Descriptif
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 some of 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. Plan of the course (subject to slight changes): 1. Introduction and examples. 2. A review of optimization (theory and algorithms). 3. A review of PDE theory and calculus of variations. 4. Parametric optimization of linear elliptic PDEs (theoretical aspects). 5. Numerical algorithms for optimization under PDE constraint. 6. Optimization for other types of PDEs. Textbook on "PDE Constrained Optimization": [https://hal.archives-ouvertes.fr/hal-05719433](https://hal.archives-ouvertes.fr/hal-05719433 "Textbook on "PDE Constrained Optimization"")
24 heures en présentiel
24 heures de travail personnel estimé pour l’étudiant.
Diplôme(s) concerné(s)
- M2 AMODS - Analyse, Modélisation, Simulation
- Diplôme d'Ingénieur de l'Ecole Nationale Supérieure de Techniques Avancées (Institut polytechnique de Paris)
