October  2020, 14(5): 913-938. doi: 10.3934/ipi.2020042

Convexification for a 1D hyperbolic coefficient inverse problem with single measurement data

Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC 28223, USA

* Corresponding author: Michael Klibanov (mklibanv@uncc.edu)

Received  February 2020 Revised  May 2020 Published  July 2020

Fund Project: The work was supported by US Army Research Laboratory and US Army Research Office grant W911NF-19-1-0044

A version of the convexification numerical method for a Coefficient Inverse Problem for a 1D hyperbolic PDE is presented. The data for this problem are generated by a single measurement event. This method converges globally. The most important element of the construction is the presence of the Carleman Weight Function in a weighted Tikhonov-like functional. This functional is strictly convex on a certain bounded set in a Hilbert space, and the diameter of this set is an arbitrary positive number. The global convergence of the gradient projection method is established. Computational results demonstrate a good performance of the numerical method for noisy data.

Citation: Alexey Smirnov, Michael Klibanov, Loc Nguyen. Convexification for a 1D hyperbolic coefficient inverse problem with single measurement data. Inverse Problems and Imaging, 2020, 14 (5) : 913-938. doi: 10.3934/ipi.2020042
References:
[1]

A. B. BakushinskiiM. 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 Analysis: Real World Applications, 34 (2017), 201-224.  doi: 10.1016/j.nonrwa.2016.08.008.

[2]

Y. Ahajjam, O. Aghzout, and J. M. Catalá-Civera, F. Peñaranda-Foix and A. Driouach, A compact UWB sub-nanosecond pulse generator for microwave radar sensor with ringing miniaturization, in 2016 5th International Conference on Multimedia Computing and Systems (ICMCS), IEEE, 2016,497–501.

[3]

L. Baudouin, M. de Buhan, S. Ervedoza and A. Osses, Carleman-based reconstruction algorithm for the waves, preprint, available at https: //hal.archives-ouvertes.fr/hal-02458787, 2020.

[4]

L. BaudouinM. de Buhan and S. Ervedoza, Convergent algorithm based on Carleman estimates for the recovery of a potential in the wave equation, SIAM Journal on Numerical Analysis, 55 (2017), 1578-1613.  doi: 10.1137/16M1088776.

[5]

L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer Science & Business Media, 2012.

[6]

L. Beilina and M. V. Klibanov, Globally strongly convex cost functional for a coefficient inverse problem, Nonlinear Analysis: Real World Applications, 22 (2015), 272-288.  doi: 10.1016/j.nonrwa.2014.09.015.

[7]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer, 2017. doi: 10.1007/978-4-431-56600-7.

[8]

M. Boulakia, M. de Buhan and E. Schwindt, Numerical reconstruction based on Carleman estimates of a source term in a reaction-diffusion equation, 2019.

[9]

A. L. Buchgeim, Uniqueness in the large of a class of multidimensional inverse problems, Sov. Math. Dokl., 24 (1981), 244-247. 

[10]

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Springer Nature, 93 (2019). doi: 10.1007/978-3-030-30351-8.

[11]

B. Engquist and A. Majda, Absorbing boundary conditions for numerical simulation of waves, Proceedings of the National Academy of Sciences, 74 (1977), 1765-1766.  doi: 10.1073/pnas.74.5.1765.

[12]

S. I. Kabanikhin, A. D. Satybaev and M. A. Shishlenin, Direct methods of solving multidimensional inverse hyperbolic problems, Walter de Gruyter, 48 (2013).

[13]

S. I. KabanikhinK. K. SabelfeldN. S. Novikov and M. A. Shishlenin, Numerical solution of the multidimensional Gelfand–Levitan equation, Journal of Inverse and Ill-Posed Problems, 23 (2015), 439-450.  doi: 10.1515/jiip-2014-0018.

[14]

S. I. KabanikhinN. S. NovikovI. V. Oseledets and M. A. Shishlenin, Fast Toeplitz linear system inversion for solving two-dimensional acoustic inverse problem, Journal of Inverse and Ill-Posed Problems, 23 (2015), 687-700.  doi: 10.1515/jiip-2015-0083.

