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In this chapter, we consider several more or less classical PDE-constrained optimization problems, written in the framework introduced in Chap. 1 , and we focus on their numerical solution using finite elements in FreeFEM and the method of interior points IpOpt respectively introduced in Sects. 1.2 and 1.2 . Starting with a classical linear quadratic example, we give various ways for solving it, which can serve as templates for the user. We also explain how to use automatic differentiation in FreeFEM. Nonlinear and time-dependent PDEs are also considered to show the great efficiency of FreeFEM to handle the constraints induced by such PDEs. A brief subsection is devoted to showing how to solve optimal shape design problems within the optimal control viewpoint. We finally propose several numerical codes which are available at FreeFEM’s website, https://freefem.org/Optim/ .

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PDE Constrained Optimization with FreeFEM

  • 章节

  • 发表日期：

    2026年

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  • Frédéric Hecht,
  • Gontran Lance,
  • Emmanuel Trélat

摘要

In this chapter, we consider several more or less classical PDE-constrained optimization problems, written in the framework introduced in Chap. 1 , and we focus on their numerical solution using finite elements in FreeFEM and the method of interior points IpOpt respectively introduced in Sects. 1.2 and 1.2 . Starting with a classical linear quadratic example, we give various ways for solving it, which can serve as templates for the user. We also explain how to use automatic differentiation in FreeFEM. Nonlinear and time-dependent PDEs are also considered to show the great efficiency of FreeFEM to handle the constraints induced by such PDEs. A brief subsection is devoted to showing how to solve optimal shape design problems within the optimal control viewpoint. We finally propose several numerical codes which are available at FreeFEM’s website, https://freefem.org/Optim/ .