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A Brief Introduction to PDE-Constrained Optimization

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Frontiers in PDE-Constrained Optimization

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 163))

Abstract

In this chapter we give a brief overview of optimization problems with partial differential equation (PDE) constraints, i.e., PDE-constrained optimization (PDECO). We start with three potentially different formulations of a general PDECO problem and focus on the so-called reduced form. We present a derivation of the optimality conditions. Later we discuss the linear and the semilinear quadratic PDECO problems. We conclude with the discretization and the convergence rates for these problems. For illustration, we make a MATLAB code available at

https://bitbucket.org/harbirantil/pde_constrained_opt that solves the semilinear PDECO problem with control constraints.

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Notes

  1. 1.

    The chain rule only requires the outer function to be Hadamard directionally differentiable and the inner function to be Hadamard (Gâteaux) directionally differentiable [79, Proposition 3.6].

  2. 2.

    Note both the choices of spaces for S are motivated by the theory of Nemytskii or superposition operators. Care must to taken to ensure their differentiability [81, Section 4.3].

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Antil, H., Leykekhman, D. (2018). A Brief Introduction to PDE-Constrained Optimization. In: Antil, H., Kouri, D.P., Lacasse, MD., Ridzal, D. (eds) Frontiers in PDE-Constrained Optimization. The IMA Volumes in Mathematics and its Applications, vol 163. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-8636-1_1

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