[15]

V. A. KhoaM. V. Klibanov and L. H. Nguyen, Convexification for a three-dimensional inverse scattering problem with the moving point source, SIAM Journal on Imaging Sciences, 13 (2020), 871-904.  doi: 10.1137/19M1303101.

[16]

M. V. Klibanov, Inverse problems in the large and Carleman bounds, Differential Equations, 20 (1984), 755-760. 

[17]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575. doi: 10.1088/0266-5611/8/4/009.

[18]

M. V. Klibanov and O. V. Ioussoupova, Uniform strict convexity of a cost functional for three-dimensional inverse scattering problem, SIAM Journal on Mathematical Analysis, 26 (1995), 147-179.  doi: 10.1137/S0036141093244039.

[19]

M. V. Klibanov, Global convexity in a three-dimensional inverse acoustic problem, SIAM Journal on Mathematical Analysis, 28 (1997), 1371-1388.  doi: 10.1137/S0036141096297364.

[20]

J. S. Lee and C. Nguyen, Novel low-cost ultra-wideband, ultra-short-pulse transmitter with MESFET impulse-shaping circuitry for reduced distortion and improved pulse repetition rate, IEEE Microwave and Wireless Components Letters, 11 (2001), 208-210. 

[21]

V. A. Khoa, G. W. Bidney, M. V. Klibanov, L. H. Nguyen, L. H. Nguyen, A. J. Sullivan and V. N. Astratov, Convexification and experimental data for a 3D inverse scattering problem with the moving point source, preprint, arXiv: 2003.11513 doi: 10.1137/19M1303101.

[22]

I. M. Gelfand and B. M. Levitan, On determining a differential equation from its spectral function, Amer. Math. Soc. Transl., 2 (1955), 253-304.  doi: 10.1090/trans2/001/11.

[23]

M. G. Krein, On a method of effective solution of an inverse boundary problem, Dokl. Akad. Nauk SSSR, 94 (1954), 987-990. 

[24]

A. L. KarchevskyM. V. KlibanovL. NguyenN. 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.

[25]

M. V. Klibanov and A. A. Timonov, Carleman estimates for coefficient inverse problems and numerical applications, Walter de Gruyter, 46 (2012). doi: 10.1515/9783110915549.

[26]

M. V. KlibanovA. V. KuzhugetS. I. Kabanikhin and D. V. Nechaev, A new version of the quasi-reversibility method for the thermoacoustic tomography and a coefficient inverse problem, Applicable Analysis, 87 (2008), 1227-1254.  doi: 10.1080/00036810802001297.

[27]

M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, Journal of Inverse and Ill-Posed Problems, 21 (2013), 477-560.  doi: 10.1515/jip-2012-0072.

[28]

M. V. Klibanov and V. G. Kamburg, Globally strictly convex cost functional for an inverse parabolic problem, Mathematical Methods in the Applied Sciences, 39 (2016), 930-940.  doi: 10.1002/mma.3531.

[29]

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, preprint, arXiv: 1602.09092. doi: 10.3934/ipi.2016032.

[30]

M. V. Klibanov, A. E. Kolesov, A. Sullivan and L. Nguyen, A new version of the convexification method for a 1D coefficient inverse problem with experimental data, Inverse Problems, 34 (2018), 115014. doi: 10.1088/1361-6420/aadbc6.

[31]

L. Baudouin, M. de Buhan, S. Ervedoza and A. Osses, Carleman-based reconstruction algorithm for the waves, 2020.

[32]

M. V. Klibanov, J. Li and W. Zhang, Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data, Inverse Problems, (2019). doi: 10.1088/1361-6420/aafecd.

[33]

M. V. KlibanovJ. 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.

[34]

M. V. Klibanov and A. E. Kolesov, Convexification of a 3-D coefficient inverse scattering problem, Computers & Mathematics with Applications, 77 (2019), 1681-1702.  doi: 10.1016/j.camwa.2018.03.016.

[35]

M. V. KlibanovA. 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. Imag. Sci., 12 (2019), 576-603.  doi: 10.1137/18M1191658.

