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Research Interests:

Inverse Problems for Partial Differential Equations

Applications: optical imaging of biological tissues, imaging of subsurface objects, such as land and sea mines, for example.

Degrees:

1972

M.S. in Applied Mathematics

Novosibirsk State University

(Novosibirsk, Russia)

1977

Ph.D. in Applied Mathematics

 Ural State University

 (Yekaterinburg, Russia)

1986

Doctor of Science in Applied Mathematics

(This is the highest scientific degree in Russia)

 Computing Center of the Siberian Branch of the Russian Academy of

 Sciences (Novosibirsk)

The above degrees were received from the elite Russian institutions.

M.V. Klibanov has been with the Department of Mathematics at UNC Charlotte since 1990, Professor since 1994.

Major Research Achievements:

 Introduction of a powerful tool of Carleman estimates in the field of inverse problems in 1981 [1,2]. Since then this approach became one of a few classical tools in the field inverse problems. While originally Carleman estimates were applied only to the uniqueness and stability results for inverse problems with single measurement [1-6], recently they found applications in numerical methods for inverse problems [7-9], it was shown in [7-9] that the use of Carleman Weight Functions makes it possible to construct strictly convex objective functions for hyperbolic and parabolic inverse problems. Next, this idea was considered in terms of global optimization [10], where the term convexification was introduced. Finally, in 2003 convexification based algorithms were introduced, see [11,12] and follow up publications.

The main reason of the work on the convexification algorithms is one of major challenging issues in the field of inverse problems: presence of multiple local minima in the conventional least squares functionals even for simple configurations, see, e.g., [10]. Because of the multiextremality of those functionals, convergence of conventional optimization techniques, such as Newton-like or gradient method, for example is not guaranteed, unless the starting vector is not located in a close proximity of the (unknown!) exact solution. This actually means local convergence. Existing methods of global optimization (e.g., simulated annealing) are stochastic in their nature and, therefore quite time consuming when the number of unknowns exceeds ten. Also, their convergence is not guaranteed for the case of inverse problems.

The essence of a convexification based algorithm is constructing of a sequence of strictly convex objective functionals. This eliminates the above issue of multiextremality. Convergence of such an algorithm to the exact solution is guaranteed independently on the starting vector. The latter is the main advantage of the convexification algorithms.

Recent federal grants:

National Science Foundation Grant (DMS-9704923) "A fast numerical method for imaging small abnormalities in diffusion

tomography," 9/1/97-8/31/00

Army Research Office Grant (DAAG55-98-1-0401) "A novel principle of data processing for hand-held ground penetrating

radars," 7/1/98-6/30/01.

Citizenship: USA

Publications:

More than 100 publications in the field of inverse problems for partial differential equations.

Most significant publications:

28. L. Beilina and M.V. Klibanov, A globally convergent numerical method for some coefficient inverse problems with resulting second order elliptic equations, submitted for publication; also available online here, (preprint number 07-311, posting date December 18, 2007) and here.

27. J. Su, H. Shan, H. Liu and M.V. Klibanov, Reconstruction method with data from a multiple-site continuous-wave source for three-dimensional optical tomography, Journal of Optical Society of America A, 23, 2388-2395, 2006.

26. C. Clason and M.V. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium, SIAM Journal on Scientific Computing,  30, 1-23, 2007.

25. M.V. Klibanov and A. Timonov, Numerical studies on the globally convergent convexification algorithm in 2D, Inverse Problems, 23, 123-138, 2007.

24. M.V. Klibanov and M. Yamamoto, Exact controllability for the non-stationary transport equation, submitted for publication. Also, available online as the preprint number 06-37 at http://www.ma.utexas.edu/mo_arc/index-06.html#end

23. M.V. Klibanov, Estimates of initial conditions of parabolic equations and inequalities via lateral Cauchy data, Inverse Problems, 22, 495-514, 2006

22. M.V. Klibanov, S.I. Kabanikhin and D.V. Nechaev, Numerical solution of the problem of computational time reversal in the quadrant, Waves in Random and Complex Media, 16, 473-494, 2006.

