1. Data assimilation - the collaborative research centre SFB1294
  2. Research
  3. Research Area A
  4. A02

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

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 algorithms that circumvent this limitation -
such as the ensemble Kalman filter (EnKF) - have become available. Even more recently, data-
driven forecast models, so called diffusion-based generative models, promise faster forecasts
and larger ensemble sizes. However, their theoretical properties in the context of sequential
data assimilation are poorly understood. In this project, we will investigate theoretically as well
as algorithmically the interplay between generative forecast models and modern data
assimilation algorithms.

  • Jiang, Z, Andreou, M, Reich, S, Chen, N (2026) A Continuous-Time Ensemble Kalman-Bucy Smoother for Causal Inference and Model Discovery arXiv preprint arXiv:2604.25157

  • Abedi, E., Bechtold, F., Rehmeier, M (2026): Non-uniqueness of nonlinear Markov processes in the sense of McKean associated with parabolic PDEs, https://doi.org/10.48550/arXiv.2604.25851

  • Gottwald, G.A., Liu, S., Marzouk, Y., Reich, S. & Tong, X.T. (2025) Localized diffusion models for high dimensional distributions generation arXiv:2505.04417

  • Kim, J. W. and Mehta, P. G. (2024): Arrow of Time in Estimation and Control: Duality Theory Beyond the Linear Gaussian Model. arXiv 2405.07650

  • Kim, J. W., Taghvaei, A., and Mehta, P. G. (2024): Divergence metrics in the study of Markov and hidden Markov processes. arXiv 2404.15779

  • Cherepanov, V., and Ertel, S. W. (2024): Neural Networks-based Random Vortex Methods for Modelling Incompressible Flows. arXiv: 2405.13691

  • Reich, S. (2023): A particle-based Algorithm for Stochastic Optimal Control. arXiv 2311.06906

  • Kim, J.W. and Reich, S. (2023): On forward-backward SDE approaches to continuousßtime minimum variance estimation. arXiv 2304.12727

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

  • Cordero-Encinar, P, Duncan, A.B., Reich, S. & Akyildiz, O.D. (2026) Sampling by averaging: A multiscale approach to score estimationarXiv 2508.15069 to appear in Advances in Neural Information Processing Systems 38 (NeurIPS 2025), 81139-81186.

  • Opper, M. & Reich, S. (2025) Digital Twins: McKean-Pontryagin control for partially observed physical twinsarXiv:2510.00967, published online in Journal of Computational Physics 10.1016/j.jcp.2026.115075

  • Calvello, E., Reich, S. and Stuart A.M. (2025): Ensemble Kalman methods: A mean field approach. Acta Numerica 34, 123-291 doi:10.1017/S0962492924000060

  • Ertel, S. W. (2025). On the mean field theory of Ensemble Kalman filters for SPDEs. SIAM/ASA Journal on Uncertainty Quantification, 13(3), 891-930. doi:10.1137/24M1658954

     

  • Kim, J. W. and Reich, S. (2025): On forward-backward SDE approaches to continuous-time minimum variance estimation. In: Chapron, B., Crisan, D., Holm, D., Mémin, E., Coughlan, J.-L. (eds) Stochastic Transport in Upper Ocean Dynamics III. STUOD 2023. Mathematics of Planet Earth, vol 13. Springer, Cham., pp. 115-136. doi: 10.1007/978-3-031-70660-8

  • Reich, S. (2025): A particle-based Algorithm for Stochastic Optimal Control. In: Chapron, B., Crisan, D., Holm, D., Mémin, E., Coughlan, J.-L. (eds) Stochastic Transport in Upper Ocean Dynamics III. STUOD 2023. Mathematics of Planet Earth, vol 13. Springer, Cham., pp. 243-268. doi: 10.1007/978-3-031-70660-8

  • Kim, J. W., Joshi, A. A., & Mehta, P. G. (2024). Backward map for filter stability analysis. In 2024 IEEE 63rd Conference on Decision and Control (CDC) (pp. 4070-4077). IEEE. doi:10.48550/arXiv.2405.01127 

  • Kim, J. W. and Mehta, P. G. (2024): Variance Decay Property for Filter Stability. IEEE Transactions on Automatic Control, doi: 10.1109/TAC.2024.3413573

  • Ertel, S.E. and Stannat, W. (2024): Analysis of the ensemble Kalman-Bucy filter for correlated observation noise. Ann. Appl. Probab. 34(1B), 1072-1107, doi: 10.1214/23-AAP1985.

