# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2021068
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## Convexification-based globally convergent numerical method for a 1D coefficient inverse problem with experimental data

 1 Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA 2 US Army Research Laboratory, 2800 Powder Mill Road, Adelphi, MD 20783-1197, USA

* Corresponding author: Michael V. Klibanov

Dedication: The authors dedicate this paper to the memory of Professor Victor Isakov, one of the world's very top experts in the field of Inverse Problems.

Received  June 2021 Revised  August 2021 Early access October 2021

Fund Project: The first author is supported by the US Army Research Laboratory and US Army Research Office grant W911NF-19-1-0044

To compute the spatially distributed dielectric constant from the backscattering computationally simulated ane experimentally collected data, we study a coefficient inverse problem for a 1D hyperbolic equation. To solve this inverse problem, we establish a new version of the Carleman estimate and then employ this estimate to construct a cost functional, which is strictly convex on a convex bounded set of an arbitrary diameter in a Hilbert space. The strict convexity property is rigorously proved. This result is called the convexification theorem and it is the central analytical result of this paper. Minimizing this cost functional by the gradient descent method, we obtain the desired numerical solution to the coefficient inverse problems. We prove that the gradient descent method generates a sequence converging to the minimizer starting from an arbitrary point of that bounded set. We also establish a theorem confirming that the minimizer converges to the true solution as the noise in the measured data and the regularization parameter tend to zero. Unlike the methods, which are based on the optimization, our convexification method converges globally in the sense that it delivers a good approximation of the exact solution without requiring a good initial guess. Results of numerical studies of both computationally simulated and experimentally collected data are presented.

Citation: 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
##### References:
 [1] A. B. Bakushinskii, M. V. Klibanov and N. A. Koshev, Carleman weight functions for a globally convergent numerical method for ill-posed Cauchy problems for some quasilinear PDEs, Nonlinear Anal. Real World Appl., 34 (2017), 201-224.  doi: 10.1016/j.nonrwa.2016.08.008. [2] L. Baudouin, M. de Buhan and S. Ervedoza, Convergent algorithm based on Carleman estimates for the recovery of a potential in the wave equation, SIAM J. Nummer. Anal., 55 (2017), 1578-1613.  doi: 10.1137/16M1088776. [3] L. Baudouin, M. de Buhan, S. Ervedoza and A. Osses, Carleman-based reconstruction algorithm for waves, SIAM J. Numerical Analysis, 59 (2021), 998-1039.  doi: 10.1137/20M1315798. [4] L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012. [5] M. Boulakia, M. de Buhan and E. Schwindt, Numerical reconstruction based on Carleman estimates of a source term in a reaction-diffusion equation, ESAIM Control Optim. Calc. Var., 27 (2021), 34pp. doi: 10.1051/cocv/2020086. [6] A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272. [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. Geosci. Remote Sens, 33 (1995), 221-223. [8] V. Isakov, Inverse Problems for Partial Differential Equations, 3$^nd$ edition, Applied Mathematical Sciences, 127. Springer, Cham, 2017. doi: 10.1007/978-3-319-51658-5. [9] A. L. Karchevsky, M. V. Klibanov, L. Nguyen, N. Pantong and A. Sullivan, The Krein method and the globally convergent method for experimental data, Appl. Numer. Math., 74 (2013), 111-127.  doi: 10.1016/j.apnum.2013.09.003. [10] A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123. Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220. [11] V. A. Khoa, G. W. Bidney, M. V. Klibanov, L. H. Nguyen, L. Nguyen, A. Sullivan and V. N. Astratov, Convexification and experimental data for a 3D inverse scattering problem with the moving point source, Inverse Problems, 36 (2020), 34pp. doi: 10.1088/1361-6420/ab95aa. [12] V. A. Khoa, M. V. Klibanov and L. H. Nguyen, Convexification for a three-dimensional inverse scattering problem with the moving point source, SIAM J. Imaging Sci., 13 (2020), 871-904.  doi: 10.1137/19M1303101. [13] M. V. Klibanov, Inverse problems and C arleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009. [14] M. V. Klibanov and O. V. Ioussoupova, Uniform strict convexity of a cost functional for three-dimensional inverse scattering problem, SIAM J. Math. Anal., 26 (1995), 147-179.  doi: 10.1137/S0036141093244039. [15] M. V. Klibanov, Global convexity in a three-dimensional inverse acoustic problem, SIAM J. Math. Anal., 28 (1997), 1371-1388.  doi: 10.1137/S0036141096297364. [16] M. V. Klibanov, Global convexity in diffusion tomography, Nonlinear World, 4 (1997), 247-265. [17] M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl., 21 (2013), 477-560.  doi: 10.1515/jip-2012-0072. [18] M. V. Klibanov, Carleman estimates for the regularization of ill-posed Cauchy problems, Appl. Numer. Math., 94 (2015), 46-74.  doi: 10.1016/j.apnum.2015.02.003. [19] M. V. Klibanov, L. H. Nguyen, A. Sullivan and L. Nguyen, A globally convergent numerical method for a 1-d inverse medium problem with experimental data, Inverse Probl. Imaging, 10 (2016), 1057-1085.  doi: 10.3934/ipi.2016032. [20] M. V. Klibanov, Carleman weight functions for solving ill-posed Cauchy problems for quasilinear PDEs, Inverse Problems, 31 (2015), 20pp. doi: 10.1088/0266-5611/31/12/125007. [21] M. V. Klibanov, Convexification of restricted Dirichlet to Neumann map, J. Inverse Ill-Posed Probl., 25 (2017), 669-685.  doi: 10.1515/jiip-2017-0067. [22] M. V. Klibanov, V. A. Khoa, A. V. Smirnov, L. H. Nguyen, G. W. Bidney, L. Nguyen, A. Sullivan and V. N. Astratov, Convexification inversion method for nonlinear SAR imaging with experimentally collected data. Preprint, arXiv: 2103.10431, 2021. [23] M. V. Klibanov and A. E. Kolesov, Convexification of a 3-D coefficient inverse scattering problem, Comput. Math. Appl., 77 (2019), 1681-1702.  doi: 10.1016/j.camwa.2018.03.016. [24] M. V. Klibanov, A. E. Kolesov and D.-L. Nguyen, Convexification method for an inverse scattering problem and its performance for experimental backscatter data for buried targets, SIAM J. Imaging Sci., 12 (2019), 576-603.  doi: 10.1137/18M1191658. [25] M. V. Klibanov, A. E. Kolesov, L. Nguyen and A. Sullivan, Globally strictly convex cost functional for a 1-D inverse medium scattering problem with experimental data, SIAM J. Appl. Math., 77 (2017), 1733-1755.  doi: 10.1137/17M1122487. [26] M. V. Klibanov, A. E. Kolesov, L. Nguyen and A. Sullivan, A new version of the convexification method for a 1-D coefficient inverse problem with experimental data, Inverse Problems, 34 (2018), 29pp. doi: 10.1088/1361-6420/aadbc6. [27] M. V. Klibanov, J. Li and W. Zhang, Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data, Inverse Problems, 35 (2019), 33pp. doi: 10.1088/1361-6420/aafecd. [28] M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, Inverse and Ill-posed Problems Series. VSP, Utrecht, 2004. doi: 10.1515/9783110915549. [29] M. V. Klibanov, J. Li and W. Zhang, Convexification for an inverse parabolic problem, Inverse Problems, 36 (2020), 32pp. doi: 10.1088/1361-6420/ab9893. [30] M. V. Klibanov, J. Li and W. Zhang, Convexification for the inversion of a time dependent wave front in a heterogeneous medium, SIAM J. Appl. Math., 79 (2019), 1722-1747.  doi: 10.1137/18M1236034. [31] M. V. Klibanov, J. Li and W. Zhang, Linear Lavrent'ev integral equation for the numerical solution of a nonlinear coefficient inverse problem, SIAM J. Appl. Math., 81 (2020), 1954-1978.  doi: 10.1137/20M1376558. [32] M. V. Klibanov, A. Smirnov, V. A. Khoa, A. Sullivan and L. Nguyen, Through-the-wall nonlinear SAR imaging, IEEE Transactions on Geoscience and Remote Sensing, 59 (2021), 7475-7486.  