American Institute of Mathematical Sciences

2018, 12(2): 493-523. doi: 10.3934/ipi.2018021

A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data

 1 Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA 2 Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA 3 Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30302, USA

* Corresponding author

Received  June 2017 Revised  September 2017 Published  February 2018

Fund Project: This work was supported by US Army Research Laboratory and US Army Research Office grant W911NF-15-1-0233 and by the Office of Naval Research grant N00014-15-1-2330

The goal of this paper is to reconstruct spatially distributed dielectric constants from complex-valued scattered wave field by solving a 3D coefficient inverse problem for the Helmholtz equation at multi-frequencies. The data are generated by only a single direction of the incident plane wave. To solve this inverse problem, a globally convergent algorithm is analytically developed. We prove that this algorithm provides a good approximation for the exact coefficient without any a priori knowledge of any point in a small neighborhood of that coefficient. This is the main advantage of our method, compared with classical approaches using optimization schemes. Numerical results are presented for both computationally simulated data and experimental data. Potential applications of this problem are in detection and identification of explosive-like targets.

Citation: 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 & Imaging, 2018, 12 (2) : 493-523. doi: 10.3934/ipi.2018021
References:
 [1] A. D. Agaltsov and R. Novikov, Riemann-Hilbert approach for two-dimensional flow inverse scattering, J. Math. Phys, 55 (2014), 103502, 25pp. [2] H. Ammari, Y. Chow and J. Zou, Phased and phaseless domain reconstructions in the inverse scattering problem via scattering coefficients, SIAM J. Appl. Math., 76 (2016), 1000-1030. doi: 10.1137/15M1043959. [3] H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities From Boundary Measurements, vol. 1846 of Lecture Notes in Mathematics, Springer, 2004. [4] A. B. Bakushinsii and M. Y. Kokurin, Iterative Methods for Approximate Solutions of Inverse Problems, Springer, New York, 2004. [5] 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. [6] 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), 513-565. [7] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Third revised edition, Pergamon Press, Oxford-New York-Paris, 1965. [8] A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Soviet Math. Doklady, 17 (1981), 244-247. [9] M. Burger and S. Osher, A survey on level set methods for inverse problems and optimal design, European J. of Appl. Math., 16 (2005), 263-301. doi: 10.1017/S0956792505006182. [10] G. Chavent, Nonlinear Least Squares for Inverse Problems: Theoretical Foundations and Step-by-Step Guide for Applications, Scientic Computation, Springer, New York, 2009. [11] Y. Chow and J. Zou, A numerical method for reconstructing the coefficient in a wave equation, Numerical Methods for PDEs, 31 (2015), 289-307. doi: 10.1002/num.21904. [12] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003. [13] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Second ed., Applied Mathematical Sciences, Springer-Verlag, Berlin, 1998. [14] M. de Buhan and M. Kray, A new approach to solve the inverse scattering problem for waves: Combining the TRAC and the Adaptive Inversion methods, Inverse Problems, 29 (2013), 085009, 24pp. [15] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996. [16] N. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg New York, 1977. [17] A. V. Goncharsky and S. Y. Romanov, Supercomputer technologies in inverse problems of ultrasound tomography, Inverse Problems, 29(2013), 075004, 22pp. [18] F. Hecht, New development in FreeF em++, J. Numerical Mathematics, 20 (2012), 251-265. [19] K. Ito, B. Jin and J. Zou, A direct sampling method for an inverse medium scattering problem, Inverse Problems, 28 (2012), 025003, 11pp. [20] K. Ito, B. Jin and J. Zou, A direct sampling method for inverse electromagnetic medium scattering, Inverse Problems, 29 (2013), 095018, 19pp. [21] S. I. Kabanikhin, K. K. Sabelfeld, N. Novikov and