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Convexification of a 3-D coefficient inverse scattering problem. (English) Zbl 1442.78007

Summary: A version of the so-called “convexification” numerical method for a coefficient inverse scattering problem for the 3D Helmholtz equation is developed analytically and tested numerically. Backscattering data are used, which result from a single direction of the propagation of the incident plane wave on an interval of frequencies. The method converges globally. The idea is to construct a weighted Tikhonov-like functional. The key element of this functional is the presence of the so-called Carleman Weight Function (CWF). This is the function which is involved in the Carleman estimate for the Laplace operator. This functional is strictly convex on any appropriate ball in a Hilbert space for an appropriate choice of the parameters of the CWF. Thus, both the absence of local minima and convergence of minimizers to the exact solution are guaranteed. Numerical tests demonstrate a good performance of the resulting algorithm. Unlikeprevious the so-called tail functions globally convergent method, we neither do not impose the smallness assumption of the interval of wavenumbers, nor we do not iterate with respect to the so-called tail functions.

MSC:

78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs

Software:

KAIRUAIN
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References:

[1] Klibanov, M. V.; Kolesov, A. E.; Nguyen, L.; Sullivan, A., Globally strictly convex cost functional for a 1-D inverse medium scattering problem with experimental data, SIAM J. Appl. Math., 77, 5, 17331755 (2017)
[2] Beilina, L.; Klibanov, M. V., A globally convergent numerical method for a coefficient inverse problem, SIAM J. Sci. Comput., 31, 1, 478-509 (2008) · Zbl 1185.65175
[3] Beilina, L.; Klibanov, M. V., Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, 407 (2012), Springer · Zbl 1255.65168
[4] Klibanov, M. V.; Nguyen, D.-L.; Nguyen, L. H.; Liu, H., A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data, Inverse Probl. Imaging, 12, 2, 493-523 (2018) · Zbl 1395.35216
[5] Kolesov, A. E.; Klibanov, M. V.; Nguyen, L. H.; Nguyen, D.-L.; Thanh, N. T., Single measurement experimental data for an inverse medium problem inverted by a multi-frequency globally convergent numerical method, Appl. Numer. Math., 120, 176-196 (2017) · Zbl 1370.65066
[6] Nguyen, D.-L.; Klibanov, M. V.; Nguyen, L. H.; Fiddy, M. A., Imaging of buried objects from multi-frequency experimental data using a globally convergent inversion method, J. Inverse Ill-Posed Probl. (2017), in press. Available online, http://dx.doi.org/10.1515/Jiip-2017-0047
[7] Nguyen, D.-L.; Klibanov, M. V.; Nguyen, L. H.; Kolesov, A. E.; Fiddy, M. A.; Liu, H., Numerical solution of a coefficient inverse problem with multi-frequency experimental raw data by a globally convergent algorithm, J. Comput. Phys., 345, 17-32 (2017) · Zbl 1378.78039
[8] Beilina, L.; Klibanov, M. V., Globally strongly convex cost functional for a coefficient inverse problem, Nonlinear Anal. RWA, 22, 272-288 (2015) · Zbl 06373713
[9] Klibanov, M. V.; Ioussoupova, O. V., Uniform strict convexity of a cost functional for three-dimensional inverse scattering problem, SIAM J. Math. Anal., 26, 1, 147-179 (1995) · Zbl 0814.35146
[10] Klibanov, M. V., Global convexity in a three-dimensional inverse acoustic problem, SIAM J. Math. Anal., 28, 6, 1371-1388 (1997) · Zbl 0889.35116
[11] Klibanov, M. V., Global convexity in diffusion tomography, Nonlinear World, 4, 247-265 (1997) · Zbl 0904.35097
[12] Klibanov, M. V.; Timonov, A., Carleman Estimates for Coefficient Inverse Problems and Numerical Applications (2004), de Gruyter: de Gruyter Utrecht · Zbl 1069.65106
