- 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 - Part 1 (M, 4.5 LP, AUSL)
|Module Number||Module Name||CP (Effort)|
|MAT-65-15A-M-7||Optimization on Manifolds - Part 1||4.5 CP (135 h)|
|CP, Effort||4.5 CP = 135 h|
|Position of the semester||1 Sem. 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.|
- About [MAT-65-15A-K-7]: Title: "Optimization on Manifolds - Part 1"; Presence-Time: 42 h; Self-Study: 93 h
Examination achievement PL1
- Form of examination: oral examination (20-30 Min.)
- Examination Frequency: irregular (by arrangement)
- Examination number: 86352 ("Optimization on Manifolds - Part 1")
Evaluation of grades
The grade of the module examination is also the module grade.
Competencies / intended learning achievements
By completing the given 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 (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)