MASTER ''Mathematical Modelling''

Institut Polytechnique de Paris and Sorbonne University

Master Sorbonne University

M2 (second year): course by G. Allaire (2026-2027)

PDE constrained optimization

The course is concerned with theoretical and numerical aspects of optimization problems under PDE (partial differential equation) type constraints. More precisely, the objective function depends indirectly on the optimization variable through the so-called state function which is the solution of a PDE where the optimization variable appears in the data. Optimization variables can be finite-dimensional parameters, functions or geometric entities (for example, the shape of the domain). The existence of optimal solutions, the calculation of the sensitivity or gradient of the objective function with respect to the optimization variable and the numerical approximation of the solutions will be treated in this course. Applications in optimal control theory, shape optimization, inverse problems, data assimilation, sensitivity analysis and uncertainty quantification will be discussed. The use of the adjoint method to calculate the derivative of the objective function will be presented in details in several contexts. Numerical simulations using the FreeFEM software will illustrate the results presented in this course.

The course does not have strict prerequisites but students must be familiar with basic PDE theory and optimization concepts.

Link to the description of the course on the web site of the Master ''Mathematical Modelling'' at Sorbonne University.

Schedule: Thursday from 13H30 to 16H30 at Ecole Polytechnique (Building 106)

Classes by G. Allaire on November 26, December 3, 10 and 17, January 7, 14, 21 and 28.


Textbook on "PDE Constrained Optimization": hal-05719433 (August 2026).

Content

  • I - Introduction and examples
  • II - A review of optimization (theory and algorithms)
  • III - A review of PDE theory and calculus of variations
  • IV - Parametric optimization of linear elliptic PDE's
  • V - Numerical algorithms for optimization under PDE constraint Some slides for chapter V (2025).
  • VI - Optimization for other types of PDE's

    • Bibliography

  • ALLAIRE G., Optimisation et contrôle, Lecture notes (in French) (2025).
  • ALLAIRE G., Conception optimale de structures, Collection: Mathématiques et Applications (in French), Vol. 58, Springer (2007).
  • NOCEDAL J., WRIGHT S., Numerical optimization, Springer Series in Operations Research and Financial Engineering, New York (2006).
  • BREZIS H., Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York (2011).
  • BONNANS J., Optimisation continue, Mathématiques appliquées pour le Master (in French) / SMAI, Dunod, Paris (2006).
  • EKELAND I., TEMAM R., Convex analysis and variational problems, Classics in Applied Mathematics 28, SIAM, Philadelphia (1999).
  • HECHT F., LANCE G., TRELAT E., PDE-constrained optimization within FreeFEM, SpringerBriefs PDEs Data Sci. Singapore: Springer, 2026. Preprint.


    • Exercises

  • Existence result for the minimization of a convex energy.
  • Existence result for the minimization of a non-convex energy.
  • Study of a parametric optimization problem.
  • Study of a control problem.
  • Study of a coupled parametric optimization problem.
  • Study of a non-linear inverse problem. Some elements of corrections.
  • Study of a coupled elliptic-parabolic optimization problem.

    • For numerical applications some FreeFEM scripts can be found here.