Geometric Aspects of Shape Optimization

We present a review of known results in shape optimization from the point of view of Geometric Analysis. This paper is devoted to the mathematical aspects of the shape optimization theory. We focus on the theory of gradient flows of objective functions and their regularizations. Shape optimization i...

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Vydané v:The Journal of geometric analysis Ročník 33; číslo 7; s. 206
Hlavní autori: Plotnikov, Pavel I., Sokolowski, Jan
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York Springer US 01.07.2023
Springer Nature B.V
Predmet:
Abstract Harmonic Analysis
Boundary conditions
Boundary value problems
Calculus of variations
Convergence
Convex and Discrete Geometry
Differential Geometric PDE Control
Differential Geometry
Dynamical Systems and Ergodic Theory
Electric fields
Electrons
Equality
Fluid flow
Fluid mechanics
Fourier Analysis
Free boundaries
Global Analysis and Analysis on Manifolds
Gradient flow
Identification
Inverse problems
Mathematics
Mathematics and Statistics
Optimization
Partial differential equations
Shape optimization
Shape Optimization and Applications
Tomography
ISSN:1050-6926, 1559-002X
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Popis
Shrnutí:We present a review of known results in shape optimization from the point of view of Geometric Analysis. This paper is devoted to the mathematical aspects of the shape optimization theory. We focus on the theory of gradient flows of objective functions and their regularizations. Shape optimization is a part of calculus of variations which uses the geometry. Shape optimization is also related to the free boundary problems in the theory of Partial Differential Equations. We consider smooth perturbations of geometrical domains in order to develop the shape calculus for the analysis of shape optimization problems. There are many applications of such a framework, in solid and fluid mechanics as well as in the solution of inverse problems. For the sake of simplicity we consider model problems, in principle in two spatial dimensions. However, the methods presented are used as well in three spatial dimensions. We present a result on the convergence of the shape gradient method for a model problem. To our best knowledge it is the first result of convergence in shape optimization. The complete proofs of some results are presented in report (Plotnikov and Sokolowski, Gradient flow for Kohn–Vogelius functional).
Bibliografia:ObjectType-Article-1
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ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-023-01252-7