TY  - JOUR
A1  - Pathiraja, Sahani Darschika
A1  - Reich, Sebastian
A1  - Stannat, Wilhelm
T1  - McKean-Vlasov SDEs in nonlinear filtering
JF  - SIAM journal on control and optimization : a publication of the Society for Industrial and Applied Mathematics
N2  - Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows us to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in [D. Crisan and J. Xiong, Stochastics, 82 (2010), pp. 53-68; J. M. Clark and D. Crisan, Probab. Theory Related Fields, 133 (2005), pp. 43-56]. We consider three filters that have been proposed in the literature and use this framework to derive Ito representations of their limiting forms as the approximation parameter delta -> 0. All filters require the solution of a Poisson equation defined on R-d, for which existence and uniqueness of solutions can be a nontrivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.
KW  - data assimilation
KW  - feedback particle filter
KW  - Poincare inequality
KW  - well-posedness
KW  - nonlinear filtering
KW  - McKean-Vlasov
KW  - mean-field equations
Y1  - 2022
U6  - https://doi.org/10.1137/20M1355197
SN  - 0363-0129
SN  - 1095-7138
VL  - 59
IS  - 6
SP  - 4188
EP  - 4215
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - de Wiljes, Jana
A1  - Reich, Sebastian
A1  - Stannat, Wilhelm
T1  - Long-Time stability and accuracy of the ensemble Kalman-Bucy Filter for fully observed processes and small measurement noise
JF  - SIAM Journal on Applied Dynamical Systems
N2  - The ensemble Kalman filter has become a popular data assimilation technique in the geosciences. However, little is known theoretically about its long term stability and accuracy. In this paper, we investigate the behavior of an ensemble Kalman-Bucy filter applied to continuous-time filtering problems. We derive mean field limiting equations as the ensemble size goes to infinity as well as uniform-in-time accuracy and stability results for finite ensemble sizes. The later results require that the process is fully observed and that the measurement noise is small. We also demonstrate that our ensemble Kalman-Bucy filter is consistent with the classic Kalman-Bucy filter for linear systems and Gaussian processes. We finally verify our theoretical findings for the Lorenz-63 system.
KW  - data assimilation
KW  - Kalman Bucy filter
KW  - ensemble Kalman filter
KW  - stability
KW  - accuracy
KW  - asymptotic behavior
Y1  - 2018
U6  - https://doi.org/10.1137/17M1119056
SN  - 1536-0040
VL  - 17
IS  - 2
SP  - 1152
EP  - 1181
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  -