  • Pathiraja, S. (2023): L2 convergence of smooth approximations of Stochastic Differential Equations with unbounded coefficients. Stochastic Analysis and Applications, 42, 354-369. doi: 0.1080/07362994.2023.2260863

  • Reich, S. (2024): Data Assimilation: A Dynamic Homotopy-Based Coupling Approach. In: Chapron, B., Crisan, D., Holm, D., Mémin, E., Radomska, A. (eds) Stochastic Transport in Upper Ocean Dynamics II. STUOD 2022. Mathematics of Planet Earth, vol 11. Springer, Cham. doi: 10.1007/978-3-031-40094-0_12

  • Kim, J. W. and Mehta, P. G. (2023): Duality for Nonlinear Filtering II: Optimal Control. IEEE Transactions on Automatic Control. doi: 10.1109/TAC.2023.3279208

  • Kim, J. W. and Mehta, P. G. (2023): Duality for Nonlinear Filtering I: Observability. IEEE Transactions on Automatic Control. doi: 10.1109/TAC.2023.3279206

  • Pathiraja, S., and van Leeuwen, P. J. (2022): Multiplicative non-Gaussian model error estimation in data assimilation. Journal of Advances in Modeling Earth Systems, 14, e2021MS002564. doi: 10.1029/2021MS002564

  • Ruchi, S., Dubinkina, S. and de Wiljes, J. (2021): Fast hybrid tempered ensemble transform filter for Bayesian elliptical problems via Sinkhorn approximation. Nonlinear Processes in Geophysics, 28(1): 23-41. doi: 10.5194/npg-28-23-2021

  • Lange, T. (2021): Derivation of Ensemble Kalman-Bucy Filters with unbounded nonlinear coefficients. Nonlinearity, Vol. 35, 1061. doi: 10.1088/1361-6544/ac4337

  • Pathiraja, S., Reich, S., and Stannat, W. (2021): McKean-Vlasov SDEs in nonlinear filtering. SIAM Journal on Control and Optimization.  doi:10.1137/20M1355197, arXiv 2007.12658

  • Pathiraja, S. and Stannat, W. (2021): Analysis of the feedback particle filter with diffusion map based approximation of the gain. Foundations of Data Science. doi:10.3934/fods.2021023 arXiv:2109.02761

  • Wormell, C.L. and Reich, S. (2021): Spectral convergence of diffusion maps: Improved error bounds and an alternative normalisation. SIAM Journal Numerical Analysis,59, 1687-1734. arXiv 2006.02037; doi:10.1137/30M1344093

  • Lange, T. and Stannat W. (2021): Mean field limit of Ensemble Square Root filters - discrete and continuous time, Foundations of Data Science. doi: 10.3934/fods.2021003

  • Lange, T. and Stannat, W. (2020): On the continuous time limit of the Ensemble Kalman Filter. Mathematics of Computation, 40(327), 233-265. arXiv 1901.05204v1; doi:10.1090/mcom/3588

  • de Wiljes, J. and Tong, X. T (2020): Analysis of a localised nonlinear Ensemble Kalman Bucy Filter with complete and accurate observations. Nonlinearity, 33(9): 4752-4782 [2]  arXiv:1908.10580v3

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

  • Reich, S. (2019): Data assimilation: The Schrödinger perspective. Acta Numerica, 28, 635-711. arXiv:1807.08351; doi: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: 1702.07241; doi: 10.1115/1.4037780