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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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1569-3945
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Volume 25, Issue 5

Issues

Convexification of restricted Dirichlet-to-Neumann map

Michael V. Klibanov
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  • Department of Mathematics and Statistics, University of North Carolina atCharlotte, Charlotte, NC 28223, USA
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Published Online: 2017-08-05 | DOI: https://doi.org/10.1515/jiip-2017-0067

Abstract

By our definition, “restricted Dirichlet-to-Neumann (DN) map” means that the Dirichlet and Neumann boundary data for a coefficient inverse problem (CIP) are generated by a point source running along an interval of a straight line. On the other hand, the conventional DN data can be generated, at least sometimes, by a point source running along a hypersurface. CIPs with restricted DN data are non-overdetermined in the n-dimensional case, with n2. We develop, in a unified way, a general and radically new numerical concept for CIPs with restricted DN data for a broad class of PDEs of second order, such as, e.g., elliptic, parabolic and hyperbolic ones. Namely, using Carleman weight functions, we construct globally convergent numerical methods. Hölder stability and uniqueness are also proved. The price we pay for these features is a well-acceptable one in the numerical analysis, that is, we truncate a certain Fourier-like series with respect to some functions depending only on the position of the point source. At least three applications are imaging of land mines, crosswell imaging and electrical impedance tomography.

Keywords: Restricted Dirichlet-to-Neumann data; convexification; global strict convexity, Carleman weight functions

MSC 2010: 35R30

References

  • [1]

    G. S. Alberti, H. Ammari, B. Jin, J.-K. Seo and W. Zhang, The linearized inverse problem in multifrequency electrical impedance tomography, SIAM J. Imaging Sci. 9 (2016), no. 4, 1525–1551. CrossrefWeb of ScienceGoogle Scholar

  • [2]

    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. Web of ScienceCrossrefGoogle Scholar

  • [3]

    L. Beilina, Energy estimates and numerical verification of the stabilized domain decomposition finite element/finite difference approach for time-dependent Maxwell’s system, Cent. Eur. J. Math. 11 (2013), no. 4, 702–733. Web of ScienceGoogle 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]

    M. I. Belishev, Recent progress in the boundary control method, Inverse Problems 23 (2007), no. 5, R1–R67. CrossrefGoogle Scholar

  • [6]

    M. I. Belishev, I. B. Ivanov, I. V. Kubyshkin and V. S. Semenov, Numerical testing in determination of sound speed from a part of boundary by the BC-method, J. Inverse Ill-Posed Probl. 24 (2016), no. 2, 159–180. Web of ScienceGoogle Scholar

  • [7]

    A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964. Google Scholar

  • [8]

    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss. 224, Springer, Berlin, 1983. Google Scholar

  • [9]

    B. Harrach and M. N. Minh, Enhancing residual-based techniques with shape reconstruction features in electrical impedance tomography, Inverse Problems 32 (2016), no. 12, Article ID 125002. Web of ScienceGoogle Scholar

  • [10]

    N. Hyvönen, P. Piiroinen and O. Seiskari, Point measurements for a Neumann-to-Dirichlet map and the Calderón problem in the plane, SIAM J. Math. Anal. 44 (2012), no. 5, 3526–3536. CrossrefGoogle Scholar

  • [11]

    B. Jin, Y. Xu and J. Zou, A convergent adaptive finite element method for electrical impedance tomography, IMA J. Numer. Anal. 37 (2017), no. 3, 1520–1550. Web of ScienceGoogle Scholar

  • [12]

    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

  • [13]

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

  • [14]

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

  • [15]

    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

  • [16]

    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

  • [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 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

  • [19]

    M. V. Klibanov and N. T. Thành, Recovering dielectric constants of explosives via a globally strictly convex cost functional, SIAM J. Appl. Math. 75 (2015), no. 2, 518–537. Web of ScienceCrossrefGoogle Scholar

  • [20]

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

  • [21]

    O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 23, American Mathematical Society, Providence, 1968. Google Scholar

  • [22]

    E. L. Lakshtanov, R. G. Novikov and B. R. Vainberg, A global Riemann-Hilbert problem for two-dimensional inverse scattering at fixed energy, Rend. Istit. Mat. Univ. Trieste 48 (2016), 21–47. Google Scholar

  • [23]

    M. M. Lavrentiev, A. V. Avdeev, M. M. Lavrentiev, Jr. and V. I. Priimenko, Inverse Problems of Mathematical Physics, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 2003. Google Scholar

  • [24]

    A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2) 143 (1996), no. 1, 71–96. CrossrefGoogle Scholar

  • [25]

    D.-L. Nguyen, M. V. Klibanov, L. H. Nguyen, A. E. Kolesov, M. A. Fiddy and H. Liu, Numerical solution of a coefficient inverse problem with multi-frequency experimental raw data by a globally convergent algorithm, J. Comput. Phys. 345 (2017), 17–32. CrossrefWeb of ScienceGoogle Scholar

  • [26]

    D.-L. Nguyen, L. H. Nguyen, M. V. Klibanov and M. A. Fiddy, Imaging of buried objects from multi-frequency experimental data using a globally convergent inversion method, preprint (2017), https://arxiv.org/abs/1705.01219; to appear in J. Inverse Ill-Posed Probl.

  • [27]

    R. G. Novikov, A multidimensional inverse spectral problem for the equation -Δψ+(v(x)-Eu(x))ψ=0, Funct. Anal. Appl. 22 (1988), no. 4, 263–272. Google Scholar

  • [28]

    R. G. Novikov and M. Santacesaria, Monochromatic reconstruction algorithms for two-dimensional multi-channel inverse problems, Int. Math. Res. Not. IMRN (2013), no. 6, 1205–1229. Google Scholar

  • [29]

    J. Sylvester and G. Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math. 39 (1986), no. 1, 91–112. CrossrefGoogle Scholar

About the article

Received: 2017-06-26

Accepted: 2017-07-06

Published Online: 2017-08-05

Published in Print: 2017-10-01


Funding Source: Army Research Laboratory

Award identifier / Grant number: W911NF-15-1-0233

Funding Source: Army Research Office

Award identifier / Grant number: W911NF-15-1-0233

Funding Source: Office of Naval Research

Award identifier / Grant number: N00014-15-1-2330

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.


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 25, Issue 5, Pages 669–685, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2017-0067.

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