WIAS Preprint No. 2244, (2016)

Reproducing kernel Hilbert spaces and variable metric algorithms in PDE constrained shape optimisation


  • Eigel, Martin
    ORCID: 0000-0003-2687-4497
  • Sturm, Kevin

2010 Mathematics Subject Classification

  • 35J15 46E22 49Q10 49K20 49K40


  • shape optimization, reproducing kernel Hilbert spaces, gradient method, variable metric, radial kernels




In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing Hilbert spaces for PDE constrained shape optimisation problems. We show that radial kernels provide convenient formulas for the shape gradient that can be efficiently used in numerical simulations. The shape gradients associated with radial kernels depend on a so called smoothing parameter that allows a smoothness adjustment of the shape during the optimisation process. Besides, this smoothing parameter can be used to modify the movement of the shape. The theoretical findings are verified in a number of numerical experiments.

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