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A New Tendency in Numerical Methods for Coefficient Inverse Problems

Abstract

Since the field of Inverse Problems is an applied one, it is insufficient just to prove some theorems. Rather it is necessary to develop reliable numerical methods. However, conventional numerical methods for Coefficient Inverse Problems (CIPs) are unreliable. The reason is that they are based on the minimization of leas squares cost functionals. These functionals are non convex. Therefore, as a rule, they have many local minima and ravines. Since any minimization procedure can stop at any local minimum, which can be far from the true solution, then these methods are unreliable and unstable. 

In the past several years Klibanov and his research team have successfully developed a radically new and very e¤ective method of solving CIPs. Furthermore, this method is verified on a variety of microwave experimental data. This is the so-called "convexification" method. In the convexification one constructs a globally strictly convex weighted Tikhonov-like functional. Therefore, the problem of local minima is avoided. The key to this functional is the presence in it of the so-called Carleman Weight Function. This is the function which is involved as the weight in the Carleman estimate for the corresponding Partial Di¤erential Operator. 

The convexification will be presented for a broad variety of CIPs. Numerical results will also be presented for both computationally simulated and experimental data. 

Many of these results can be found in the book [1], which will be published in 2021. 

[1]. M.V. Klibanov and J. Li, Inverse Problems and Carleman Estimates: Global Uniqueness, Global Convergence and Experimental Data, De Gruyter, to be published in 2021.


About the speaker

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.