21. M.V. Klibanov, Lipschitz stability for hyperbolic inequalities in octants with the lateral Cauchy data and refocusing in time reversal, Journal of Inverse and Ill-Posed Problems, 13, 353-363,2005.

20. M.V. Klibanov and A. Timonov, Global uniqueness for a 3d/2d inverse conductivity problem via the modified method of Carleman estimates, J. Inverse and Ill-Posed Problems, 13, 149-174, 2005.

19. M.V. Klibanov, On the recovery of a 2-D Function from the modulus of its Fourier transform, Journal of Mathematical Analysis and Applications, 323, 818-843, 2006..

18. M.V. Klibanov, Distributed modeling of propagation of computer viruses/worms by Partial Differential Equations, accepted for publication in Applicable Analysis, 2004.

17. H. Egger, H. W. Engl and M.V. Klibanov, H. Egger, H. W. Engl and M.V. Klibanov, Global uniqueness and Holder stability for recovering a nonlinear source term in a parabolic equation, Inverse Problems, 21, 271-290, 2005..

16. M.V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an accoustic equation, Applicable Analysis, 85, 515-538, 2006.

15. M.V. Klibanov and A. Timonov, A unified framework for constructing globally convergent algorithms for multidimensional coefficient inverse problems, Applicable Analysis, 2004, 83, 933-955

14. M.V. Klibanov and A. Timonov, On the mathematical treatment of time reversal, Inverse Problems, 2003, 19, 1299-1318.

13. M.V. Klibanov, Global uniqueness of a multidimensional inverse problem for a nonlinear parabolic equation by a Carleman estimate , Inverse Problems, 20, 1003-1032, 2004.

12. M.V. Klibanov and A. Timonov, A globally convergent convexification algorithm for the inverse problem of electromagnetic frequency sounding in one dimension, Numerical Methods and Programming, 2003, 4, 52-81. This journal is available online free of charge at http://num-meth.srcc.msu.su

11. M.V. Klibanov and A. Timonov, A sequential minimization algorithm based on the convexification approach, Inverse Problems, 2003, 19, 331-354.

10. M.V. Klibanov and A. Timonov, A new slant on inverse problems of electromagnetic frequency sounding: convexification� of a multiextremal objective function via Carleman weight function, Inverse Problems, 2001, 17, 1865-1888.

9. M.V. Klibanov, Global convexity in diffusion tomography, Nonlinear World, 4, 1997, 247-265.

8. M. V. Klibanov, Global convexity in 3-dimensional inverse acoustic problem, SIAM J. Math. Anal., 1997, 28,1371-1380.

7. M.V. Klibanov and O.V. Ioussoupova, Uniform strict convexity of a cost functional for three-dimensional inverse scattering problem, SIAM J. Math. Anal., 1995, 26, 147-179.

6. M. V. Klibanov, Inverse problems and Carleman estimates in the last two decades, in D. Colton, H. W. Engl, A. Louis, J. McLaughlin and W. Rundell (editors), Solution Methods For Inverse Problems, Springer, New York, 2000.

5. M.V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 1992, 8, 575-596.

4. M.V. Klibanov, A class of inverse problems for nonlinear parabolic equations, Siberian Mathematical J., 27, 698-708, 1987.

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

2. M.V. Klibanov, Uniqueness in the large of some multidimensional inverse problems, in Non-Classical Problems of Mathematical Physics, Proc. Computing Center of the Siberian Branch of the Russian Academy of Science, Novosibrisk, 1981, 101-114 (In Russian).

1. A.L. Buhgeim and M. V. Klibanov, Global uniqueness of a class of inverse problems, Soviet Math. Doklady, 1981, 24, 244-247.