Office: FRET 360B
Office: FRET 360B
Research Interests: Inverse Problems for Partial Differential Equations, Ill-Posed Problems.
Substantial experience in the interdisciplinary research including microwaves and nano science.
Total Publications: 143
Total citations: 1432
My citations on Google Scholar:
390 citations of the first publication on the Bukhgeim-Klibanov method, 1981.
Information from Google Scholar:
|Citation indices||All||Since 2011|
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.
10 Long Standing Important Problems Being Solved:
- 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), see 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 2 is an example of those methods.
- In 1997 has started, for the first time, the development of globally convergent numerical methods for coefficient inverse problems. In 2008 this development got a new start and has been continued up to now. 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? One of our globally convergent methods is completely verified on experimental data. All other numerical methods for coefficient inverse problems 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.
- In 1985 has proved, for the first time, uniqueness of the so-called Phase Problems in Optics. This is the problem of recovering a complex valued function with the compact support from the modulus of its Fourier transform.
- 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.
- 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.
- In particular, items 3 and 4 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 2015 addressed a long standing problem posed in 1930 by one of most distinguished mathematicians of the 20st century Sergey L. Sobolev. This is a problem about the construction of the fundamental solution for a 3-d hyperbolic equation with a variable coefficient in the principal part of the operator. Sobolev in 1930 has constructed this solution assuming the regularity of geodesic lines in the entire space R^3. In my work the regularity is not required. In addition, unlike the solution of Sobolev, my solution can be easily computed.
- In 2016 has proposed the first rigorous numerical method for solving ill-posed Cauchy problems for quasilinear PDEs. The method is based on the construction of a globally strictly convex cost functional. The key element of this functional is the presence of the Carleman Weight Function. Previously ill-posed Cauchy problems were solved only for linear PDEs.
- In 2016 has proposed a new mathematical model for the Black-Scholes equation. It was shown on real market data for 368 stock options that this model indeed helps to be quite profitable.
- 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.
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.
B1. M.V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, The Netherlands, 2004.
B2. L. Beilina and M.V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012.
- Article in Journal of Inverse and Ill-Posed Problems, 19, 533-536, 2011. The title “Michael V. Klibanov”.
- A. Hasanov, Biography of Michael V. Klibanov, Ph.D. and Doctor of Science in physics and mathematics and an outstanding expert in inverse problems, Applicable Analysis, 90, 1453-1459, 2011.
- These two papers are laudations of myself because of my 60st anniversary (2010).
- 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
I was awarded a number of federal grants for the period of 2005-2015. They were awarded for two topics: (1) globally convergent numerical methods and (2) inverse scattering problems without the phase information.