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PDE-Constrained Shape Optimization: Towards Product Shape Spaces and Stochastic Models

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Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

Abstract

Shape optimization models with one or more shapes are considered in this chapter. Of particular interest for applications are problems in which a so-called shape functional is constrained by a partial differential equation (PDE) describing the underlying physics. A connection can be made between a classical view of shape optimization and the differential geometric structure of shape spaces. To handle problems where a shape functional depends on multiple shapes, a theoretical framework is presented, whereby the optimization variable can be represented as a vector of shapes belonging to a product shape space. The multi-shape gradient and multi-shape derivative are defined, which allows for a rigorous justification of a steepest descent method with Armijo backtracking. As long as the shapes as subsets of a hold-all domain do not intersect, solving a single deformation equation is enough to provide descent directions with respect to each shape. Additionally, a framework for handling uncertainties arising from inputs or parameters in the PDE is presented. To handle potentially high-dimensional stochastic spaces, a stochastic gradient method is proposed. A model problem is constructed, demonstrating how uncertainty can be introduced into the problem and the objective can be transformed by use of the expectation. Finally, numerical experiments in the deterministic and stochastic case are devised, which demonstrate the effectiveness of the presented algorithms.

This work has been partly supported by the state of Hamburg within the Landesforschungsförderung under project “Simulation-Based Design Optimization of Dynamic Systems Under Uncertainties” (SENSUS) with project number LFF-GK11, and by the German Academic Exchange Service (DAAD) within the program “Research Grants-Doctoral Programmes in Germany, 2017/18.” ____

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Author information

Authors and Affiliations

  1. Weierstrass Institute, Berlin, Germany

    Caroline Geiersbach

  2. Institute for Analysis and Numerics, University of Münster, Münster, Germany

    Estefania Loayza-Romero

  3. Faculty of Mechanical Engineering and Civil Engineering, Helmut-Schmidt-University/University of the Federal Armed Forces Hamburg, Hamburg, Germany

    Kathrin Welker

Authors
  1. Caroline Geiersbach
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  2. Estefania Loayza-Romero
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  3. Kathrin Welker
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Corresponding author

Correspondence to Caroline Geiersbach .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, The University of Liverpool, Liverpool, UK

    Ke Chen

  2. Dept Appl Math & Theoretical Phys, University of Cambridge, Cambridge, UK

    Carola-Bibiane Schönlieb

  3. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Xue-Cheng Tai

  4. Department of Applied Mathematics & Stat, Johns Hopkins University, Baltimore, MD, USA

    Laurent Younces

Section Editor information

  1. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

    Xuecheng Tai Ph.D

  2. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom

    Carola-Bibiane Schönlieb Ph.D

  3. Department of Mathematical Sciences, University of Liverpool Mathematical Sciences Building, Liverpool, UK

    Ke Chen Ph.D

  4. Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, MD, USA

    Laurent Younes

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Geiersbach, C., Loayza-Romero, E., Welker, K. (2022). PDE-Constrained Shape Optimization: Towards Product Shape Spaces and Stochastic Models. In: Chen, K., Schönlieb, CB., Tai, XC., Younces, L. (eds) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer, Cham. https://doi.org/10.1007/978-3-030-03009-4_120-1

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  • DOI: https://doi.org/10.1007/978-3-030-03009-4_120-1

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