MASTER ''Mathematical Modelling''

Institut Polytechnique de Paris and Sorbonne University

Master Sorbonne University

M2 (second year): course by G. Allaire (2025-2026)

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)

Classes by G. Allaire on November 27, December 4 (morning and afternoon) and 18 (no class on the 11th), January 8, 15, 22 and 29.


Content

  • I - Introduction and motivating examples
  • II - A review of optimization (theory and algorithms)
  • III - A review of PDE's and calculus of variations
  • IV - Parametric optimization of linear elliptic PDE's (theoretical aspects)
  • V - Parametric optimization of linear elliptic PDE's (numerical aspects)
  • VI - Extension to non-linear, parabolic, hyperbolic PDE's
  • VII - Geometric or shape optimization
  • VIII - Introduction to topology optimization (if time permits)

    • 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).