Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

IMPACT FACTOR 2018: 0.881
5-year IMPACT FACTOR: 1.170

CiteScore 2018: 0.91

SCImago Journal Rank (SJR) 2018: 0.430
Source Normalized Impact per Paper (SNIP) 2018: 0.969

Mathematical Citation Quotient (MCQ) 2018: 0.66

See all formats and pricing
More options …
Volume 27, Issue 4


On the travel time tomography problem in 3D

Michael V. Klibanov
  • Corresponding author
  • Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC 28223, USA
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2019-07-16 | DOI: https://doi.org/10.1515/jiip-2019-0036


Numerical issues for the 3D travel time tomography problem with non-overdetemined data are considered. Truncated Fourier series with respect to a special orthonormal basis of functions depending on the source position is used. In addition, truncated trigonometric Fourier series with respect to two out of three spatial variables are used. First, the Lipschitz stability estimate is obtained. Next, a globally convergent numerical method is constructed using a Carleman estimate for an integral operator.

Keywords: Global convergence; semi-finite dimensional mathematical model; Carleman weight function; weighted Tikhonov-like functional

MSC 2010: 35R25; 35R30


  • [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. CrossrefWeb of ScienceGoogle Scholar

  • [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. Numer. Anal. 55 (2017), no. 4, 1578–1613. CrossrefWeb of ScienceGoogle Scholar

  • [3]

    L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012. Google Scholar

  • [4]

    L. Beilina and M. V. Klibanov, Globally strongly convex cost functional for a coefficient inverse problem, Nonlinear Anal. Real World Appl. 22 (2015), 272–288. Web of ScienceCrossrefGoogle Scholar

  • [5]

    I. N. Bernšteĭn 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), no. 2, 302–305. Google Scholar

  • [6]

    A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR 260 (1981), no. 2, 269–272. Google Scholar

  • [7]

    J.-P. Guillement and R. G. Novikov, Inversion of weighted Radon transforms via finite Fourier series weight approximations, Inverse Probl. Sci. Eng. 22 (2014), no. 5, 787–802. CrossrefWeb of ScienceGoogle Scholar

  • [8]

    G. Herglotz, Ãœber die Elastizitaet der Erde bei Beruecksichtigung ihrer variablen Dichte, Z. Math. Phys. 52 (1905), 275–299. Google Scholar

  • [9]

    S. I. Kabanikhin, Projection-difference Methods for Determining of Hyperbolic Equations Coefficients (in Russian), Nauka, Novosibirsk, 1988. Google Scholar

  • [10]

    S. I. Kabanikhin, K. K. Sabelfeld, N. S. Novikov and M. A. Shishlenin, Numerical solution of the multidimensional Gelfand—Levitan equation, J. Inverse Ill-Posed Probl. 23 (2015), no. 5, 439–450. Web of ScienceGoogle Scholar

  • [11]

    S. I. Kabanikhin, A. D. Satybaev and M. A. Shishlenin, Direct Methods of solving Multidimensional Inverse Hyperbolic Problems, VSP, Utrecht, 2005. Google Scholar

  • [12]

    M. V. Klibanov, Global convexity in a three-dimensional inverse acoustic problem, SIAM J. Math. Anal. 28 (1997), no. 6, 1371–1388. CrossrefGoogle Scholar

  • [13]

    M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl. 21 (2013), no. 4, 477–560. Web of ScienceGoogle Scholar

  • [14]

    M. V. Klibanov, Carleman weight functions for solving ill-posed Cauchy problems for quasilinear PDEs, Inverse Problems 31 (2015), no. 12, Article ID 125007. Web of ScienceGoogle Scholar

  • [15]

    M. V. Klibanov, Convexification of restricted Dirichlet-to-Neumann map, J. Inverse Ill-Posed Probl. 25 (2017), no. 5, 669–685. Web of ScienceGoogle Scholar

  • [16]

    M. V. Klibanov, Travel time tomography with formally determined incomplete data in 3D, preprint (2019), https://arxiv.org/abs/1904.06610.

