December  2019, 13(6): 1367-1393. doi: 10.3934/ipi.2019060

Travel time tomography with formally determined incomplete data in 3D

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

Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC, 28223, mklibanv@uncc.edu

Received  April 2019 Revised  July 2019 Published  October 2019

For the first time, a globally convergent numerical method is developed and Lipschitz stability estimate is obtained for the challenging problem of travel time tomography in 3D for formally determined incomplete data. The semidiscrete case is considered meaning that finite differences are involved with respect to two out of three variables. First, Lipschitz stability estimate is derived, which implies uniqueness. Next, a weighted globally strictly convex Tikhonov-like functional is constructed using a Carleman-like weight function for a Volterra integral operator. The gradient projection method is constructed to minimize this functional. It is proven that this method converges globally to the exact solution if the noise in the data tends to zero.

Citation: Michael V. Klibanov. Travel time tomography with formally determined incomplete data in 3D. Inverse Problems and Imaging, 2019, 13 (6) : 1367-1393. doi: 10.3934/ipi.2019060
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]

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.

[3]

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

[4]

I. N. Bernšte$\mathop {\rm{i}}\limits^ ˇ $n and M. L. Gerver, A problem of integral geometry for a family of geodesics and an inverse kinematic seismics problem, Dokl. Akad. Nauk SSSR, 243 (1978), 302-305. 

[5]

A. L. Bukhge$\mathop {\rm{i}}\limits^ˇ $m and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272. 

[6]

J.-P. Guillement and R. G. Novikov, Inversion of weighted Radon transforms via finite Fourier series weight approximation, Inverse Problems in Science and Engineering, 22 (2014), 787-802.  doi: 10.1080/17415977.2013.823417.

[7]

G. Herglotz, $\tilde{\rm A}$oeber die Elastizitaet der Erde bei Beruecksichtigung ihrer variablen Dichte, Zeitschr. fur Math. Phys., 52 (1905), 275-299. 

[8]

S. I. Kabanikhin, A. D. Satybaev and M. A. Shishlenin, Direct Methods of Solving Multidimensional Hyperbolic Inverse Problems, Inverse and Ill-posed Problems Series. VSP, Utrecht, 2005.

[9]

S. I. KabanikhinK. K. SabelfeldN. 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.

[10]

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

[11]

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.

[12]

M. V. Klibanov, Global convexity in a three-dimensional inverse acoustic problem, SIAM J. Math. Anal., 28 (1997), 1371-1388.  doi: 10.1137/S0036141096297364.

[13]

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.

[14]

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.  doi: 10.1515/jip-2012-0072.

[15]

M. V. Klibanov and V. G. Romanov, Reconstruction procedures for two inverse scattering problems without the phase information, SIAM J. Appl. Math., 76 (2016), 178-196.  doi: 10.1137/15M1022367.

[16]

M. V. Klibanov, Convexification of restricted Dirichlet-to-Neumann map, J. Inverse and Ill-Posed Problems, 25 (2017), 669-685.  doi: 10.1515/jiip-2017-0067.

[17]

M. V. KlibanovA. E. Kolesov and D.-L. Nguyen, Convexification method for a coefficient inverse problem and its performance for experimental backscatter data for buried targets, SIAM J. Imaging Sciences, 12 (2019), 576-603.  doi: 10.1137/18M1191658.

[18]

M. V. Klibanov, J. Z. Li and W. L. Zhang, Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data, Inverse Problems, 35 (2019), 035005, 33 pp. doi: 10.1088/1361-6420/aafecd.

[19]

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, 29 pp. doi: 10.1088/1361-6420/aadbc6.

[20]

M. V. Klibanov and L. H. Nguyen, PDE-based numerical method for a limited angle x-ray tomography, Inverse Problems, 35 (2019), 045009, 32 pp. doi: 10.1088/1361-6420/ab0133.

[21]

R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric and integral geometry, Dokl. Akad. Nauk SSSR, 232 (1977), 32-35. 

