Dynamik auf Mannigfaltigkeit

Die Kunst der Mathematik besteht darin, diesen einen speziellen Fall zu finden, der alle Elemente der Verallgemeinerung enthält.
- David Hilbert

todos

Differential geometry

1. Geodesic equations on a manifold
2. Christoffel symbols as function of metric tensor
3. Metric tensor of some standard manifolds: 2-sphere, Torus, Möbius strip, and Klien bottle
4. Numerical simulation of the geodesics for above manifolds

Probability and statistics on manifolds

1. Maximum likelihood estimation on Riemannian Manifolds
2. Fisher-Rao metric
3. Central limit theorem on Riemannian manifolds
4. Hypothesis testing on Riemannian manifolds
5. Geodesic regression
6. Intrinsic mean and covariance
7. Manifold-valued stochastic processes

Kalman Filter

1. Understand the works of Thiele, Swerling, Wiener, and Stratonovich as historical prospective
2. Original derivation by Rudolf Kalman
3. Existing approaches of Kalman filters on Riemannian manifolds

Stochastic differential equations

1. Wiener process
2. Stochastic integrals
3. Ito calculus
4. Kushner-Stratonovich equation
5. Fokker-Planck equation
6. Brownian motion
7. Numerical sumulations of SDEs
8. Filtering problem in SDEs
9. Stratonovich, Ito-vectos, and Ito-jet projection

Numerical simulations

1. Anderssen, Hoog, Weiss - On the numerical solution of Brownian motion processes (1973)
2. Julier - A New Method for the Nonlinear Transformation of Means and Covariances in Filters and Estimators (2000)
3. Tenne, Singh - The Higher Order Unscented Filter (2003)
4. Hendeby - On Nonlinear Transformations of Stochastic Variables and its Application to Nonlinear Filtering (2007)
5. Lee et al.- Propagation of Uncertainty in Rigid Body Attitude Flows (2007)
6. Lee et al.- Global Symplectic Uncertainty Propagation on SO(3) (2008)
7. Terenaju et al. - Uncertainty Propagation for Nonlinear Dynamic Systems Using Gaussian Mixture Models (2008)
8. Sun, Kumar - Nonlinear bayesian filtering based on Fokker-Planck equation and tensor decomposition (2015)
9. Armstrong, Brigo - Stochastic filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples (2016)
10. Ito, Xiong - Gaussian Filters for Nonlinear Filtering Problems (2000)
11. Higham - An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations (2001)
12. Numerical simulations of multiple non-trivial SDEs