Mesh Dependence in PDE-Constrained Optimisation

1. Front Matter

   Pages i-viii

   PDF

2. Introduction to PDE-constrained optimisation

   • Tobias Schwedes, David A. Ham, Simon W. Funke, Matthew D. Piggott

   Pages 1-52

3. Mesh dependence in PDE-constrained optimisation

   • Tobias Schwedes, David A. Ham, Simon W. Funke, Matthew D. Piggott

   Pages 53-78

4. An application: Optimising the layout of tidal turbine arrays

   • Tobias Schwedes, David A. Ham, Simon W. Funke, Matthew D. Piggott

   Pages 79-107

5. Back Matter

   Pages 109-110

   PDF

This book provides an introduction to PDE-constrained optimisation using finite elements and the adjoint approach. The practical impact of the mathematical insights presented here are demonstrated using the realistic scenario of the optimal placement of marine power turbines, thereby illustrating the real-world relevance of best-practice Hilbert space aware approaches to PDE-constrained optimisation problems.

Many optimisation problems that arise in a real-world context are constrained by partial differential equations (PDEs). That is, the system whose configuration is to be optimised follows physical laws given by PDEs. This book describes general Hilbert space formulations of optimisation algorithms, thereby facilitating optimisations whose controls are functions of space. It demonstrates the importance of methods that respect the Hilbert space structure of the problem by analysing the mathematical drawbacks of failing to do so. The approaches considered are illustrated using the optimisation problem arising in tidal array layouts mentioned above.

This book will be useful to readers from engineering, computer science, mathematics and physics backgrounds interested in PDE-constrained optimisation and their real-world applications.

• 49K20, 65K10, 65L60, 68U20, 35L25, 46E30
• gradient-based optimisation
• mesh-independent optimisation
• Riesz's representation
• tidal farm layout
• renewable energy
• discretisation dependent optimisation
• PDE-constraint optimisation
• derivative based methods
• finite elements
• adjoint methods
• partial differential equations

“The book is quite an interesting supplementary read for people starting to work in the field of PDE-constrained optimal control. More experienced researchers in this field may use it as a source of ideas for explaining things while teaching about PDE-constrained optimization.” (Volker H. Schulz, SIAM Review, Vol. 61 (2), 2019)

• Imperial College, London, United Kingdom

  Tobias Schwedes

• Department of Mathematics, Imperial College, London, United Kingdom

  David A. Ham

• Biomedical Computing, Simula Research Laboratory, Oslo, Norway

  Simon W. Funke

• Earth Science & Engineering, Imperial College, London, United Kingdom

  Matthew D. Piggott

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