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Carleman Estimates for Globally Convergent Numerical Method for Coefficient Inverse Problems

ABSTRACT


The powerful tool of Carleman estimates was first introduced in the field of Inverse Problems in the work of A.L. Bukhgeim and M.V. Klibanov, "Uniqueness in the large of a class of multidimensional inverse problems" Soviet Mathematics. Doklady, 17, 244-247, 1981. Currently this publication has more than 500 citations, which is quite rare for a mathematical paper. Initially this idea was used only for proofs of uniqueness and stability theorems for Coefficient Inverse Problems. In fact, it still remains the unique tool for such proofs for a very broad class of Coefficient Inverse Problems with non overdetermined data: other tools are unknown. More recently, however, Klibanov with coauthors has actively started to use that idea for constructions of the so-called convexification numerical method for a broad class of Coefficient Inverse Problems. Global convergence is guaranteed and numerical results confirm this. So, in this talk we will present the convexification for many problems.



BRIEF BIOGRAPHY

Michael Victor Klibanov has graduated from Novosibirsk State University (NSU), Novosibirsk, Russia, in 1972. NSU is one of very top Russian universities. He got MS in Mathematics. In  1977 he got PhD in Mathematics from Urals State University, Yekaterinburg, Russia. In 1986 he got the highest scientific degree, Doctor of Science in Mathematics from Computing Center of the Siberian Branch of Russian Academy of Science, Novosibirsk. Through his entire career Klibanov works solely on inverse problems. The paper of A.L. Bukhgeim and M.V. Klibanov, "Uniqueness in the large of a class of multidimensional inverse problems" Soviet Mathematics. Doklady, 17, 244-247, 1981.  became one of very few classical papers in the field of Inverse Problems. In this paper the powerful tool of Carleman estimates was introduced in the field for the first time. While previously Klibanov has worked only on the uniqueness issue, currently he develops globally convergent numerical methods for Coefficient Inverse Problems without overdetermination.
He has published total 170 papers and his works were cited 2406 times. The latter is a very high number for a mathematician. Since 1990 Klibanov is with University of North Carolina at Charlotte, USA.