Lecture plan

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 Tröltzsch's book on optimal control with PDE constraints (you may also use the German version of the same book):

  • Tr … F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, AMS Graduate Studies in Mathematics, v. 112, 2010.

If you need some additional background on PDEs, I suggest to use Evans' book:

  • Ev … L.C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics, v. 19, 2010.

Parts of the lecture are based on other references, though they are not necessarily suited for self-study. Specifically, I have used:

  • Da … B. Dacorogna, Direct Methods in the Calculus of Variations, 2nd Edition, Applied Mathematical Sciences, v. 78, Springer 2008.
  • FL … I. Fonseca, G. Leoni, Modern Methods in the Calculus of Variations: \(L^p\) Spaces, Springer 2007.
  • HPUU … M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, v. 23, Springer, 2009.
  • IK … K. Ito, K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control, SIAM, 2008.
  • CNQ … X. Chen, Z. Nashed, L. Qi … Smoothing methods and semismooth methods for nondifferentiable operator equations, SIAM Numer. Anal. 38(4), pp. 1200-1216, 2000.
  • SGGHL … O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, F. Lenzen, Variational Methods in Imaging, Applied Mathematical Sciences, v. 167, Springer 2009.
Date Topics Notes Reading material / References
Week 2 Introduction
Lebesgue measure
Lebesgue integration
Tue, Jan 12
Fri Jan 15

Ev App E, FL 2
Ev App E, FL 2
Week 3 \(L^p\)-spaces
Regular domains
Weak derivatives
Sobolev spaces
Tue Jan 19
Fri, Jan 22
Tr 2.2.1, FL 2
Tr 2.2.2
Tr 2.2.3, Ev 5.2
Tr 2.2.3, Ev 5.2, 5.5, 5.6
Week 4 Sobolev spaces
Weak solutions of PDEs
Lax-Milgram theorem
Browder-Minty theorem
Tue, Jan 26

Fri, Jan 29
Tr 2.2.3, Ev 5.5, 5.6, 5.8.1
Tr 2.3, Ev 6.2
Tr 2.3, Ev 6.2
Tr 4.2, Ev 9.1
Week 5 Browder-Minty for PDEs
Variational problems
Variation of a functional
Euler–Lagrange equations
Coercivity and lower semi-continuity
Non-existence of minimisers
Weak topology
Tue, Feb 02



Fri, Feb 05
Tr 4.2, Ev 9.1
Ev 8.1
Ev 8.1
Ev 8.1, 8.2
Ev 8.2

Tr 2.4.2
Week 6 Weak compactness
Weak lower semi-continuity
Existence for variational problems
Gâteaux and Fréchet differentiability
Tue, Feb 09

Fri, Feb 12
Tr 2.4.2
Ev 8.2, FL 6.4
Ev 8.2
Tr 2.6
Week 7 Fréchet differentiability
Gradients
Differentiability of superposition operators
Euler–Lagrange equations
Lavrentiev phenomenon
Tue, Feb 16


Fri, Feb 19
Tr 2.6
Tr 2.6, HPUU 2.2.11)
Tr 4.3
Ev 8.2, Da 3.4
Da 4.7
Week 8 Linear–quadratic control problems
Existence and uniqueness
Gradient of the reduced functional
Lagrange multipliers
Adjoint equation
Tue, Feb 23


Fri, Feb 26
Tr 2.5
Tr 2.5
Tr 2.8
Tr 2.10
Tr 2.8, 2.10
Week 9 Revision: elliptic control problems,Parabolic
control Problems, Introduction and derivation
of adjoint equation
Fri, Mar 5
Week 10 Parabolic control Problems
nec. optimality conditions
Introduction to FEniCS
box constraints, proj. gradient method
Tue, Mar 9

Fri Mar 12
Week 11 Rothe's method for parabolic equations

error estimates for elliptic control problems
Tue, Mar 16 - notes

Tue, Mar 16 - slides
Week 12 Non-linear elliptic control problems
Existence of minimisers
Optimality conditions
Tue Mar 23
Fri Mar 26
Tr 4.6
Week 13 Easter holidays
Week 14 No lecture on Tuesday
Optimality conditions
Newton's method in Banach spaces

Fri Apr 09

Tr 4.3
Tr 4.11
Week 15 Newton's method for control problems
Primal-dual active set method
Semi-smooth Newton methods
Tue Apr 13

Fri Apr 16
Tr 4.11
IK 8.4
CNQ
Week 16 Inverse problems
Tikhonov regularisation
Stability and convergence
Project presentations
Tue Apr 20 SGGHL 3.1

SGGHL 3.2
Week 17 Question session
1)
This concerns specifically the convergence of gradient descent with Armijo line search.
2021-04-20, Markus Grasmair