[36]

A. Einstein, Onthe electrodynamics of moving bodies, Annalen der Physik, 322 (1905), 891-921. 

[37]

M. V. Klibanov, J. Li and W. Zhang, Convexification for an inverse parabolic problem, preprint, arXiv: 2001.01880.

[38]

J. Korpela, M. Lassas and L. Oksanen, Regularization strategy for an inverse problem for a 1+ 1 dimensional wave equation, Inverse Problems, 32 (2016), 065001. doi: 10.1088/0266-5611/32/6/065001.

[39]

J. Korpela, M. Lassas and L. Oksanen, Discrete regularization and convergence of the inverse problem for 1+ 1 dimensional wave equation, preprint, arXiv: 1803.10541. doi: 10.3934/ipi.2019027.

[40]

A. V. Kuzhuget, et al., Blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm, Inverse Problems, 28 (2012), 095007.

[41]

T. T. Le and L. H. Nguyen, A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data, preprint, arXiv: 1910.05584

[42]

B. T. Polyak, Introduction to optimization. Optimization software, Inc., Publications Division, New York, 1 (1987).

[43]

V. G. Romanov, Inverse Problems of Mathematical Physics, Walter de Gruyter GmbH & Co KG, 2018.

[44]

J. A. ScalesM. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems, J. Comput. Phys., 103 (1992), 258-268. 

[45]

A. N. Tikhonov, A. Goncharsky, V. Stepanov and A. G. Yagola, Numerical methods for the solution of ill-posed problems, Springer Science & Business Media, 328 (2013). doi: 10.1007/978-94-015-8480-7.

show all references

References:
[1]

A. B. BakushinskiiM. 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 Analysis: Real World Applications, 34 (2017), 201-224.  doi: 10.1016/j.nonrwa.2016.08.008.

[2]

Y. Ahajjam, O. Aghzout, and J. M. Catalá-Civera, F. Peñaranda-Foix and A. Driouach, A compact UWB sub-nanosecond pulse generator for microwave radar sensor with ringing miniaturization, in 2016 5th International Conference on Multimedia Computing and Systems (ICMCS), IEEE, 2016,497–501.

[3]

L. Baudouin, M. de Buhan, S. Ervedoza and A. Osses, Carleman-based reconstruction algorithm for the waves, preprint, available at https: //hal.archives-ouvertes.fr/hal-02458787, 2020.

[4]

L. BaudouinM. de Buhan and S. Ervedoza, Convergent algorithm based on Carleman estimates for the recovery of a potential in the wave equation, SIAM Journal on Numerical Analysis, 55 (2017), 1578-1613.  doi: 10.1137/16M1088776.

[5]

L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer Science & Business Media, 2012.

[6]

L. Beilina and M. V. Klibanov, Globally strongly convex cost functional for a coefficient inverse problem, Nonlinear Analysis: Real World Applications, 22 (2015), 272-288.  doi: 10.1016/j.nonrwa.2014.09.015.

[7]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer, 2017. doi: 10.1007/978-4-431-56600-7.

[8]

M. Boulakia, M. de Buhan and E. Schwindt, Numerical reconstruction based on Carleman estimates of a source term in a reaction-diffusion equation, 2019.

[9]

A. L. Buchgeim, Uniqueness in the large of a class of multidimensional inverse problems, Sov. Math. Dokl., 24 (1981), 244-247. 

[10]

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Springer Nature, 93 (2019). doi: 10.1007/978-3-030-30351-8.

[11]

B. Engquist and A. Majda, Absorbing boundary conditions for numerical simulation of waves, Proceedings of the National Academy of Sciences, 74 (1977), 1765-1766.  doi: 10.1073/pnas.74.5.1765.

[12]

S. I. Kabanikhin, A. D. Satybaev and M. A. Shishlenin, Direct methods of solving multidimensional inverse hyperbolic problems, Walter de Gruyter, 48 (2013).