M. A. Shishlenin, Numerical solution of the multidimensional G elfand-Levitan equation, J. Inverse and Ill-Posed Problems, 23 (2015), 439-450. [22] A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009. [23] M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse and Ill-Posed Problems, 21 (2013), 477-560. [24] M. V. Klibanov, M. A. Fiddy, L. Beilina, N. Pantong and J. Schenk, Picosecond scale experimental verification of a globally convergent algorithm for a coefficient inverse problem, Inverse Problems, 26 (2010), 045003, 30pp. [25] 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 Problems and Imaging, 10 (2016), 1057-1085. doi: 10.3934/ipi.2016032. [26] M. V. Klibanov and V. G. Romanov, Two reconsrtuction procedures for a 3-D phaseless inverse scattering problem for the generalized Helmholtz equation, Inverse Problems, 32 (2016), 015005, 16pp. [27] A. Kolesov, M. V. Klibanov, L. H. Nguyen, D.-L. Nguyen and N. T. Thành, Single measurement experimental data for an inverse medium problem inverted by a multi-frequency globally convergent numerical method, Applied Numerical Mathematics, 120 (2017), 176-196. doi: 10.1016/j.apnum.2017.05.007. [28] 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. [29] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. [30] A. Lechleiter and D.-L. Nguyen, A trigonometric Galerkin method for volume integral equations arising in TM grating scattering, Adv. Comput. Math., 40 (2014), 1-25. doi: 10.1007/s10444-013-9295-2. [31] J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, Multiscale Model. Simul., 12 (2014), 927-952. doi: 10.1137/13093409X. [32] J. Li, H. Liu and Q. Wang, Enhanced multilevel linear sampling methods for inverse scattering problems, J. Comput. Phys., 257 (2014), 554-571. doi: 10.1016/j.jcp.2013.09.048. [33] D. -L. Nguyen, M. V. Klibanov, L. H. Nguyen and M. A. Fiddy, Imaging of buried objects from multi-frequency experimental data using a globally convergent inversion method, To appear on Journal of Inverse and Ill-Posed Problems. [34] D.-L. Nguyen, M. V. Klibanov, L. Nguyen, A. E. Kolesov, M. A. Fiddy and H. Liu, Numerical solution for a coefficient inverse problem with multi-frequency experimental raw data by a globally convergent algorithm, Journal of Computational Physics, 345 (2017), 17-32. doi: 10.1016/j.jcp.2017.05.015. [35] R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta \psi +(v(x)-Eu(x))\psi = 0$, Funct. Anal. Appl., 22 (1988), 263-272. [36] R. G. Novikov, The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator, J. Functional Analysis, 103 (1992), 409-463. doi: 10.1016/0022-1236(92)90127-5. [37] R. G. Novikov, An iterative approach to non-overdetermined inverse scattering at fixed energy, Sbornik: Mathematics, 206 (2015), 120-134. [38] L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd edition, Cambridge University Press, Cambridge, UK, 2012. [39] V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press, Utrecht, 1987. [40] V. G. Romanov, Inverse problems for differential equations with memory, Eurasian J. Math. Comput. Appl., 2 (2014), 51-80. [41] J. A. Scales, M. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems, J. Computational Physics, 103 (1992), 258-268. [42] M. Soumekh, Syntetic Aperture Radar Signal Processing, John Wiley&Sons, New York, 1999. [43] N. T. Thành, L. Bellina, 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 (2014), 757-786. doi: 10.1137/140972469. [44] N. T. Thành, L. Bellina, M. V. Klibanov and M. A. Fiddy, Reconstruction of the refractive index from experimental backscattering data using a globally convergent inverse method, SIAM J. Sci. Comput., 36 (2014), B273-B293. doi: 10.1137/130924962. [45] A. N. Tikhonov, A. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer Academic Publishers Group, Dordrecht, 1995. [46] B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, New York, Gordon and Breach Science Publishers, 1989. [47] G. Vainikko, Fast solvers of the L ippmann-Schwinger equation, in Direct and Inverse Problems of Mathematical Physics (ed. D. Newark), Int. Soc. Anal. Appl. Comput. 5, Kluwer, Dordrecht, 2000,423-440. [48] M. Yamamoto, Carleman estimates for parabolic equations. Topical Review, Inverse Problems, 25 (2009), 123013, 75pp.

