Gradient Flows in Shape Optimization Theory

The problem of identifying an inclusion is considered. The inclusion is an unknown subdomain of a given physical region. Available information on the inclusion is given by measurements on the boundary of this region. This class of problems includes single-measurement electrical impedance tomography...

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Bibliographic Details
Published inDoklady. Mathematics Vol. 108; no. 2; pp. 387 - 391
Main Authors Plotnikov, P. I., Sokolowski, J.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.10.2023
Springer Nature B.V
Subjects
Electrical impedance
Existence theorems
Gradient flow
Inverse problems
Mathematics
Mathematics and Statistics
Regularization
Shape optimization
Euler elastica
shape optimization
Willmore flow
inverse problems
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Summary:The problem of identifying an inclusion is considered. The inclusion is an unknown subdomain of a given physical region. Available information on the inclusion is given by measurements on the boundary of this region. This class of problems includes single-measurement electrical impedance tomography and other inverse problems. The shape identification problem can be solved by minimizing an objective function characterizing the deviation of a given configuration from an admissible solution of the problem. The best choice of such an objective function is the Kohn–Vogelius energy functional. The standard regularization of the Kohn–Vogelius functional is considered, which is obtained by adding to the functional a linear combination of the perimeter of the inclusion and the Willmore curvature functional evaluated for an admissible inclusion boundary. In the two-dimensional case, a nonlocal theorem on the existence of strong solutions is proved for the gradient flow dynamical system generated for such a regularization of the Kohn–Vogelius functional.
ISSN:1064-5624
1531-8362
DOI:10.1134/S1064562423700990