- Gateaux and Fréchet differentiability,
- differentiable finite-dimensional manifolds: charts, atlas, tangent vectors and tangent spaces, vector fields, differential of a mapping,
- Riemannian metric, geodesics, Christoffel symbols, exponential map and logarithmic map,
- Hopf-Rinow theorem,
- first-order optimization, descent methods,
- examples of manifolds: spheres, hyperbolic spaces, positive definite matrices, probability simplex, Grassmann manifolds, Stiefel manifolds, special Euclidean group, rotation group,
- linear connection and parallel transport.
Optimization on Manifolds (M, 9.0 LP, AUSL)
|Module Number||Module Name||CP (Effort)|
|MAT-65-15-M-7||Optimization on Manifolds||9.0 CP (270 h)|
|CP, Effort||9.0 CP = 270 h|
|Position of the semester||2 Sem. from irreg.|
|Level|| Master (Advanced)|
+ further Lecturers of the department Mathematics
|Area of study||[MAT-SPAS] Analysis and Stochastics|
|Reference course of study||[MAT-88.105-SG] M.Sc. Mathematics|
|Livecycle-State||[AUSL] Phase-out period|
|Type/SWS||Course Number||Title||Choice in |
|SL||SL is |
required for exa.
Optimization on Manifolds - Part 1
|P||42 h||93 h||-||-||PL1||4.5||irreg.|
Optimization on Manifolds - Part 2
|P||28 h||107 h||-||-||PL1||4.5||irreg.|
Examination achievement PL1
- Form of examination: oral examination (20-30 Min.)
- Examination Frequency: irregular (by arrangement)
- Examination number: 86351 ("Optimization on Manifolds")
Evaluation of grades
The grade of the module examination is also the module grade.
- Levi-Civita connection,
- second-order optimization methods, Newton-like methods,
- Jacobi fields,
- Hadamard manifolds: introduction, proximal point methods, optimization methods from convex analysis,
• (possible outlooks: Lie groups and Lie algebras, integration on manifolds).
Competencies / intended learning achievements
They understand the mathematical background of the algorithms and are able to analyze them and to apply them to concrete questions. They understand the proofs presented in the lecture and are able to comprehend and explain them. In particular, they are able to outline the conditions and assumptions that are necessary for the validity of the statements. By completing exercises, the students have developed a skilled, precise and independent handling of the terms, propositions and techniques taught in the lecture. Moreover, they have performed implementations for the application of the algorithms in image processing.
- P.-A. Absil, R. Mahony, R. Sepulchre: Optimization Algorithms on Matrix Manifolds,
- J. Jost: Riemannian Geometry and Geometric Analysis,
- J. M. Lee: Introduction to Smooth Manifolds,
- S. Helgason: Differential Geometry, Lie Groups and Symmetric Spaces,
- D. Gromoll, W. Klingenberg, and W. Meyer: Riemannsche Geometrie im Großen.
Requirements for attendance of the module (informal)
- [MAT-10-1-M-2] Fundamentals of Mathematics (M, 28.0 LP)
- [MAT-14-11-M-3] Introduction to Numerical Methods (M, 9.0 LP)
- [MAT-12-23-K-3] Introduction to Functional Analysis (2V+1U, 4.5 LP)
- [MAT-12-27-K-3] Vector Analysis (2V+1U, 4.5 LP)