A02 – Long-time stability and accuracy of ensemble transform filter algorithms

This project is concerned with the sequential estimation of the states and parameters of stochastic processes from partial and noisy observations. We will investigate existing sequential filter algorithms, develop new ones and study their theoretical properties in particular in the setting of small Monte Carlo sample sizes.

More specifically, sequential Monte Carlo (SMC) methods provide a standard tool for sequential state and parameter estimation. However, SMC methods are applicable only to low dimensional problems in practice. Recently, other sequential filter algorithms have become available, such as the ensemble Kalman filter (EnKF), which circumvent this limitation; but their theoretical properties are poorly understood. In this project, we will study the long-time stability and accuracy of the EnKF and related sequential ensemble transform filter algorithms.

Figure 1. Simulation results for the Lorenz-63 model. The reference trajectory is plotted in blue and its estimate from noisy continuous-time measurements using the ensemble Kalman-Bucy filter in red. It can be seen that the ensemble Kalman-Bucy filter, implemented with 4 ensemble members, is able to successfully track the reference solution of this chaotic dynamical system.

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

  1. partially observed processes and ensemble sizes smaller than the dimension of state space and
  2. 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.


  1. 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

  2. Second-order accurate ensemble transform particle filters

    Jana de Wiljes, Walter Acevedo and Sebastian Reich,  University of Potsdam, 2016

  3. Probabilistic Forecasting and Bayesian Data Assimilation

    Colin Cotter and Sebastian Reich, Cambridge University Press, 2015

  • Wormell, C.L. and Reich, S. (2020): Spectral convergence of diffusion maps: Improved error bounds and an alternative normalisation. arXiv 2006.02037

  • Lange, T. and Stannat, W. (2019). On the continuous time limit of Ensemble Square Root FiltersarXiv 1910.12493

  • de Wiljes, J. and Tong, X. T. (2019). Analysis of a localised nonlinear Ensemble Kalman Bucy Filter with complete and accurate observations. arXiv:1908.10580

  • Lange, T. and Stannat, W. (2019). On the continuous time limit of the Ensemble Kalman Filter. arXiv: 1901.05204v1

  • Angwenyi, D., de Wiljes, J. and Reich, S. (2017). Interacting particle filters for simultaneous state and parameter estimation. arXiv:1709.09199

  • de Wiljes, J., Pathiraja, S. and Reich, S. (2020). Ensemble transform algorithms for nonlinear smoothing problems. SIAM J. Scientific Computing, 42, A87-A114. arXiv:1901.06300doi: 10.1137/19M1239544

  • Pathiraja, S. and Reich, S. (2019). Discrete gradients for computational Bayesian inference. Journal of Computational Dynamics, 6, 385-400. arXiv:1901.06300doi: 10.3934/jcd.2019019

  • Reich, S. (2019). Data assimilation: The Schrödinger perspective. Acta Numerica, 28, 635-711. arXiv:1807.08351doi:10.1017/S0962492919000011

  • Leeuwen, P. J. v., Künsch, H.-R., Nerger, L., Potthast, R. and Reich, S. (2019). Particle filters for high-dimensional geoscience applications: a review. Quarterly J Royal Meteorlog. Soc., 145, 2335-2365. arXiv: 1807.10434v2 doi: 10.1002/qj.3551

  • Morzfeld, M. and Reich, S. (2018). Data assimilation: mathematics for merging models and data. Snapshots of modern mathematics from Oberwolfach, 11. doi: 10.14760/SNAP-2018-011-EN

  • de Wiljes, J., Reich, S. and Stannat, W. (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(2), 1152-1181. arXiv: 1612.06065; doi: 10.1137/17M1119056

  • Taghvaei, A., de Wiljes, J., Mehta, P. G. and Reich, S. (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. arXiv: 1702.07241doi: 10.1115/1.4037780