[13]

S. I. KabanikhinK. K. SabelfeldN. S. Novikov and M. A. Shishlenin, Numerical solution of the multidimensional Gelfand–Levitan equation, Journal of Inverse and Ill-Posed Problems, 23 (2015), 439-450.  doi: 10.1515/jiip-2014-0018.

[14]

S. I. KabanikhinN. S. NovikovI. V. Oseledets and M. A. Shishlenin, Fast Toeplitz linear system inversion for solving two-dimensional acoustic inverse problem, Journal of Inverse and Ill-Posed Problems, 23 (2015), 687-700.  doi: 10.1515/jiip-2015-0083.

[15]

V. A. KhoaM. V. Klibanov and L. H. Nguyen, Convexification for a three-dimensional inverse scattering problem with the moving point source, SIAM Journal on Imaging Sciences, 13 (2020), 871-904.  doi: 10.1137/19M1303101.

[16]

M. V. Klibanov, Inverse problems in the large and Carleman bounds, Differential Equations, 20 (1984), 755-760. 

[17]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575. doi: 10.1088/0266-5611/8/4/009.

[18]

M. V. Klibanov and O. V. Ioussoupova, Uniform strict convexity of a cost functional for three-dimensional inverse scattering problem, SIAM Journal on Mathematical Analysis, 26 (1995), 147-179.  doi: 10.1137/S0036141093244039.

[19]

M. V. Klibanov, Global convexity in a three-dimensional inverse acoustic problem, SIAM Journal on Mathematical Analysis, 28 (1997), 1371-1388.  doi: 10.1137/S0036141096297364.

[20]

J. S. Lee and C. Nguyen, Novel low-cost ultra-wideband, ultra-short-pulse transmitter with MESFET impulse-shaping circuitry for reduced distortion and improved pulse repetition rate, IEEE Microwave and Wireless Components Letters, 11 (2001), 208-210. 

[21]

V. A. Khoa, G. W. Bidney, M. V. Klibanov, L. H. Nguyen, L. H. Nguyen, A. J. Sullivan and V. N. Astratov, Convexification and experimental data for a 3D inverse scattering problem with the moving point source, preprint, arXiv: 2003.11513 doi: 10.1137/19M1303101.

[22]

I. M. Gelfand and B. M. Levitan, On determining a differential equation from its spectral function, Amer. Math. Soc. Transl., 2 (1955), 253-304.  doi: 10.1090/trans2/001/11.

[23]

M. G. Krein, On a method of effective solution of an inverse boundary problem, Dokl. Akad. Nauk SSSR, 94 (1954), 987-990. 

[24]

A. L. KarchevskyM. V. KlibanovL. NguyenN. 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.

[25]

M. V. Klibanov and A. A. Timonov, Carleman estimates for coefficient inverse problems and numerical applications, Walter de Gruyter, 46 (2012). doi: 10.1515/9783110915549.

[26]

M. V. KlibanovA. V. KuzhugetS. I. Kabanikhin and D. V. Nechaev, A new version of the quasi-reversibility method for the thermoacoustic tomography and a coefficient inverse problem, Applicable Analysis, 87 (2008), 1227-1254.  doi: 10.1080/00036810802001297.

[27]

M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, Journal of Inverse and Ill-Posed Problems, 21 (2013), 477-560.  doi: 10.1515/jip-2012-0072.

[28]

M. V. Klibanov and V. G. Kamburg, Globally strictly convex cost functional for an inverse parabolic problem, Mathematical Methods in the Applied Sciences, 39 (2016), 930-940.  doi: 10.1002/mma.3531.

[29]

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, preprint, arXiv: 1602.09092. doi: 10.3934/ipi.2016032.

[30]

M. V. Klibanov, A. E. Kolesov, A. Sullivan and L. Nguyen, A new version of the convexification method for a 1D coefficient inverse problem with experimental data, Inverse Problems, 34 (2018), 115014. doi: 10.1088/1361-6420/aadbc6.

[31]

L. Baudouin, M. de Buhan, S. Ervedoza and A. Osses, Carleman-based reconstruction algorithm for the waves, 2020.

[32]

M. V. Klibanov, J. Li and W. Zhang, Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data, Inverse Problems, (2019). doi: 10.1088/1361-6420/aafecd.

