Lectures and learning material
Here you can find a tentative overview of the lecture. This plan will be continuously updated. The recordings of the online lectures can be found on blackboard.
Generally, the lecture uses both Tröltzsch's book on optimal control with PDE constraints (you may also use the German version of the same book) and the new book by A. Manzoni, A. Quarteroni, S. Salsa:
- Tr … F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, AMS Graduate Studies in Mathematics, v. 112, 2010.
- MQS … A. Manzoni, A. Quarteroni, S. Salsa, Optimal Control of Partial Differential Equations: Analysis, Approximation, and Applications, Springer, 2021. Individual chapters are available at Springer link.
Additional sources we might use:
- HPUU … M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, v. 23, Springer, 2009. Available online via Springer link
- DlR J.C. De los Reyes, Numerical PDE-Constrained Optimization, 2015. Available online via Springer link
If you need some additional background on PDEs, I suggest to have a look at Evans' book or the book by Renardy/Rogers:
- Ev … L.C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics, v. 19, 2010.
- RR … M. Renardy, R.C. Rogers, An Introduction to Partial Differential Equations, Text in Applied Mathematics, Springer, 2005 Springer Link
A nice well-written book treating both theoretical and numerical methods for partial differential equations:
- LT … S. Larsson, V. Thomée, Partial Differential Equations with Numerical Methods, 2003.
Juputer notebooks
All Jupyter notebooks are uploaded to a public github repository, you can either download individual notebooks from there or just clone (and regularly pull) the entire git repository. Below you'll also find individual links to specific notebooks used in the tutorial sessions.
Lecture plan
| Date | Topics | Notes | Notebooks | Reading material | |
|---|---|---|---|---|---|
| Week 2 | Introduction Crash course in measure and integration theory | Chapter_01.pdf, Chapter_02.pdf | Tr 1.2, MQS 1.1, 1.2 HPUU 1.2.1.1, 1.2.2.1–1.2.2.4 | ||
| Week 3 | Cancelled | ||||
| Week 4 | Finite dimensional optimal control Weak solution of PDEs and Lax-Milgram theorem | Lecture_03.pdf , Lecture_04.pdf , FA toolbox | Tr 1.4 TR 2.1-2.3 | ||
| Week 5 | Weak convergence, reduced cost functional Existence of optimal controls | lecture_5.pdf lecture_6.pdf | Tr 2.4, 2.5 | ||
| Week 6 | Frechet and Gateaux differentiability First order optimality conditions | lecture_7.pdf lecture_8.pdf | Tr 2.6 Tr 2.7, 2.8 | ||
| Week 7 | Box constraints and projection formula | lecture_9.pdf | Tr 2.8 | ||
| Week 8 | Crash Course in finite element method, Julia and Gridap | ||||
| Week 9 | Optimal control for elliptic PDEs | ||||
| Week 10 | Parabolic optimal control problems - theory | Solution theory for parabolic eqns Parabolic control problems | no further reference | ||
| Week 11 | Numerical method for parabolic state problems | ||||
| Week 12 | Numerical method for parabolic state problems/Introduction to numerical methods for OCP | ||||
| Week 13 | Numerical methods for OCP | ||||
| Week 14 | Numerical methods for OCP | ||||
| Week 15 | Easter holidays | ||||
| Week 16 | |||||
| Week 17 |
Lecture material
Week 8 - Crash course in FEM Julia and Gridap
- A nice intro to the finite element method can be found as part of the FEniCS course lecture_00_fem_introduction.pdf, explaining how to pass from strong PDEs to a finite-dimensional linear system.
- A rather concise introduction to FEMs including the basic error theory is available in Chapter 5 of [LT].
- Jupyter notebooks for the Julia crash course can be found in the JuliaCrashCourse subfolder of the git repository.
- Jupyter notebooks for the FEM crash course can be found in the FEMCrashCourse subfolder of the git repository.
Week 10 - Parabolic optimal control problems - theory
- Parabolic control problems, see also MQS: 7.1-7.5, Tr: 3.5, 3.6.
- Optimization II project presentation - Optimal dosage planning for Laser thermotherapy Project presentation
Week 11 - 12 - Numerical method for parabolic state problems
- [LT, 8.3] for basic energy/stability estimates for the continuous parabolic problem
- [LT, 10.1] the semi-discretization (in space) of the heat equation
- [LT, 10.1] the analysis of a completely discrete scheme for the heat equation
- heat_equation.ipynb for a how to implement the most basic theta-method based solver for the heat equation in Gridap
- heat_equation_part_II.ipynb for more sophisticated used of time-discretization in Gridap
Keywords
- Energy-based estimates
- semi-discretization in space or time, method of lines, Rothe's method
- Ritz projector, discrete Laplace operator, L2 projection
- Theta method and its relation to implicit/explicit Euler and Crank-Nicolson
- A priori error estimates for the semi-discrete (in space) and fully-discrete problem, convergence rates
Week 12 - 14 Numerical method for OCP
- Introduction to numerical methods for OCS: discretize then optimize (DtO) vs. optimize then discretize (OtP). Example of an advection-dominant example where OtD is not equal DtO. [MQS] 6.1, 6.2.1-6.2.4
- Descent-based methods for unconstrained OCP. [MQS] 6.3, [DlR] 4.1
- Projected descent/gradient methods for (control) constrained OCPs. [MQS] 6.4, [Dlr] 5.3
Week 15 - 16 Project work/Nonlinear problems
Fridays sessions were reserved for project work. During the remaining Monday lectures, we had a glance at nonlinear variationl problems.
- (Closest point) projection on closed convex subsets in Hilbert spaces: We proved existence/uniqueness, basic properties including variational inequality satisfied by the projection mapping, (non-strict) monotonicity and that the projection is non-expansive. [MQS] 5.2.2. Mind the error in [MQS] claiming that the projection is "strict" monotone (it is not!)
- Nonlinear equations of monotone type: coercivity, monotonicity, Browder-Minty theorem. [DlR] 2.3.3, [RR] 10.3