Research Interests: Inverse Problems for Partial Differential Equations, Ill-Posed Problems.
Substantial experience in the interdisciplinary research including microwaves and nano science.
Data from MathSciNet:
Total number of publications: 176
Total number of citations: 2633
My citations on Google Scholar:
560 citations of the first publication on the Bukhgeim-Klibanov method, 1981.
A.L. Bukhgeim and M.V. Klibanov, Global uniqueness of a class of multidimensional inverse problems, Soviet Mathematics Doklady, 24, 244-247, 1981.
Information from Google Scholar:
|Citation indices||All||Since 2013|
Federal Funding mostly from US Army Research Office: More than $3,300,000 for the period of 2005-2021. The main topic: Globally Convergent Numerical Methods for Coefficient Inverse Problems.
1972, MS in Mathematics, Diploma with Honor, Novosibirsk State University, Novosibirsk, Russia. This is one of three top Russian universities.
1977, Ph.D. in Mathematics, subject area “Inverse Problems for Partial Differential Equations”, Urals State University, Ekaterinburg, Russia.
Scientific Mentor: Mikhail Mikhailovich Lavrent’ev (1932-2010), a Member of Russian Academy of Science, one of founders of the field of Inverse Problems
1986, Doctor of Science in Mathematics, subject area “Inverse Problems for Partial Differential Equations”, Computing Center of The Siberian Branch of The Russian Academy of Science, Novosibirsk.
1977-1990, Associate Professor, Department of Mathematics of The Samara State University, Samara, Russia.
1990-present, Associate Professor and then Full Professor (since 1994), Department of Mathematics and Statistics of The University of North Carolina at Charlotte.
Editorial Board Member:
Applicable Analysis, Inverse Problems in Science and Engineering, Numerical Methods and Programming.
My research is both applied and interdisciplinary oriented. It combines a strong theory with numerical results, which are based on this theory. In particular, I have many publications which describe the work of our numerical methods on experimental data.
MAIN MATHEMATICAL ACHIEVEMENTS:
- In a breakthrough work of 1981 has introduced, for the first time, the powerful tool of Carleman estimates in the field of Multidimensional Coefficient Inverse Problems (MCIPs). Publication: A.L. Buhgeim and M.V. Klibanov, Global uniqueness of a class of multidimensional inverse problems, Soviet Mathematics Doklady, 24, 244-247, 1981. This tool has allowed, for the first time, to prove global uniqueness theorems for broad classes of MCIPs with the non over-determined data. Contrary to this, only local uniqueness theorems were proven for such MCIPs prior to that. This idea is called nowadays “The Bukhgeim-Klibanov method”. This method has generated many publications of many authors. Currently, this remains the single method, enabling one to prove global uniqueness and stability theorems for MCIPs with non over-determined data. In addition, this idea has generated effective numerical methods, which I have developed later with some of coauthors. Item 3 is an example of those methods. Recent survey on this method: M.V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse and Ill-Posed Problems, 21, 477-560, 2013.
- In 1997 has started, for the first time, the development of globally convergent numerical methods (GCM) for coefficient inverse problems (CIPs). This research addresses the following crucial question: How to rigorously obtain at least one point in a sufficiently small neighborhood of the exact solution without any advanced knowledge of this neighborhood? Two GCMs have been developed: (1) The so-called tail functions method and (2) The convexification method. Both are currently verified on experimental data. Both work with non-redundant data. All other numerical methods for coefficient inverse problems which work with non redundant data, converge only locally. The convergence of a locally convergent method to the exact solution is rigorously guaranteed only if the starting point of iterations is located in a sufficiently small neighborhood of the exact solution. This means that locally convergent numerical methods are inherently unstable, since solutions they deliver critically depend on the starting points for iterations.
- My current interest is in the convexification. The convexification constructs a weighted Tikhonov-like functional for a CIP. The weight is the Carleman Weight Function. This is the function which is involved in the Carleman estimate for the underlying Partial Differential Operator. Given a ball B(R) of an arbitrary radius R>0 in an appropriate Hilbert space, one can choose the parameter of the CWF in such a way that the above functional becomes strictly convex on B(R). The strict convexity, in turn, implies convergence to the exact solution of the gradient projection method. It is CRUCIAL that the starting point of this method is an arbitrary point of B(R). Since, in turn R>0 is an arbitrary number, then this is the GLOBAL convergence.
- In 2014 has proved, for the first time, uniqueness theorems for 3-d Inverse Scattering Problems without the phase information. Applications are in imaging of nanostructures and living biological cells. The first publication: M.V. Klibanov, Phaseless inverse scattering problems in three dimensions, SIAM J. Appl. Math., 74, 392-410, 2014.
- In 2015 has developed (jointly with V.G. Romanov), for the first time, reconstruction methods for 3-d Inverse Scattering Problems without the phase information. One of those methods was computationally implemented. The first publication: M.V. Klibanov and V.G. Romanov, Reconstruction procedures for two inverse scattering problems without the phase information, SIAM J. Appl. Math., 76, 178-196, 2016.
- In particular, items 4 and 5 have solved a long standing problem posed by K. Chadan and P.C. Sabatier in in their book Inverse Problems in Quantum Scattering Theory, Springer-Verlag, New York, 1977.
- In 2016 has proposed the first rigorous numerical method for solving ill-posed Cauchy problems for quasilinear PDEs. The method is an an adaptation of the convexification. Previously ill-posed Cauchy problems were solved only for linear PDEs. Publication: M.V. Klibanov, Carleman weight functions for solving ill-posed Cauchy problems for quasilinear PDEs, Inverse Problems, 31, 125007, 2015.
- In 2016 has proposed a new mathematical model for the Black-Scholes equation. It was shown on real market data for 368 randomly chosen stock options that this model indeed helps to be quite profitable. Publication: M.V. Klibanov, A.V. Kuzhuget and K.V. Golubnichiy, An ill-posed problem for the Black–Scholes equation for a profitable forecast of prices of stock options on real market data, Inverse Problems, 32, 015010, 2016.
- In 1991 has introduced, for the first time, the powerful tool of Carleman estimates in the Control Theory via proving the Lipschitz stability property for hyperbolic equations with the Cauchy data at the lateral side of the time cylinder and the book (initial conditions at t=0 are absent in this case). This idea became a common place since then in the Control Theory. Publication: M. V. Klibanov and J. Malinsky, Newton-Kantorovich method for 3-dimensional potential inverse scattering problem and stability for the hyperbolic Cauchy problem with time dependent data, Inverse Problems 7, 577–596, 1991.
B1. M.V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, De Gruyter, 2012.
B2. L. Beilina and M.V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012.
- In 2017, Sobolev Institute of Mathematics has awarded to me Golden Medal. The writing on this medal says “For distinguished impact in mathematics”. This is a life time achievement of myself. Other awardees of this Golden Medal are World distinguished mathematicians. https://inside.uncc.edu/news-features/2017-09-20/mathematician-receives-lifetime-achievement-award
- My paper “Inverse problems and Carleman estimates”, Inverse Problems, 8, 575-596, 1992, is in the list of 30 top cited papers of “Inverse Problems” on the 30st anniversary of this journal, see http://iopscience.iop.org/0266-5611/page/top-30-cited