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References:
 [1] A. D. Agaltsov and R. Novikov, Riemann-Hilbert approach for two-dimensional flow inverse scattering, J. Math. Phys, 55 (2014), 103502, 25pp. [2] H. Ammari, Y. Chow and J. Zou, Phased and phaseless domain reconstructions in the inverse scattering problem via scattering coefficients, SIAM J. Appl. Math., 76 (2016), 1000-1030. doi: 10.1137/15M1043959. [3] H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities From Boundary Measurements, vol. 1846 of Lecture Notes in Mathematics, Springer, 2004. [4] A. B. Bakushinsii and M. Y. Kokurin, Iterative Methods for Approximate Solutions of Inverse Problems, Springer, New York, 2004. [5] 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. [6] 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), 513-565. [7] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Third revised edition, Pergamon Press, Oxford-New York-Paris, 1965. [8] A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Soviet Math. Doklady, 17 (1981), 244-247. [9] M. Burger and S. Osher, A survey on level set methods for inverse problems and optimal design, European J. of Appl. Math., 16 (2005), 263-301. doi: 10.1017/S0956792505006182. [10] G. Chavent, Nonlinear Least Squares for Inverse Problems: Theoretical Foundations and Step-by-Step Guide for Applications, Scientic Computation, Springer, New York, 2009. [11] Y. Chow and J. Zou, A numerical method for reconstructing the coefficient in a wave equation, Numerical Methods for PDEs, 31 (2015), 289-307. doi: 10.1002/num.21904. [12] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003. [13] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Second ed., Applied Mathematical Sciences, Springer-Verlag, Berlin, 1998. [14] M. de Buhan and M. Kray, A new approach to solve the inverse scattering problem for waves: Combining the TRAC and the Adaptive Inversion methods, Inverse Problems, 29 (2013), 085009, 24pp. [15] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996. [16] N. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg New York, 1977. [17] A. V. Goncharsky and S. Y. Romanov, Supercomputer technologies in inverse problems of ultrasound tomography, Inverse Problems, 29(2013), 075004, 22pp. [18] F. Hecht, New development in FreeF em++, J. Numerical Mathematics, 20 (2012), 251-265. [19] K. Ito, B. Jin and J. Zou, A direct sampling method for an inverse medium scattering problem, Inverse Problems, 28 (2012), 025003, 11pp. [20] K. Ito, B. Jin and J. Zou, A direct sampling method for inverse electromagnetic medium scattering, Inverse Problems, 29 (2013), 095018, 19pp. [21] S. I. Kabanikhin, K. K. Sabelfeld, N. Novikov and M. A. Shishlenin, Numerical solution of the multidimensional G elfand-Levitan equation, J. Inverse and Ill-Posed Problems, 23 (2015), 439-450. [22] A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512. doi: 10.1088/0266-5611/14/6/009. [23] M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse and Ill-Posed Problems, 21 (2013), 477-560. [24] M. V. Klibanov, M. A. Fiddy, L. Beilina, N. Pantong and J. Schenk, Picosecond scale experimental verification of a globally convergent algorithm for a coefficient inverse problem, Inverse Problems, 26 (2010), 045003, 30pp. [25] 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 Problems and Imaging, 10 (2016), 1057-1085. doi: 10.3934/ipi.2016032. [26] M. V. Klibanov and V. G. Romanov, Two reconsrtuction procedures for a 3-D phaseless inverse scattering problem for the generalized Helmholtz equation, Inverse Problems, 32 (2016), 015005, 16pp. [27] A. Kolesov, M. V. Klibanov, L. H. Nguyen, D.-L. Nguyen and N. T. Thành, Single measurement experimental data for an inverse medium problem inverted by a multi-frequency globally convergent numerical method, Applied Numerical Mathematics, 120 (2017), 176-196. doi: 10.1016/j.apnum.2017.05.007. [28] 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. [29] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. [30] A. Lechleiter and D.-L. Nguyen, A trigonometric Galerkin method for volume integral equations arising in TM grating scattering, Adv. Comput. Math., 40 (2014), 1-25. doi: 10.1007/s10444-013-9295-2. [31] J. Li, H. Liu and J. Zou, Locating multiple multiscale acoustic scatterers, Multiscale Model. Simul., 12 (2014), 927-952. doi: 10.1137/13093409X. [32] J. Li, H. Liu and Q. Wang, Enhanced multilevel linear sampling methods for inverse scattering problems, J. Comput. Phys., 257 (2014), 554-571. doi: 10.1016/j.jcp.2013.09.048. [33] D. -L. Nguyen, M. V. Klibanov, L. H. Nguyen and M. A. Fiddy, Imaging of buried objects from multi-frequency experimental data using a globally convergent inversion method, To appear on Journal of Inverse and Ill-Posed Problems. [34] D.