 1.483
Tools
Metrics
Other articles
by authors
[Back to Top]