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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.

Mathematical optimization. --- Differential equations, Partial. --- Partial differential equations --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematics. --- Environmental sciences. --- Partial differential equations. --- Computer mathematics. --- Calculus of variations. --- Mathematics of Planet Earth. --- Environmental Science and Engineering. --- Partial Differential Equations. --- Continuous Optimization. --- Calculus of Variations and Optimal Control; Optimization. --- Computational Science and Engineering. --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Differential equations, partial. --- Computer science. --- Informatics --- Science --- Math --- Computer mathematics --- Electronic data processing --- Mathematics --- Isoperimetrical problems --- Variations, Calculus of --- Environmental science --- Computer science