Abstract
Motivated by the challenge of incorporating data into misspecified and multiscale dynamical models, we study a McKean–Vlasov equation that contains the data stream as a common driving rough path. This setting allows us to prove well-posedness as well as continuity with respect to the driver in an appropriate rough-path topology. The latter property is key in our subsequent development of a robust data assimilation methodology: We establish propagation of chaos for the associated interacting particle system, which in turn is suggestive of a numerical scheme that can be viewed as an extension of the ensemble Kalman filter to a rough-path framework. Finally, we discuss a data-driven method based on subsampling to construct suitable rough path lifts and demonstrate the robustness of our scheme in a number of numerical experiments related to parameter estimation problems in multiscale contexts.
Funding Statement
This research has been partially funded by Deutsche Forschungsgemeinschaft (DFG) through the grants CRC 1114 ‘Scaling Cascades in Complex Systems’ (project number 235221301) and CRC 1294 ‘Data Assimilation’ (project number 318763901). This work was commenced while T.N. was at TU Berlin, supported by DFG Research Unit FOR2402 “Rough paths and SPDEs” and M.C. was at WIAS Berlin, supported by the Einstein Center Berlin, ECMath Project “Stochastic methods for the analysis of lithium-ion batteries”.
Acknowledgments
We would like to thank R. Chew, A. Djurdjevac, G. Hastermann and R. Klein for very stimulating discussions. N.N. would like to thank M. Engel for organising a seminar on homogenisation at Imperial College London in 2016 leading to Section 6.3, as well as for sharing his insight into the topic.
Citation
Michele Coghi. Torstein Nilssen. Nikolas Nüsken. Sebastian Reich. "Rough McKean–Vlasov dynamics for robust ensemble Kalman filtering." Ann. Appl. Probab. 33 (6B) 5693 - 5752, December 2023. https://doi.org/10.1214/23-AAP1957
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