[22]

R. G. Mukhometov and V. G. Romanov, On the problem of determining an isotropic Riemannian metric in the $n$-dimensional space, Dokl. Akad. Nauk SSSR, 243 (1978), 41-44. 

[23]

L. Pestov and G. Uhlmann, Two dimensional simple Riemannian manifolds are boundary distance rigid, Annals of Mathematics, 161 (2005), 1093-1110.  doi: 10.4007/annals.2005.161.1093.

[24] V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Press, b.v., Utrecht, 1987. 
[25]

V. G. Romanov, Inverse problems for differential equations with memory, Eurasian J. of Mathematical and Computer Applications, 2 (2014), 51-80. 

[26]

A. A. Samarskii, The Theory of Difference Schemes, Monographs and Textbooks in Pure and Applied Mathematics, 240. Marcel Dekker, Inc., New York, 2001. doi: 10.1201/9780203908518.

[27]

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

[28]

U. Schröder and T. Schuster, An iterative method to reconstruct the refractive index of a medium from time-off-light measurements, Inverse Problems, 32 (2016), 085009, 35 pp. doi: 10.1088/0266-5611/32/8/085009.

[29]

A. V. Smirnov, M. V. Klibanov and L. H. Nguyen, A numerical method for an inverse source problem for the radiative transfer equation with absorption and scattering terms, SIAM J. Sci. Comp, accepted for publication, (2019), arXiv: 1904.00547.

[30]

P. Stefanov, G. Uhlmann and A. Vasy, Local and Global Boundary Rigidity and the Geodesic X-Ray Transform in the Normal Gauge, (2017), arXiv: 1702.03638v2.

[31]

A. N. Tikhonov, A. V. 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.

[32]

M. M. Vajnberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Israel Program for Scientific Translations, Jerusalem-London, 1973.

[33]

L. Volgyesi and M. Moser, The inner structure of the Earth, Periodica Polytechnica Chemical Engineering, 26 (1982), 155-204. 

[34]

E. Wiechert and K. Zoeppritz, Uber erdbebenwellen, Nachr. Koenigl. Geselschaft Wiss. Gottingen, 4 (1907), 415-549. 

[35]

H. K. Zhao and Y. M. Zhong, A hybrid adaptive phase space method for reflection traveltime tomography, SIAM J. Imaging Sciences, 12 (2019), 28-53.  doi: 10.1137/18M117426X.

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]

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.

[3]

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

[4]

I. N. Bernšte$\mathop {\rm{i}}\limits^ ˇ $n and M. L. Gerver, A problem of integral geometry for a family of geodesics and an inverse kinematic seismics problem, Dokl. Akad. Nauk SSSR, 243 (1978), 302-305. 

[5]

A. L. Bukhge$\mathop {\rm{i}}\limits^ˇ $m and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272. 

[6]

J.-P. Guillement and R. G. Novikov, Inversion of weighted Radon transforms via finite Fourier series weight approximation, Inverse Problems in Science and Engineering, 22 (2014), 787-802.  doi: 10.1080/17415977.2013.823417.

[7]

G. Herglotz, $\tilde{\rm A}$oeber die Elastizitaet der Erde bei Beruecksichtigung ihrer variablen Dichte, Zeitschr. fur Math. Phys., 52 (1905), 275-299. 

[8]

S. I. Kabanikhin, A. D. Satybaev and M. A. Shishlenin, Direct Methods of Solving Multidimensional Hyperbolic Inverse Problems, Inverse and Ill-posed Problems Series. VSP, Utrecht, 2005.

[9]

S. I. KabanikhinK. K. SabelfeldN. 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.

[10]

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

[11]

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.

[12]

M. V. Klibanov, Global convexity in a three-dimensional inverse acoustic problem, SIAM J. Math. Anal., 28 (1997), 1371-1388.  doi: 10.1137/S0036141096297364.

[13]

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.

[14]

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.  doi: 10.1515/jip-2012-0072.

[15]

M. V. Klibanov and V. G. Romanov, Reconstruction procedures for two inverse scattering problems without the phase information, SIAM J. Appl. Math., 76 (2016), 178-196.  doi: 10.1137/15M1022367.