doi: 10.1109/TGRS.2021.3055805. [33] M. V. Klibanov and J. Li, Inverse Problems and Carleman Estimates: Global Uniqueness, Global Convergence and Experimental Data, De Gruyter, 2021. [34] J. Korpela, M. Lassas and L. Oksanen, Regularization strategy for an inverse problem for a 1+1 dimensional wave equation, Inverse Problems, 32 (2016), 24pp. doi: 10.1088/0266-5611/32/6/065001. [35] A. 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. [36] 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 Trans. Geosci. Remote Sens, 51 (2013), 2937-2948.  doi: 10.1109/TGRS.2012.2211885. [37] R. Lattès and J. L. Lions, The Method of Quasireversibility: Applications to Partial Differential Equations, American Elsevier Publishing Co., Inc., New York 1969. [38] M. M. Lavrent'ev, On an inverse problem for the wave equation, Soviet Mathematics Doklady, 5 (1964), 970-972. [39] M. M. Lavrent'ev, V. G. Romanov and S. P. Shishatskiĭ, Ill-Posed Problems of Mathematical Physics and Analysis, Translations of Mathematical Monographs. AMS, Providence: RI, 1986. [40] T. T. Le and L. H. Nguyen, A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data, J. Inverse and Ill-posed Problems, 2020. doi: 10.1515/jiip-2020-0028. [41] T. T. Le and L. H. Nguyen, The gradient descent method for the convexification to solve boundary value problems of quasi-linear PDEs and a coefficient inverse problem, preprint, arXiv: 2103.04159, 2021. [42] B. M. Levitan, Inverse Sturm–Liouville Problems, O. Efimov. VSP, Zeist, 1987. [43] M. Minoux, Mathematical Programming: Theory and Algorithms, John Wiley & Sons, Ltd., Chichester, 1986. [44] C. Montalto, Stable determination of a simple metric, a covector field and a potential from the hyperbolic D irichlet-to- Neumann map, Comm. Partial Differential Equations, 39 (2014), 120-145.  doi: 10.1080/03605302.2013.843429. [45] L. H. Nguyen, An inverse space-dependent source problem for hyperbolic equations and the L ipschitz-like convergence of the quasi-reversibility method, Inverse Problems, 35 (2019), 28pp. doi: 10.1088/1361-6420/aafe8f. [46] L. H. Nguyen, Q. Li and M. V. Klibanov., A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media, Inverse Probl. Imaging, 13 (2009), 1067-1094.  doi: 10.3934/ipi.2019048. [47] 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, (2007), 1–8. [48] V. G. Romanov, Inverse Problems of Mathematical Physics, De Gruyter, 1986. [49] J. A. Scales, M. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems., J. Computational Physics, 103 (1992), 258-268. [50] A. V. Smirnov, M. V. Klibanov and L. H. Nguyen, Convexification for a 1D hyperbolic coefficient inverse problem with single measurement data, Inverse Probl. Imaging, 14 (2020), 913-938.  doi: 10.3934/ipi.2020042. [51] A. V. Smirnov, M. V. Klibanov, A. Sullivan and L. H. Nguyen, Convexifcation for an inverse problem for a 1d wave equation with experimental data, Inverse Problems, 36 (2020), 32pp. doi: 10.1088/1361-6420/abac9a. [52] A. N. Tikhonov, A. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Mathematics and its Applications, 328. Kluwer Academic Publishers Group, Dordrecht, 1995. doi: 10.1007/978-94-015-8480-7. [53] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 75pp. doi: 10.1088/0266-5611/25/12/123013.

show all references

##### References:
 [1] A. B. Bakushinskii, M. V. Klibanov and N. A. Koshev, Carleman weight functions for a globally convergent numerical method for ill-posed Cauchy problems for some quasilinear PDEs, Nonlinear Anal. Real World Appl., 34 (2017), 201-224.  doi: 10.1016/j.nonrwa.2016.08.008. [2] L. Baudouin, M. de Buhan and S. Ervedoza, Convergent algorithm based on Carleman estimates for the recovery of a potential in the wave equation, SIAM J. Nummer. Anal., 55 (2017), 1578-1613.  doi: 10.1137/16M1088776. [3] L. Baudouin, M. de Buhan, S. Ervedoza and A. Osses, Carleman-based reconstruction algorithm for waves, SIAM J. Numerical Analysis, 59 (2021), 998-1039.  