| Abstract: |
| We consider a high-dimensional time-continuously partially observed linear system of SDEs and the Ensemble Kalman filter (EnKF) to evolve the filtering distribution.
To reduce computational complexity, we propose a dynamical low-rank approximation (DLR-EnKF), where particles evolve in a relatively small, online-computed, time-varying subspace at reduced cost. This allows for a significantly larger ensemble size compared with standard EnKF at equivalent cost, thereby lowering the Monte Carlo error and improving filter accuracy. Some theoretical properties, including a propagation-of-chaos result, will be presented.
We then extend the framework to high-dimensional non-linear dynamical systems for efficient joint state-parameter estimation. We discuss, in particular, a time-integration strategy that combines the Basis Update & Galerkin scheme with forecast/analysis discretisation, and a DEIM-based hyper-reduction technique for efficient evaluation of nonlinear terms. We demonstrate the effectiveness, robustness, and computational advantages of the proposed approach on benchmark problems, including a state/parameter estimation problem for a reduced one-dimensional blood flow model of the human arterial system observed in only three spatial locations. |
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