In a preliminary study, we proved stability and accuracy of the EnKF for fully observed processes and ensemble sizes larger than the dimension of state space. Consider the stochastically perturbed Lorenz-63 system as an example. Its dimension of state space is three. We implemented the EnKF with 4 ensemble members and observed all three components continuously in time with the variance of the Brownian measurement error process of size R=1.0. A numerical demonstration of the filter behavioris shown in Figure 1 above.
The project will advance these results by extending them to
- partially observed processes and ensemble sizes smaller than the dimension of state space and
- to the wider class of second-order accurate ensemble filter methods. Here we will start from fully observed systems and will consider partially observed processes later.
- Long-time stability and accuracy of the ensemble Kalman-Bucy filter for fully observed processes and small measurement noise
Jana de Wiljes, Sebastian Reich and Wilhelm Stannat, University of Potsdam, 2016
- Second-order accurate ensemble transform particle filters
Jana de Wiljes, Walter Acevedo and Sebastian Reich, University of Potsdam, 2016
- Probabilistic Forecasting and Bayesian Data Assimilation
Colin Cotter and Sebastian Reich, Cambridge University Press, 2015
D. Angwenyi, J. de Wiljes and S. Reich (2017). Interacting particle filters for simultaneous state and parameter estimation. arXiv:1709.09199
J. de Wiljes, S. Reich and W. Stannat (2017). Long-time stability and accuracy of the ensemble Kalman-Bucy filter for fully observed processes and smallmeasurement noise. arXiv:1612.06065
M. Morzfeld and S. Reich (2018). Data assimilation: mathematics for merging models and data. Snapshots of modern mathematics from Oberwolfach, 11. doi: 10.14760/SNAP-2018-011-EN
J. de Wiljes, S. Reich and W. Stannat (2018). Long-Time Stability and Accuracy of the Ensemble Kalman--Bucy Filter for Fully Observed Processes and Small Measurement Noise. SIAM Journal on Applied Dynamical Systems, 17, 1152-1181. doi: 10.1137/17M1119056 (free PDF)
J. de Wiljes, W. Acevedo and S. Reich (2017). A second-order accurate ensemble transform particle filter. SIAM Journal on Scientific Computing, 39, A1834-A1850. doi:10.1137/16M1095184 (arXiv:1608.08179)
A. Taghvaei, J. de Wiljes, P. G. Mehta and S. Reich (2017). Kalman filter and its modern extensions for the continuous-time nonlinear filtering problem. ASME Journal of Dynamical Systems, Measurement, and Control, 140(3), 030904. doi:10.1115/1.4037780 (arXiv:1702.07241)