• FU-Students only need to register via Campus Management.
• Non-FU-students are required to register via MyCampus/Whiteboard.

Winter Term 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.	-	-
07.02.	-	-
14.02.	-	-

Literature:

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

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

• I. Reid: Estimation II. Lecture Notes 2 (2001)
• H.W. Sorenson: Least-squares estimation: from Gauss to Kalman (1970)

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“):

• Julier, Uhlmann: Unscented Filtering and Nonlinear Estimation
• Menegaz, et.al.: A Systematization of the Unscented Kalman Filter Theory

Ensemble Kalman filter (approximate distributions by particles):

• Evensen: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophysical Res., Vol.99, NO. C5, pages 10,143-10,162, May 15, 1994.
• Evensen: The Ensemble Kalman Filter: theoretical formulation and practical implementation. Ocean Dynamics 53: 343–367 (2003)
• Houtekamer & Mitchell 1998: Data Assimilation Using an Ensemble Kalman Filter Technique. Monthly Weather Rev., Vol.126: issue 3, p.796-811, (1998)

Large-sample asymptotics of EnKF:

• E. Kwiatkowski and J. Mandel: Convergence of the Square Root Ensemble Kalman Filter in the Large Ensemble Limit. 2015
• F. Le Gland, V. Monbet and V.D. Tran: Large sample asymptotics for the ensemble Kalman filter. Chapter 22 in The Oxford Handbook of Nonlinear Filtering. pp. 598–631 (2011)
• J. Mandel, L. Cobb, and J.D. Beezley: On the convergence of the ensemble Kalman filter. Applications of Mathematics. vol. 56, no. 6. pp. 533–541 (2011)

Particle filters:

• Doucet, Johansen: A Tutorial on Particle Filtering and Smoothing: Fifteen years later
• Doucet, de Freitas, Gordon: An Introduction to Sequential Monte Carlo Methods

Curse of dimensionality for particle filters:

• Snyder, Bengtsson, Bickel, Anderson: Obstacles to High-Dimensional Particle Filtering. Monthly Weather Review, Vol.135: Issue 12, p.4629-4640 (2008)
• van Leeuwen: Nonlinear data assimilation in geosciences: an extremely efficient particle filter. Quarterly J. Royal Meteorological Society, Vol.136, Issue 653, p.1991-1999 (2010)

Local particle filters to beat the curse of dimension:

• Rebeschini, van Handel: Can local particle filters beat the curse of dimensionality? Ann. Appl. Probab. 25(5), p.2809-2866 (2015)

Ensemble Kalman filter for nonlinear systems:

• Kelly, Law, Stuart: Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time. Nonlinearity 27, p.2579 (2014)

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:

• R. Atar, O. Zeitouni - Lyapunov exponents for finite state nonlinear filtering (SIAM Journal on Control and Optimization, 1997)

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

• R. van Handel - Filtering, Stability, and Robustness (PhD thesis, 2007), Chapter 3, only the case \tilde h = h