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On Computation of the Shape Hessian of the Cost Functional Without Shape Sensitivity of the State Variable

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Abstract

A framework for calculating the shape Hessian for the domain optimization problem, with a partial differential equation as the constraint, is presented. First and second order approximations of the cost with respect to geometry perturbations are arranged in an efficient manner that allows the computation of the shape derivative and Hessian of the cost without the necessity to involve the shape derivative of the state variable. In doing so, the state and adjoint variables are only required to be Hölder continuous with respect to geometry perturbations.

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Acknowledgments

The second author was supported in part by the Austrian Science Fund Fond zur Forderung der Wissenschaftlichen Forschung (FWF) under grant SFB F32 (SFB “Mathematical Optimization and Applications in Biomedical Sciences”).

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Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, 4040 , Linz, Austria

    H. Kasumba

  2. Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens Universität Graz, Heinrichstraße 36, 8010, Graz, Austria

    K. Kunisch

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  1. H. Kasumba
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  2. K. Kunisch
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Correspondence to H. Kasumba.

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Communicated by Günter Leugering.

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Kasumba, H., Kunisch, K. On Computation of the Shape Hessian of the Cost Functional Without Shape Sensitivity of the State Variable. J Optim Theory Appl 162, 779–804 (2014). https://doi.org/10.1007/s10957-013-0520-4

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  • DOI: https://doi.org/10.1007/s10957-013-0520-4

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