doi: 10.1137/20M1315798. [4] L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012. [5] M. Boulakia, M. de Buhan and E. Schwindt, Numerical reconstruction based on Carleman estimates of a source term in a reaction-diffusion equation, ESAIM Control Optim. Calc. Var., 27 (2021), 34pp. doi: 10.1051/cocv/2020086. [6] A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272. [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. Geosci. Remote Sens, 33 (1995), 221-223. [8] V. Isakov, Inverse Problems for Partial Differential Equations, 3$^nd$ edition, Applied Mathematical Sciences, 127. Springer, Cham, 2017. doi: 10.1007/978-3-319-51658-5. [9] A. L. Karchevsky, M. V. Klibanov, L. Nguyen, N. Pantong and A. Sullivan, The Krein method and the globally convergent method for experimental data, Appl. Numer. Math., 74 (2013), 111-127.  doi: 10.1016/j.apnum.2013.09.003. [10] A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123. Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420036220. [11] V. A. Khoa, G. W. Bidney, M. V. Klibanov, L. H. Nguyen, L. Nguyen, A. Sullivan and V. N. Astratov, Convexification and experimental data for a 3D inverse scattering problem with the moving point source, Inverse Problems, 36 (2020), 34pp. doi: 10.1088/1361-6420/ab95aa. [12] V. A. Khoa, M. V. Klibanov and L. H. Nguyen, Convexification for a three-dimensional inverse scattering problem with the moving point source, SIAM J. Imaging Sci., 13 (2020), 871-904.  doi: 10.1137/19M1303101. [13] M. V. Klibanov, Inverse problems and C arleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009. [14] M. V. Klibanov and O. V. Ioussoupova, Uniform strict convexity of a cost functional for three-dimensional inverse scattering problem, SIAM J. Math. Anal., 26 (1995), 147-179.  doi: 10.1137/S0036141093244039. [15] M. V. Klibanov, Global convexity in a three-dimensional inverse acoustic problem, SIAM J. Math. Anal., 28 (1997), 1371-1388.  doi: 10.1137/S0036141096297364. [16] M. V. Klibanov, Global convexity in diffusion tomography, Nonlinear World, 4 (1997), 247-265. [17] M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl., 21 (2013), 477-560.  doi: 10.1515/jip-2012-0072. [18] M. V. Klibanov, Carleman estimates for the regularization of ill-posed Cauchy problems, Appl. Numer. Math., 94 (2015), 46-74.  doi: 10.1016/j.apnum.2015.02.003. [19] M. V. Klibanov, L. H. Nguyen, A. Sullivan and L. Nguyen, A globally convergent numerical method for a 1-d inverse medium problem with experimental data, Inverse Probl. Imaging, 10 (2016), 1057-1085.  doi: 10.3934/ipi.2016032. [20] M. V. Klibanov, Carleman weight functions for solving ill-posed Cauchy problems for quasilinear PDEs, Inverse Problems, 31 (2015), 20pp. doi: 10.1088/0266-5611/31/12/125007. [21] M. V. Klibanov, Convexification of restricted Dirichlet to Neumann map, J. Inverse Ill-Posed Probl., 25 (2017), 669-685.  doi: 10.1515/jiip-2017-0067. [22] M. V. Klibanov, V. A. Khoa, A. V. Smirnov, L. H. Nguyen, G. W. Bidney, L. Nguyen, A. Sullivan and V. N. Astratov, Convexification inversion method for nonlinear SAR imaging with experimentally collected data. Preprint, arXiv: 2103.10431, 2021. [23] M. V. Klibanov and A. E. Kolesov, Convexification of a 3-D coefficient inverse scattering problem, Comput. Math. Appl., 77 (2019), 1681-1702.  doi: 10.1016/j.camwa.2018.03.016. [24] M. V. Klibanov, A. E. Kolesov and D.-L. Nguyen, Convexification method for an inverse scattering problem and its performance for experimental backscatter data for buried targets, SIAM J. Imaging Sci., 12 (2019), 576-603.  doi: 10.1137/18M1191658. [25] M. V. Klibanov, A. E. Kolesov, L. Nguyen and A. Sullivan, Globally strictly convex cost functional for a 1-D inverse medium scattering problem with experimental data, SIAM J. Appl. Math., 77 (2017), 1733-1755.  doi: 10.1137/17M1122487. [26] M. V. Klibanov, A. E. Kolesov, L. Nguyen and A. Sullivan, A new version of the convexification method for a 1-D coefficient inverse problem with experimental data, Inverse Problems, 34 (2018), 29pp. doi: 10.1088/1361-6420/aadbc6. [27] M. V. Klibanov, J. Li and W. Zhang, Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data, Inverse Problems, 35 (2019), 33pp. doi: 10.1088/1361-6420/aafecd. [28] M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, Inverse and Ill-posed Problems Series. VSP, Utrecht, 2004. doi: 10.1515/9783110915549. [29] M. V. Klibanov, J. Li and W. Zhang, Convexification for an inverse parabolic problem, Inverse Problems, 36 (2020), 32pp. doi: 10.1088/1361-6420/ab9893. [30] M. V. Klibanov, J. Li and W. Zhang, Convexification for the inversion of a time dependent wave front in a heterogeneous medium, SIAM J. Appl. Math., 79 (2019), 1722-1747.  