© 2017

Mesh Dependence in PDE-Constrained Optimisation

An Application in Tidal Turbine Array Layouts

  • Provides an introduction to PDE-constraint optimisation, including adjoint methods and optimisation on Hilbert spaces

  • Applies mathematical insights to example problems, for instance to optimisation problems arising from tidal array layouts

  • Provides numerical solutions and implementation code to these problems


Part of the Mathematics of Planet Earth book series (MPE)

Also part of the SpringerBriefs in Mathematics of Planet Earth book sub series (SBMPE-WCO)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Tobias Schwedes, David A. Ham, Simon W. Funke, Matthew D. Piggott
    Pages 1-52
  3. Tobias Schwedes, David A. Ham, Simon W. Funke, Matthew D. Piggott
    Pages 53-78
  4. Tobias Schwedes, David A. Ham, Simon W. Funke, Matthew D. Piggott
    Pages 79-107
  5. Back Matter
    Pages 109-110

About this book


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

Authors and affiliations

  1. 1.Imperial CollegeLondonUnited Kingdom
  2. 2.Department of MathematicsImperial CollegeLondonUnited Kingdom
  3. 3.Biomedical ComputingSimula Research LaboratoryOsloNorway
  4. 4.Earth Science & EngineeringImperial CollegeLondonUnited Kingdom

Bibliographic information


“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)