Lecture plan

Here you can find a tentative overview of the lecture. This plan will be continuously updated. My handwritten notes for the lecture are available on blackboard.

  • Tr … F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, AMS Graduate Studies in Mathematics, v. 112, 2010.
  • 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.
  • Ev … L.C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics, v. 19, 2010.
  • SGGHL … O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, F. Lenzen, Variational Methods in Imaging, Applied Mathematical Sciences, v. 167, Springer 2009.
  • EHN … H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Mathematics and Its Applications, v. 375, Kluwer Academic Publishers, 1996.
Date Topics Slides Reading material / References
Week 2 Introduction
Lebesgue measure
Lebesgue integration
Ev App E
Ev App E
Week 3 \(L^p\)-spaces
Integral functionals on \(L^p\)
Weak topology and weak convergence
Tr 2.2
Week 4 Existence of minimisers of integral functionals on \(L^p\)
Sobolev spaces
Calculus of variations

Tr 2.2, Ev 5.2
Ev 8.1, 8.2
Week 5 Euler–Lagrange equations
Differentiation in Banach spaces
Precise derivation of Euler–Lagrange equations
Ev 8.1
Tr, 2.6
Ev 8.2
Week 6 Galerkin methods
Lavrentiev phenomenon
Weak formulation of PDEs
Lax–Milgram Theorem
Browder–Minty theory

Tr 2.3
Tr 2.4, Ev 6.2
Tr 4.2
Week 7 Existence and stability for monotone, quasi-linear PDEs
Introduction to optimal control problems
Existence of optimal controls
Review of finite dimensional constrained optimisation
Tr 4.2

Tr 4.4
Tr 1.4
Week 8 Presentation of projects
Optimality system for elliptic problems
Reduced functional and gradient
Adjoint equation
Lagrangian formulation

Tr 2.8, 2.10
Tr 2.8
Tr 2.8
Tr 2.10
Week 9 Parabolic control problems
Problems with control constraints
Gradient projection method
Tr, Ch. 3.4, 3.5
Tr, Ch. 2.8
Tr, Ch. 2.12
Week 10 Optimal control with non-linear PDEs
Fréchet differentiability of superposition operators
Newton's method in Banach spaces
Newton's method and SQP for optimal control
Tr 4.6
Tr 4.3

Tr 4.11
Week 11 Higher order differentiability conditions
Active set methods
Semi-smooth Newton methods
Relation between active set and semi-smooth Newton methods
Tr 4.3
Tr 4.11
IK 8.4
Week 12 Quarantine
Week 13 Inverse problems
Parameter identification problems for PDEs
Ill-posed problems and regularisation
Tikhonov regularisation
Reconstruction errors
SGGHL 3.1, EHN 3.1, 5.1
Week 14 Parameter choice methods
Discrepancy principle
Iterative regularisation
Landweber method
EHN 4.3, 4.5

EHN 6.1
Week 16 Student presentations
Week 17 Summary Summary
2020-04-20, Markus Grasmair