[13] Klibanov, M. V.; Kamburg, V. G., Globally strictly convex cost functional for an inverse parabolic problem, Math. Methods Appl. Sci., 39, 4, 930-940 (2016) · Zbl 1336.35368
[14] Klibanov, M. V.; Nguyen, L. H.; Sullivan, A.; Nguyen, L., A globally convergent numerical method for a 1-d inverse medium problem with experimental data, Inverse Probl. Imaging, 10, 4, 1057-1085 (2016) · Zbl 1355.34040
[15] Klibanov, M. V.; Thành, N. T., Recovering dielectric constants of explosives via a globally strictly convex cost functional, SIAM J. Appl. Math., 75, 2, 518-537 (2015) · Zbl 1328.35315
[16] Chavent, G., Nonlinear Least Squares for Inverse Problems - Theoretical Foundations and Step-by-Step Guide for Applications (2009), Springer · Zbl 1191.65062
[17] Goncharsky, A.; Romanov, S., Supercomputer technologies in inverse problems of ultrasound tomography, Inverse Problems, 29, 075004 (2013) · Zbl 1278.65144
[18] Goncharsky, A. V.; Romanov, S. Y., Iterative methods for solving coefficient inverse problems of wave tomography in models with attenuation, Inverse Problems, 33, 2, 025003 (2017) · Zbl 1357.92041
[19] Scales, J. A.; Smith, M. L.; Fischer, T. L., Global optimization methods for multimodal inverse problems, J. Comput. Phys., 103, 2, 258-268 (1992) · Zbl 0765.65062
[20] Lakhal, A., Kairuain-algorithm applied on electromagnetic imaging, Inverse Problems, 29, 095001 (2010) · Zbl 1290.78010
[21] Lakhal, A., A direct method for nonlinear ill-posed problems, Inverse Problems, 34, 2, 025002 (2018) · Zbl 1453.65125
[22] Klibanov, M. V.; Koshev, N. A.; Li, J.; Yagola, A. G., Numerical solution of an ill-posed Cauchy problem for a quasilinear parabolic equation using a Carleman weight function, J. Inverse Ill-Posed Probl., 24, 761-776 (2016) · Zbl 1351.35263
[23] Bakushinskii, A. B.; Klibanov, M. V.; Koshev, N. A., Carleman weight functions for a globally convergent numerical method for ill-posed Cauchy problems for some quasilinear PDEs, Nonlinear Anal. RWA, 34, 201-224 (2017) · Zbl 1353.65098
[24] Klibanov, M. V., Carleman weight functions for solving ill-posed Cauchy problems for quasilinear PDEs, Inverse Problems, 31, 12, 125007 (2015) · Zbl 1417.35227
[25] Bukhgeim, A.; Klibanov, M., Uniqueness in the large of a class of multidimensional inverse problems, Soviet Math. Doklady, 17, 244-247 (1981) · Zbl 0497.35082
[26] Klibanov, M. V., Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl., 21, 4, 477-560 (2013) · Zbl 1273.35005
[27] Baudouin, L.; Buhan, M.d.; Ervedoza, S., Convergent algorithm based on Carleman estimates for the recovert of a potential in the wave equation, SIAM J. Numer. Anal., 55, 1578-1613 (2017) · Zbl 1366.93070
[28] Ammari, H.; Garnier, J.; Jing, W.; Kang, H.; Lim, M.; Solna, K.; Wang, H., Mathematical and statistical methods for multistatic imaging, Lect. Notes Math., 2098, 125-157 (2013)
[29] Ammari, H.; Chow, Y.; Zou, J., The concept of heterogeneous scattering and its applications in inverse medium scattering, SIAM J. Math. Anal., 46, 2905-2935 (2014) · Zbl 1301.35073
[30] Ammari, H.; Chow, Y.; Zou, J., Phased and phaseless domain reconstruction in inverse scattering problem via scattering coefficients, SIAM J. Appl. Math., 76, 1000-1030 (2016) · Zbl 1338.35490
[31] Bao, G.; Li, P.; Lin, J.; Triki, F., Inverse scattering problems with multi-frequencies, Inverse Problems, 31, 093001 (2015) · Zbl 1332.78019
[32] de Buhan, M.; Kray, M., A new approach to solve the inverse scattering problem for waves: combining the TRAC and the Adaptive Inversion methods, Inverse Problems, 29, 085009 (2013) · Zbl 1279.65115
[33] Chow, Y. T.; Zou, J., A numerical method for reconstructing the coefficient in a wave equation, Numer. Methods Partial Differential Equations, 31, 289-307 (2015) · Zbl 1309.65108
