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 , which will be published in 2021.
. 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.
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