[33]

M. V. KlibanovJ. 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.

[34]

M. V. Klibanov and A. E. Kolesov, Convexification of a 3-D coefficient inverse scattering problem, Computers & Mathematics with Applications, 77 (2019), 1681-1702.  doi: 10.1016/j.camwa.2018.03.016.

[35]

M. V. KlibanovA. 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. Imag. Sci., 12 (2019), 576-603.  doi: 10.1137/18M1191658.

[36]

A. Einstein, Onthe electrodynamics of moving bodies, Annalen der Physik, 322 (1905), 891-921. 

[37]

M. V. Klibanov, J. Li and W. Zhang, Convexification for an inverse parabolic problem, preprint, arXiv: 2001.01880.

[38]

J. Korpela, M. Lassas and L. Oksanen, Regularization strategy for an inverse problem for a 1+ 1 dimensional wave equation, Inverse Problems, 32 (2016), 065001. doi: 10.1088/0266-5611/32/6/065001.

[39]

J. Korpela, M. Lassas and L. Oksanen, Discrete regularization and convergence of the inverse problem for 1+ 1 dimensional wave equation, preprint, arXiv: 1803.10541. doi: 10.3934/ipi.2019027.

[40]

A. V. Kuzhuget, et al., Blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm, Inverse Problems, 28 (2012), 095007.

[41]

T. T. Le and L. H. Nguyen, A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data, preprint, arXiv: 1910.05584

[42]

B. T. Polyak, Introduction to optimization. Optimization software, Inc., Publications Division, New York, 1 (1987).

[43]

V. G. Romanov, Inverse Problems of Mathematical Physics, Walter de Gruyter GmbH & Co KG, 2018.

[44]

J. A. ScalesM. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems, J. Comput. Phys., 103 (1992), 258-268. 

[45]

A. N. Tikhonov, A. Goncharsky, V. Stepanov and A. G. Yagola, Numerical methods for the solution of ill-posed problems, Springer Science & Business Media, 328 (2013). doi: 10.1007/978-94-015-8480-7.

Figure 1.  The rectangle $ D(x,t) = \left\{ (\xi ,\tau ):\left\vert \xi \right\vert <\tau <t-\left\vert x-\xi \right\vert \right\} $ and the triangle $ Tr $
Figure 3.  Numerical reconstructions (the black marked dots) of functions $ a(x) $ (the solid lines). Noise level $ \xi = 0.1 $.
Figure 2.  The comparison of noiseless and noisy data. Figure 2(A) shows the norm of the functional (6.2) for each iteration of the gradient descent for the test function depicted on Figure 3(A). Figure 2(D) corresponds to our test for $ \lambda $ = 0; see the text.
Figure 4.  Limiting testing of different values of the parameter $ \lambda $ for the test function of Test 1, see comments in the text. The data are noiseless.
Figure 5.  Testing of different values of the parameter $ \alpha , $ see comments in the text. Solid line is the correct function of Test 1. The data are noiseless.
Table 6.2.  Summary of numerical results. Here $ \left\Vert \cdot\right\Vert _{\infty } $ denotes the $ L_{\infty } $
Test $ n^{\ast} $ Error $ \Vert J^h_{\lambda,\beta,\mu}(w_0) \Vert_{\infty} $ $ \Vert J^h_{\lambda,\beta,\mu}(w_{n^{\ast}}) \Vert_{\infty} $
1 30 0.1628 2570 2.7465
2 33 0.2907 34.42 0.22
3 51 0.0804 3.12 0.0007
4 41 0.3222 0.82 0.0003
Test $ n^{\ast} $ Error $ \Vert J^h_{\lambda,\beta,\mu}(w_0) \Vert_{\infty} $ $ \Vert J^h_{\lambda,\beta,\mu}(w_{n^{\ast}}) \Vert_{\infty} $
1 30 0.1628 2570 2.7465
2 33 0.2907 34.42 0.22
3 51 0.0804 3.12 0.0007
4 41 0.3222 0.82 0.0003
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