-L. Nguyen, M. V. Klibanov, L. Nguyen, A. E. Kolesov, M. A. Fiddy and H. Liu, Numerical solution for a coefficient inverse problem with multi-frequency experimental raw data by a globally convergent algorithm, Journal of Computational Physics, 345 (2017), 17-32. doi: 10.1016/j.jcp.2017.05.015. [35] R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta \psi +(v(x)-Eu(x))\psi = 0$, Funct. Anal. Appl., 22 (1988), 263-272. [36] R. G. Novikov, The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator, J. Functional Analysis, 103 (1992), 409-463. doi: 10.1016/0022-1236(92)90127-5. [37] R. G. Novikov, An iterative approach to non-overdetermined inverse scattering at fixed energy, Sbornik: Mathematics, 206 (2015), 120-134. [38] L. Novotny and B. Hecht, Principles of Nano-Optics, 2nd edition, Cambridge University Press, Cambridge, UK, 2012. [39] V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press, Utrecht, 1987. [40] V. G. Romanov, Inverse problems for differential equations with memory, Eurasian J. Math. Comput. Appl., 2 (2014), 51-80. [41] J. A. Scales, M. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems, J. Computational Physics, 103 (1992), 258-268. [42] M. Soumekh, Syntetic Aperture Radar Signal Processing, John Wiley&Sons, New York, 1999. [43] N. T. Thành, L. Bellina, 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 (2014), 757-786. doi: 10.1137/140972469. [44] N. T. Thành, L. Bellina, M. V. Klibanov and M. A. Fiddy, Reconstruction of the refractive index from experimental backscattering data using a globally convergent inverse method, SIAM J. Sci. Comput., 36 (2014), B273-B293. doi: 10.1137/130924962. [45] A. N. Tikhonov, A. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer Academic Publishers Group, Dordrecht, 1995. [46] B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, New York, Gordon and Breach Science Publishers, 1989. [47] G. Vainikko, Fast solvers of the L ippmann-Schwinger equation, in Direct and Inverse Problems of Mathematical Physics (ed. D. Newark), Int. Soc. Anal. Appl. Comput. 5, Kluwer, Dordrecht, 2000,423-440. [48] M. Yamamoto, Carleman estimates for parabolic equations. Topical Review, Inverse Problems, 25 (2009), 123013, 75pp.
For $k = 6.48$, we present in (a) the absolute value of the noisy backscattered field on the rectangle $(-5, 5)^2\times \{z = -7.6\}$ and in (b) the absolute value of the propagated data on the rectangle $(-5, 5)^2\times \{z = -0.75 \}$
Visualizations of the exact coefficient $c(\mathbf{x})$ in (120) (left) and the reconstructed coefficient $c_{comp}(\mathbf{x})$ (right) for the case of complete data with 15% artificial noise. The first row is the projection of $c(\mathbf{x})$ and $c_{comp}(\mathbf{x})$ on $\{y = 0\}$. The last row is a 3D isosurface, with isovalue 2.45, of the exact and reconstructed geometry of the target using MATLAB
Reconstruction result for the coefficient $c(\mathbf{x})$ in (120) with backscatter data. The left picture is the projection of $c_{comp}(\mathbf{x})$ on $\{y = 0\}$. The right one is the reconstructed geometry of the target
Visualizations of the exact coefficient $c(\mathbf{x})$ (left) in (123) and the reconstructed coefficient $c_{comp}(\mathbf{x })$ (right) for the case of backscatter data. The first row is the projection of $c(\mathbf{x})$ and $c_{comp}(\mathbf{x})$ on $\{y = 0\}$. The last row is a 3D visualization of the exact and reconstructed geometry of the target using MATLAB's isosurface. The isovalue is chosen as 50% of the maximal value of $c_{comp}(\mathbf{x})$
Visualizations of exact (left) and reconstructed (right) geometry of the target using the isosurface command in MATLAB
Measured and computed dielectric constants $c$ of the targets
 Target Measured $c$ (std. dev.) Computed $c_{\max }$ Relative error A piece of yellow pine 5.30 (1.6%) 5.44 2.6% A piece of wet wood 8.48 (4.9%) 7.60 10.3% A geode 5.44 (1.1%) 5.55 2.0% A tennis ball 3.80 (13.0%) 4.00 5.2% A baseball not available 4.76 n/a
 Target Measured $c$ (std. dev.) Computed $c_{\max }$ Relative error A piece of yellow pine 5.30 (1.6%) 5.44 2.6% A piece of wet wood 8.48 (4.9%) 7.60 10.3% A geode 5.44 (1.1%) 5.55 2.0% A tennis ball 3.80 (13.0%) 4.00 5.2% A baseball not available 4.76 n/a
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