  • [17]

    M. V. Klibanov and V. G. Kamburg, Globally strictly convex cost functional for an inverse parabolic problem, Math. Methods Appl. Sci. 39 (2016), no. 4, 930–940. Web of ScienceCrossrefGoogle Scholar

  • [18]

    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), no. 1, 576–603. CrossrefWeb of ScienceGoogle Scholar

  • [19]

    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), no. 5, 1733–1755. CrossrefGoogle Scholar

  • [20]

    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), no. 11, Article ID 115014. Web of ScienceGoogle Scholar

  • [21]

    M. V. Klibanov, J. Li and W. Zhang, Convexification for the inversion of a time dependent wave front in a heterogeneous medium, preprint (2018), https://arxiv.org/abs/1812.11281.

  • [22]

    M. V. Klibanov, J. Li and W. Zhang, Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data, Inverse Problems 35 (2019), no. 3, Article ID 035005. Web of ScienceGoogle Scholar

  • [23]

    M. V. Klibanov and L. H. Nguyen, PDE-based numerical method for a limited angle x-ray tomography, Inverse Problems 35 (2019), no. 4, Article ID 045009. Web of ScienceGoogle Scholar

  • [24]

    M. V. Klibanov and V. G. Romanov, Reconstruction procedures for two inverse scattering problems without the phase information, SIAM J. Appl. Math. 76 (2016), no. 1, 178–196. Web of ScienceCrossrefGoogle Scholar

  • [25]

    M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004. Google Scholar

  • [26]

    R. G. Muhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry, Dokl. Akad. Nauk SSSR 232 (1977), no. 1, 32–35. Google Scholar

  • [27]

    R. G. Muhometov and V. G. Romanov, On the problem of determining an isotropic Riemannian metric in the n-dimensional space, Dokl. Acad. Sci. USSR 19 (1978), 1330–1333. Google Scholar

  • [28]

    L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math. (2) 161 (2005), no. 2, 1093–1110. CrossrefGoogle Scholar

  • [29]

    V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science, Utrecht, 1987. Google Scholar

  • [30]

    V. G. Romanov, A problem on determining the permittivity coefficient in a stationary system of Maxwell equations, Dokl. Akad. Nauk 474 (2017), no. 4, 413–417. Google Scholar

  • [31]

    J. A. Scales, M. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems, J. Comput. Phys 103 (1992), 258–268. CrossrefGoogle Scholar

  • [32]

    U. Schröder and T. Schuster, An iterative method to reconstruct the refractive index of a medium from time-of-flight measurements, Inverse Problems 32 (2016), no. 8, Article ID 085009. Google Scholar

  • [33]

    P. Stefanov, G. Uhlmann and A. Vasy, Inverting the local geodesic X-ray transform on tensors, J. Anal. Math. 136 (2018), no. 1, 151–208. Web of ScienceCrossrefGoogle Scholar

  • [34]

    A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-posed Problems, Math. Appl. 328, Kluwer Academic, Dordrecht, 1995. Google Scholar

  • [35]

    M. M. Vaĭnberg, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, John Wiley & Sons, Washington, 1973. Google Scholar

  • [36]

    L. Volgyesi and M. Moser, The inner structure of the Earth, Period. Polytech. Chem. Eng. 26 (1982), 155–204. Google Scholar

  • [37]

    E. Wiechert and K. Zoeppritz, Über Erdbebenwellen, Nachr. Koenigl. Geselschaft Wiss. Gottingen 4 (1907), 415–549. Google Scholar

  • [38]

    H. Zhao and Y. Zhong, A hybrid adaptive phase space method for reflection traveltime tomography, SIAM J. Imaging Sci. 12 (2019), no. 1, 28–53. Web of ScienceCrossrefGoogle Scholar

About the article

Received: 2019-05-21

Accepted: 2019-06-03

Published Online: 2019-07-16

Published in Print: 2019-08-01

This work was supported by US Army Research Laboratory and US Army Research Office grant W911NF-19-1-0044.

Citation Information: Journal of Inverse and Ill-posed Problems, Volume 27, Issue 4, Pages 591–607, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2019-0036.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in