doi: 10.1137/18M1236034. [31] M. V. Klibanov, J. Li and W. Zhang, Linear Lavrent'ev integral equation for the numerical solution of a nonlinear coefficient inverse problem, SIAM J. Appl. Math., 81 (2020), 1954-1978.  doi: 10.1137/20M1376558. [32] M. V. Klibanov, A. Smirnov, V. A. Khoa, A. Sullivan and L. Nguyen, Through-the-wall nonlinear SAR imaging, IEEE Transactions on Geoscience and Remote Sensing, 59 (2021), 7475-7486.  doi: 10.1109/TGRS.2021.3055805. [33] M. V. Klibanov and J. Li, Inverse Problems and Carleman Estimates: Global Uniqueness, Global Convergence and Experimental Data, De Gruyter, 2021. [34] J. Korpela, M. Lassas and L. Oksanen, Regularization strategy for an inverse problem for a 1+1 dimensional wave equation, Inverse Problems, 32 (2016), 24pp. doi: 10.1088/0266-5611/32/6/065001. [35] A. 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. [36] 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 Trans. Geosci. Remote Sens, 51 (2013), 2937-2948.  doi: 10.1109/TGRS.2012.2211885. [37] R. Lattès and J. L. Lions, The Method of Quasireversibility: Applications to Partial Differential Equations, American Elsevier Publishing Co., Inc., New York 1969. [38] M. M. Lavrent'ev, On an inverse problem for the wave equation, Soviet Mathematics Doklady, 5 (1964), 970-972. [39] M. M. Lavrent'ev, V. G. Romanov and S. P. Shishatskiĭ, Ill-Posed Problems of Mathematical Physics and Analysis, Translations of Mathematical Monographs. AMS, Providence: RI, 1986. [40] T. T. Le and L. H. Nguyen, A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data, J. Inverse and Ill-posed Problems, 2020. doi: 10.1515/jiip-2020-0028. [41] T. T. Le and L. H. Nguyen, The gradient descent method for the convexification to solve boundary value problems of quasi-linear PDEs and a coefficient inverse problem, preprint, arXiv: 2103.04159, 2021. [42] B. M. Levitan, Inverse Sturm–Liouville Problems, O. Efimov. VSP, Zeist, 1987. [43] M. Minoux, Mathematical Programming: Theory and Algorithms, John Wiley & Sons, Ltd., Chichester, 1986. [44] C. Montalto, Stable determination of a simple metric, a covector field and a potential from the hyperbolic D irichlet-to- Neumann map, Comm. Partial Differential Equations, 39 (2014), 120-145.  doi: 10.1080/03605302.2013.843429. [45] L. H. Nguyen, An inverse space-dependent source problem for hyperbolic equations and the L ipschitz-like convergence of the quasi-reversibility method, Inverse Problems, 35 (2019), 28pp. doi: 10.1088/1361-6420/aafe8f. [46] L. H. Nguyen, Q. Li and M. V. Klibanov., A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media, Inverse Probl. Imaging, 13 (2009), 1067-1094.  doi: 10.3934/ipi.2019048. [47] 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, (2007), 1–8. [48] V. G. Romanov, Inverse Problems of Mathematical Physics, De Gruyter, 1986. [49] J. A. Scales, M. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems., J. Computational Physics, 103 (1992), 258-268. [50] A. V. Smirnov, M. V. Klibanov and L. H. Nguyen, Convexification for a 1D hyperbolic coefficient inverse problem with single measurement data, Inverse Probl. Imaging, 14 (2020), 913-938.  doi: 10.3934/ipi.2020042. [51] A. V. Smirnov, M. V. Klibanov, A. Sullivan and L. H. Nguyen, Convexifcation for an inverse problem for a 1d wave equation with experimental data, Inverse Problems, 36 (2020), 32pp. doi: 10.1088/1361-6420/abac9a. [52] A. N. Tikhonov, A. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Mathematics and its Applications, 328. Kluwer Academic Publishers Group, Dordrecht, 1995. doi: 10.1007/978-94-015-8480-7. [53] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 75pp. doi: 10.1088/0266-5611/25/12/123013.
