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19208111 Masterseminar Stochastics "Stochastische Filtertheorie"

Winter semester 2021/2022

Lecturer: Prof. Dr. Nicolas Perkowski and Prof. Dr. Péter Koltai


  • Time and place: Monday, 16:00-17:30, online

Prerequisites:  Stochastics I und II.
Target group:  BMS students, Master students and advanced Bachelor students.

Contents:  The goal in filtering is to estimate the current state of a (possibly random) dynamical system given dynamic, noisy measurements, i.e. to filter out the observation noise. We will discuss classical and modern mathematical methods, starting with the Kalman filter that was used in the Apollo missions of the 1960s. In particular, we will put special emphasis on challenges and properties of different filters. Among others, we will treat the following subjects:

  • Kalman filter
  • Extended / ensemble Kalman filter
  • Particle filters & Markov chain Monte Carlo
  • Curse of dimension
  • Robustness of filters
  • Ergodic filters, long-time stability

Talks

Date Speaker Subject
18.10. - -
25.10. - First meeting, discussion of subjects
01.11. - -
08.11. - -
15.11. A. Schröder The Kalman Filter
22.11. J. Bayer The Extended Kalman Filter
29.11. L. Bazahica The Unscented Kalman Filter
06.12. L. Ye Sequential Monte Carlo Methods and Particle Filters
13.12. - -
2022    
03.01. - -
10.01. H. Karanbash The Ensemble Kalman Filter
17.01. - -
24.01. K. Riechers Convergence of the Ensemble Kalman Filter
31.01. W. Xu Continuous time filtering equations
07.02. L. Ye Curse of dimension for particle filters
14.02. N.N. TBA

Literature:

Among others the following. Further literature will be made available during the seminar.

Kalman filter (classical, linear problem, discrete time):

Extended Kalman filter (local linearisation in every step):

  • B.D.O. Anderson and J.B. Moore: Optimal Filtering. Dover Publications (1979)

Unscented Kalman filter (approximate moments by sample points — „sigma points“):

Ensemble Kalman filter (approximate distributions by particles):

Large-sample asymptotics of EnKF:

Particle filters:

Curse of dimensionality for particle filters:

Local particle filters to beat the curse of dimension:

Ensemble Kalman filter for nonlinear systems:

Filtering equations in continuous time:

  • A. Bain, D. Crisan - Fundamentals of Stochastic Filtering (Springer, 2009), Chapter I.3

Continuous time filtering equations with finite-dimensional solutions:

  • A. Bain, D. Crisan - Fundamentals of Stochastic Filtering (Springer, 2009), Chapter I.

Ergodic behavior of the filter, robustness to mis-specification of the initial condition:

Model robustness of the nonlinear filter (continuous time, finite state space):