[34] Chow, Y. T.; Ito, K.; Liu, K.; Zou, J., Direct sampling method in diffuse optical tomography, SIAM J. Sci. Comput., 37, A1658-A1684 (2015) · Zbl 1320.65195
[35] Ito, K.; Jin, B.; Zou, J., A direct sampling method for inverse electromagnetic medium scattering, Inverse Problems, 29, 9, 095018 (2013) · Zbl 1290.78008
[36] Jin, B.; Zhou, Z., A finite element method with singularity reconstruction for fractional boundary value problems, ESAIM Math. Model. Numer. Anal., 49, 1261-1283 (2015) · Zbl 1332.65115
[37] Kabanikhin, S.; Satybaev, A.; Shishlenin, M., Direct Methods of Solving Multidimensional Inverse Hyperbolic Problem (2004), VSP: VSP Utrecht · Zbl 1069.65105
[38] Kabanikhin, S.; Sabelfeld, K.; Novikov, N.; Shishlenin, M., Numerical solution of the multidimensional Gelfand-Levitan equation, J. Inverse Ill-Posed Probl., 23, 439-450 (2015) · Zbl 1326.65124
[39] Kabanikhin, S.; Novikov, N.; Osedelets, I.; Shishlenin, M., Fast Toeplitz linear system inversion for solving two-dimensional acoustic inverse problem, J. Inverse Ill-Posed Probl., 23, 687-700 (2015) · Zbl 1327.65184
[40] Lakhal, A., A decoupling-based imaging method for inverse medium scattering for Maxwell’s equations, Inverse Problems, 26, 015007 (2010) · Zbl 1185.35328
[41] Li, J.; Liu, H.; Wang, Q., Enhanced multilevel linear sampling methods for inverse scattering problems, J. Comput. Phys., 257, 554-571 (2014) · Zbl 1349.76624
[42] Li, J.; Li, P.; Liu, H.; Liu, X., Recovering multiscale buried anomalies in a two-layered medium, Inverse Problems, 31, 105006 (2015) · Zbl 1330.78012
[43] Liu, H.; Wang, Y.; Yang, C., Mathematical design of a novel gesture-based instruction/input device using wave detection, SIAM J. Imaging Sci., 9, 822-841 (2016) · Zbl 1432.94016
[45] Klibanov, M. V.; Romanov, V., Two reconstruction procedures for a 3-D phaseless inverse scattering problem for the generalized Helmholtz equation, Inverse Problems, 32, 0150058 (2016)
[46] Romanov, V., Inverse Problems of Mathematical Physics (1987), VNU Science Press: VNU Science Press Utrecht
[47] Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order (1984), Springer: Springer New York · Zbl 0691.35001
[48] Romanov, V., Inverse problems for differential equations with memory, Eurasian J. Math. Comput. Appl., 2, 4, 51-80 (2014)
[49] Klibanov, M. V., Carleman estimates for the regularization of ill-posed Cauchy problems, Appl. Numer. Math., 94, 46-74 (2015) · Zbl 1325.65148
[50] Tikhonov, A.; Goncharsky, A.; Stepanov, V.; Yagola, A., Numerical Methods for the Solution of Ill-Posed Problems (1995), Kluwer: Kluwer London · Zbl 0831.65059
[51] Thành, N. T.; Beilina, L.; Klibanov, M. V.; Fiddy, M. A., Imaging of buried objects from experimental backscattering time-dependent measurements using a globally convergent inverse algorithm, SIAM J. Imaging Sci., 8, 1, 757-786 (2015) · Zbl 1432.35259
[52] Vainikko, G., Fast solvers of the Lippmann-Schwinger equation, (Newark, D., Direct and Inverse Problems of Mathematical Physics. Direct and Inverse Problems of Mathematical Physics, Int. Soc. Anal. Appl. Comput., vol. 5 (2000), Kluwer: Kluwer Dordrecht), 423 · Zbl 0962.65097
[53] Lechleiter, A.; Nguyen, D.-L., A trigonometric Galerkin method for volume integral equations arising in TM grating scattering, Adv. Comput. Math., 40, 1-25 (2014) · Zbl 1304.78010
[55] Novotny, L.; Hecht, B., Principles of Nano-Optics (2012), Cambridge University Press: Cambridge University Press Cambridge
[57] Klibanov, M.; Santosa, F., A computational quasi-reversibility method for Cauchy problems for Laplace’s equation, SIAM J. Appl. Math., 51, 1653-1675 (1991) · Zbl 0769.35005
[58] Kuzhuget, A. V.; Klibanov, M., Global convergence for a 1-D inverse problem with application to imaging of land mines, Appl. Anal., 89, 125-157 (2010) · Zbl 1205.65258
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