Module Handbook

  • Dynamischer Default-Fachbereich geändert auf MAT

Course MAT-65-15A-K-7

Optimization on Manifolds - Part 1 (2V+1U, 4.5 LP, AUSL)

Course Type

SWS Type Course Form CP (Effort) Presence-Time / Self-Study
- K Lecture with exercise classes (V/U) 4.5 CP 93 h
2 V Lecture 28 h
1 U Exercise class (in small groups) 14 h
(2V+1U) 4.5 CP 42 h 93 h

Basedata

SWS 2V+1U
CP, Effort 4.5 CP = 135 h
Position of the semester 1 Sem. irreg.
Level [7] Master (Advanced)
Language [EN] English
Lecturers
Area of study [MAT-SPAS] Analysis and Stochastics
Livecycle-State [AUSL] Phase-out period

Notice

This course was offered for the last time in SS 2018.

Contents

  • 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.

Literature

  • 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.

Materials

Further literature will be announced in the lecture.

References to Course [MAT-65-15A-K-7]

Module Name Context
[MAT-65-15A-M-7] Optimization on Manifolds - Part 1 P: Obligatory 2V+1U, 4.5 LP, AUSL
[MAT-65-15-M-7] Optimization on Manifolds P: Obligatory 2V+1U, 4.5 LP, AUSL