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On Shape Optimization with Parabolic State Equation

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Trends in PDE Constrained Optimization

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 165))

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Abstract

The present paper intends to summarize the main results of Harbrecht and Tausch (Inverse Probl 27:065013, 2011; SIAM J Sci Comput 35:A104–A121, 2013) on the numerical solution of shape optimization problems for the heat equation. This is carried out by means of a specific problem, namely the reconstruction of a heat source which is located inside the computational domain under consideration from measurements of the heat flux through the boundary. We arrive at a shape optimization problem by tracking the mismatch of the heat flux at the boundary. For this shape functional, the Hadamard representation of the shape gradient is derived by use of the adjoint method. The state and its adjoint equation are expressed as parabolic boundary integral equations and solved using a Nyström discretization and a space-time fast multipole method for the rapid evaluation of thermal potentials. To demonstrate the similarities to shape optimization problems for elliptic state equations, we consider also the related stationary shape optimization problem which involves the Poisson equation. Numerical results are given to illustrate the theoretical findings.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Universität Basel, Rheinsprung 21, 4051, Basel, Switzerland

    Helmut Harbrecht

  2. Department of Mathematics, Southern Methodist University, Dallas, TX, USA

    Johannes Tausch

Authors
  1. Helmut Harbrecht
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  2. Johannes Tausch
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Corresponding author

Correspondence to Helmut Harbrecht .

Editor information

Editors and Affiliations

  1. Department Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

    Günter Leugering

  2. Institut für Dynamik komplexer technisch, Max-Planck-Institut, Magdeburg, Germany

    Peter Benner

  3. Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Dortmund, Germany

    Sebastian Engell

  4. Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Andreas Griewank

  5. Mathematisches Institut, Universität Basel, Basel, Switzerland

    Helmut Harbrecht

  6. Fachbereich Mathematik Optimierung und Approximation, Universität Hamburg, Hamburg, Germany

    Michael Hinze

  7. Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany

    Rolf Rannacher

  8. Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany

    Stefan Ulbrich

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Harbrecht, H., Tausch, J. (2014). On Shape Optimization with Parabolic State Equation. In: Leugering, G., et al. Trends in PDE Constrained Optimization. International Series of Numerical Mathematics, vol 165. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-05083-6_14

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