[16]

M. V. Klibanov, Convexification of restricted Dirichlet-to-Neumann map, J. Inverse and Ill-Posed Problems, 25 (2017), 669-685.  doi: 10.1515/jiip-2017-0067.

[17]

M. V. KlibanovA. E. Kolesov and D.-L. Nguyen, Convexification method for a coefficient inverse problem and its performance for experimental backscatter data for buried targets, SIAM J. Imaging Sciences, 12 (2019), 576-603.  doi: 10.1137/18M1191658.

[18]

M. V. Klibanov, J. Z. Li and W. L. Zhang, Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data, Inverse Problems, 35 (2019), 035005, 33 pp. doi: 10.1088/1361-6420/aafecd.

[19]

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, 29 pp. doi: 10.1088/1361-6420/aadbc6.

[20]

M. V. Klibanov and L. H. Nguyen, PDE-based numerical method for a limited angle x-ray tomography, Inverse Problems, 35 (2019), 045009, 32 pp. doi: 10.1088/1361-6420/ab0133.

[21]

R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric and integral geometry, Dokl. Akad. Nauk SSSR, 232 (1977), 32-35. 

[22]

R. G. Mukhometov and V. G. Romanov, On the problem of determining an isotropic Riemannian metric in the $n$-dimensional space, Dokl. Akad. Nauk SSSR, 243 (1978), 41-44. 

[23]

L. Pestov and G. Uhlmann, Two dimensional simple Riemannian manifolds are boundary distance rigid, Annals of Mathematics, 161 (2005), 1093-1110.  doi: 10.4007/annals.2005.161.1093.

[24] V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Press, b.v., Utrecht, 1987. 
[25]

V. G. Romanov, Inverse problems for differential equations with memory, Eurasian J. of Mathematical and Computer Applications, 2 (2014), 51-80. 

[26]

A. A. Samarskii, The Theory of Difference Schemes, Monographs and Textbooks in Pure and Applied Mathematics, 240. Marcel Dekker, Inc., New York, 2001. doi: 10.1201/9780203908518.

[27]

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

[28]

U. Schröder and T. Schuster, An iterative method to reconstruct the refractive index of a medium from time-off-light measurements, Inverse Problems, 32 (2016), 085009, 35 pp. doi: 10.1088/0266-5611/32/8/085009.

[29]

A. V. Smirnov, M. V. Klibanov and L. H. Nguyen, A numerical method for an inverse source problem for the radiative transfer equation with absorption and scattering terms, SIAM J. Sci. Comp, accepted for publication, (2019), arXiv: 1904.00547.

[30]

P. Stefanov, G. Uhlmann and A. Vasy, Local and Global Boundary Rigidity and the Geodesic X-Ray Transform in the Normal Gauge, (2017), arXiv: 1702.03638v2.

[31]

A. N. Tikhonov, A. V. 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.

[32]

M. M. Vajnberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Israel Program for Scientific Translations, Jerusalem-London, 1973.

[33]

L. Volgyesi and M. Moser, The inner structure of the Earth, Periodica Polytechnica Chemical Engineering, 26 (1982), 155-204. 

[34]

E. Wiechert and K. Zoeppritz, Uber erdbebenwellen, Nachr. Koenigl. Geselschaft Wiss. Gottingen, 4 (1907), 415-549. 

[35]

H. K. Zhao and Y. M. Zhong, A hybrid adaptive phase space method for reflection traveltime tomography, SIAM J. Imaging Sciences, 12 (2019), 28-53.  doi: 10.1137/18M117426X.

Figure 1.  An illustration for complete and incomplete data in the 2D case, see details in [24]. To simplify, we assume in this figure that the geodesics are straight lines. Thus, we deal in this figure with the data of Radon transform, generated by the function "radon" of MATLAB. a) The true function $ m\left( \mathbf{x}\right) $ to be imaged. b) The complete data of the Radon transform of the function of a). c) The incomplete data of the Radon transform of a) in the case when the source runs along an interval of a straight line, as in this paper below.
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