The schematic for the data generating and collecting device. A device, called radar, emits an acoustic source and then collect the time-dependent backscattering wave. In the physical experiment, we consider two cases: (a) the target is placed in the air and (b) the target is buried a few centimeters under the ground.
Illustration of the process of correcting the data near $(x = 0, t = 0).$ We know that when $x$ is small, $c(x) = 1$. Therefore by (2.5), $u(x, t) = \frac{1}{2}$ for $t<| \tau (x)|.$ We, therefore, set $u(x, t) = \frac{1}{2}$ for $x$ and $t$ small. Figure 2a and figure 2b are the graphs of the function $u(0, t)$ before and after, respectively, this reassignment. These functions are taken from Test 3 in subsection 6.3.
it The functions $q^{(0)}(x, 0)$ and $Q^{(0)}(x, 0)$, $x \in (\epsilon, M)$ computed by solving (6.3) and using (6.5). The data to compute these functions are taken from the data for Test 1 with $5\%$ noise.
The true spatially distributed dielectric constant function $c_{\mathrm{true}}$, its initial version $c_{\mathrm{init}}$ computed by (6.6) and its final reconstruction $c_{\mathrm{comp}}$ by our convexification method. It is evident that in all cases, the initial solution $c_{\mathrm{init}}$ computed by (6.6) already carries some information of $c_{\mathrm{true}}$. The following iterative steps significantly improve the positions of "inclusions" and their values. Especially, in test 5 (Figure 4e), the convexification method successfully reconstructs the curves in the inclusion in the left.
The case when the target is in the air. The raw and preprocessed data in the first row correspond to the wave scattered from a bush. (A) The time-dependent raw data, (B) The time-dependent backscattering wave after preprocessing, (C) Computed dielectric constant and its maximal value is 6.76. The raw and preprocessed data in the second row correspond to the wave scattered from a wood stake. (D) The time-dependent raw data, (E) The time-dependent backscattering wave after preprocessing, (F) Computed dielectric constant and its maximal value is 2.2. The computed dielectric constants for these two tests meet the expectation since they belong to intervals of their true value, see the last three rows of Table 1.
The case when the target is buried under the ground. The raw and preprocessed data in the first row correspond to the wave scattered from a metal box. (A) The time-dependent raw data, (B) The time-dependent backscattering wave after preprocessing, (C) Computed dielectric constant and its maximal value is 5.2. The raw and preprocessed data in the second row correspond to the wave scattered from a metal cylinder. The raw and preprocessed data in the third row correspond to the wave scattered from a plastic cylinder. (D) The time-dependent raw data, (E) The time-dependent backscattering wave after preprocessing, (F) Computed dielectric constant and its maximal value is 4.7. Unlike the tests in the first two rows, we choose the upper envelop in third case when preprocessing the data because $c_{\mathrm{target}} < c_{\mathrm{bckgr}}.$. (G) The time-dependent raw data, (H) The time-dependent backscattering wave after preprocessing, (I) Computed dielectric constant and its minimal value is 0.37. The computed dielectric constants for these three tests meet the expectation since they belong to intervals of their true value, see the last three rows of Table 1.
Computed dielectric constants of five targets
 Target $c_{\mathrm{bckgr}}$ computed $c_{\mathrm{rel}}$ $c_{\mathrm{ bckgr}}$ computed $c_{\text{target}}$ True $c_{\text{target}}$ Bush 1 6.76 1 6.76 $[3, 20]$ Wood stake 1 2.22 1 2.22 $[2, 6]$ Metal box 4 5.2 $[3, 5]$ $[15.6, 26]$ $[10, 30]$ Metal cylinder 4 4.7 $[3, 5]$ $[14.1, 23.5]$ $[10, 30]$ Plastic cylinder 4 0.37 $[3, 5]$ $[1.11, 1.85]$ $\left[ 1.1, 3.2 \right]$
 Target $c_{\mathrm{bckgr}}$ computed $c_{\mathrm{rel}}$ $c_{\mathrm{ bckgr}}$ computed $c_{\text{target}}$ True $c_{\text{target}}$ Bush 1 6.76 1 6.76 $[3, 20]$ Wood stake 1 2.22 1 2.22 $[2, 6]$ Metal box 4 5.2 $[3, 5]$ $[15.6, 26]$ $[10, 30]$ Metal cylinder 4 4.7 $[3, 5]$ $[14.1, 23.5]$ $[10, 30]$ Plastic cylinder 4 0.37 $[3, 5]$ $[1.11, 1.85]$ $\left[ 1.1, 3.2 \right]$
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