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.
- Available online via Springer link
- 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.
- Available online via Springer link
- 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 | Introduction | 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 |
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| Week 5 | Euler–Lagrange equations Differentiation in Banach spaces Precise derivation of Euler–Lagrange equations | Ev 8.1 Tr, 2.6 Ev 8.2 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 CNQ IK 8.4 |
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| 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 L-curve Quasi-optimality Iterative regularisation Landweber method | EHN 4.3, 4.5 EHN 6.1 |
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| Week 16 | Student presentations | ||
| Week 17 | Summary | Summary |