Optimization of Complex Systems WiSe 2019/20
The modeling of complex phenomena arising in engineering, life sciences, and other disciplines of science usually leads to partial differential equations (PDEs). Often one wants to achieve a desired behavior of the solutions of these PDEs which can be achieved by designing appropriate control functions in order to manipulate the solutions. This leads to optimization problems with PDE constraints. In this course we will discuss the solution of such infinite-dimensional optimization problems. We will derive necessary and sufficient optimality conditions and discuss efficient discretization techniques as well as the numerical solution of the arising large-scale linear systems of equations.
This course is divided into two parts in which the following topics will be discussed (preliminary plan):
Part 1 (first half of the semester)
- examples of PDE constrained optimization problems
- recap of constrained optimization techniques in finite dimensions
- weak solutions of elliptic PDEs
- existence of optimal solutions
- necessary and sufficient optimality conditions
Part 2 (second half of the semester)
- discretization of optimal control problems (finite element methods)
- numerical solution of the high-dimensional discretized problems (such as the MINRES method for saddle-point systems and preconditioning)
- optimal control of parabolic PDEs, solution of linear systems with tensor product structure
- optimal control of semi-linear PDEs (optional)
- basic courses in linear algebra and calculus
Attendents with the recommended prequisites will have an advantage in following the course, but all necessary concepts in optimization and functional analysis will be introduced in the lecture. Participants without any knowledge in optimization or functional analysis will have the chance to follow the lecture and are highly welcome!
- basic course in optimization
- basic knowledge of functional analysis
- numerical analysis/numerical linear algebra
- Lecture: Wednesday, 2:15-3:45pm in Geom H6 and Thursday, 12:15-1:45pm in Geom H6
- Exercise: Monday, 2:15-3:45pm in Geom 432
I will not prepare own lecture notes for this course, but there is a bunch of very good books and lecture notes that I will also use for my own preparation:
- F. Tröltzsch. Optimale Steuerung partieller Differentialgleichungen: Theorie, Verfahren und Anwendungen. Vieweg+Teubner, Wiesbaden, 2nd edition, 2009. ISBN 978-3-8348-0885-1.
- F. Tröltzsch. Optimal Control of Partial Differential Equations: Theory, Methods and Applications, volume 112 of Grad. Stud. Math. American Mathematical Society, Providence, RI, 2010. ISBN 978-0-8218-4904-0.
- J. C. De los Reyes. Numerical PDE-Constrained Optimization. Springer Briefs Opt., Springer International Publishing, 2015. ISBN 978-3-319-13394-2.
- Lecture notes on optimal control of PDEs by Christian Meyer (in German).
Depending on whether you decide to take only the first part or both parts together, the final oral exam will have a different duration:
Exams can be taken in English or German (your choice). The exact dates and place will be announced later.
- part 1 only: 30 minutes,
- part 1 and 2: 40 minutes.
Every student should at least present one solution on the blackboard.
- 1st exercise sheet, discussion on October 28.
- 2nd exercise sheet, discussion on November 4.
- 3rd exercise sheet, discussion on November 18. (Note the correction in Problem 5.)
- 4th exercise sheet, discussion on November 25. (Note two corrected typos.)
- 5th exercise sheet, discussion on December 2.
- 6th exercise sheet, discussion on December 9.
- 7th exercise sheet, discussion on December 16.
- 8th exercise sheet, discussion on January 6.
- 9th exercise sheet, discussion on January 20.
- 10th exercise sheet, discussion on January 27.