Topological Derivatives of Shape Functionals. Part III: Second-Order Method and Applications

The framework of asymptotic analysis in singularly perturbed geometrical domains presented in the first part of this series of review papers can be employed to produce two-term asymptotic expansions for a class of shape functionals. In Part II (Novotny et al. in J Optim Theory Appl 180(3):1–30, 2019...

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Veröffentlicht in:	Journal of optimization theory and applications Jg. 181; H. 1; S. 1 - 22
Hauptverfasser:	Novotny, Antonio André, Sokołowski, Jan, Żochowski, Antoni
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.04.2019 Springer Nature B.V Springer Verlag
Schlagworte:	Algorithms Applications of Mathematics Approximation Asymptotic methods Asymptotic series Calculus of Variations and Optimal Control; Optimization Derivatives Domains Engineering Functionals Inverse reconstruction Invited Paper Iterative methods Mathematics Mathematics and Statistics Newton methods Numerical methods Operations Research/Decision Theory Optimization Robustness (mathematics) Shape optimization Theory of Computation Topology optimization 49J20 Applications in inverse problems 35J15 35Q74 49M15 Second-order method 49N45 Topological derivatives
ISSN:	0022-3239, 1573-2878
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Abstract	The framework of asymptotic analysis in singularly perturbed geometrical domains presented in the first part of this series of review papers can be employed to produce two-term asymptotic expansions for a class of shape functionals. In Part II (Novotny et al. in J Optim Theory Appl 180(3):1–30, 2019 ), one-term expansions of functionals are required for algorithms of shape-topological optimization. Such an approach corresponds to the simple gradient method in shape optimization. The Newton method of shape optimization can be replaced, for shape-topology optimization, by two-term expansions of shape functionals. Thus, the resulting approximations are more precise and the associated numerical methods are much more complex compared to one-term expansion topological derivative algorithms. In particular, numerical algorithms associated with first-order topological derivatives of shape functionals have been presented in Part II (Novotny et al. 2019 ), together with an account of their applications currently found in the literature, with emphasis on shape and topology optimization. In this last part of the review, second-order topological derivatives are introduced. Second-order algorithms of shape-topological optimization are used for numerical solution of representative examples of inverse reconstruction problems. The main feature of these algorithms is that the method is non-iterative and thus very robust with respect to noisy data as well as independent of initial guesses.
AbstractList	The framework of asymptotic analysis in singularly perturbed geometrical domains presented in the first part of this series of review papers can be employed to produce two-term asymptotic expansions for a class of shape functionals. In Part II (Novotny et al. in J Optim Theory Appl 180(3):1–30, 2019 ), one-term expansions of functionals are required for algorithms of shape-topological optimization. Such an approach corresponds to the simple gradient method in shape optimization. The Newton method of shape optimization can be replaced, for shape-topology optimization, by two-term expansions of shape functionals. Thus, the resulting approximations are more precise and the associated numerical methods are much more complex compared to one-term expansion topological derivative algorithms. In particular, numerical algorithms associated with first-order topological derivatives of shape functionals have been presented in Part II (Novotny et al. 2019 ), together with an account of their applications currently found in the literature, with emphasis on shape and topology optimization. In this last part of the review, second-order topological derivatives are introduced. Second-order algorithms of shape-topological optimization are used for numerical solution of representative examples of inverse reconstruction problems. The main feature of these algorithms is that the method is non-iterative and thus very robust with respect to noisy data as well as independent of initial guesses. The framework of asymptotic analysis in singularly perturbed geometrical domains presented in the first part of this series of review papers can be employed to produce two-term asymptotic expansions for a class of shape functionals. In Part II (Novotny et al. in J Optim Theory Appl 180(3):1–30, 2019), one-term expansions of functionals are required for algorithms of shape-topological optimization. Such an approach corresponds to the simple gradient method in shape optimization. The Newton method of shape optimization can be replaced, for shape-topology optimization, by two-term expansions of shape functionals. Thus, the resulting approximations are more precise and the associated numerical methods are much more complex compared to one-term expansion topological derivative algorithms. In particular, numerical algorithms associated with first-order topological derivatives of shape functionals have been presented in Part II (Novotny et al. 2019), together with an account of their applications currently found in the literature, with emphasis on shape and topology optimization. In this last part of the review, second-order topological derivatives are introduced. Second-order algorithms of shape-topological optimization are used for numerical solution of representative examples of inverse reconstruction problems. The main feature of these algorithms is that the method is non-iterative and thus very robust with respect to noisy data as well as independent of initial guesses.
Author	Żochowski, Antoni Novotny, Antonio André Sokołowski, Jan
Author_xml	– sequence: 1 givenname: Antonio André surname: Novotny fullname: Novotny, Antonio André organization: Laboratório Nacional de Computação Científica LNCC/MCT, Coordenação de Matemática Aplicada e Computacional – sequence: 2 givenname: Jan orcidid: 0000-0002-3560-7342 surname: Sokołowski fullname: Sokołowski, Jan email: Jan.Sokolowski@univ-lorraine.fr organization: UMR 7502 Laboratoire de Mathématiques, Institut Élie Cartan, Université de Lorraine, Systems Research Institute, Polish Academy of Sciences – sequence: 3 givenname: Antoni surname: Żochowski fullname: Żochowski, Antoni organization: Systems Research Institute, Polish Academy of Sciences
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Snippet	The framework of asymptotic analysis in singularly perturbed geometrical domains presented in the first part of this series of review papers can be employed to...
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SubjectTerms	Algorithms Applications of Mathematics Approximation Asymptotic methods Asymptotic series Calculus of Variations and Optimal Control; Optimization Derivatives Domains Engineering Functionals Inverse reconstruction Invited Paper Iterative methods Mathematics Mathematics and Statistics Newton methods Numerical methods Operations Research/Decision Theory Optimization Robustness (mathematics) Shape optimization Theory of Computation Topology optimization
Title	Topological Derivatives of Shape Functionals. Part III: Second-Order Method and Applications
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