Dynamical Low-Rank Ensemble Kalman filter for State/Parameter estimation

TL;DR

This work advances joint state–parameter estimation for high-dimensional dynamical systems by introducing the Dynamical Low-Rank Ensemble Kalman Filter (DLR-ENKF), which evolves a time-adaptive low-rank subspace for the state while updating augmented parameters via EnKF-like rules. The method combines a robust forecast/analysis time-discretisation (inspired by BUG) with DEIM/CUR hyper-reduction to efficiently evaluate nonlinear terms in the augmented dynamics. Numerical tests on Fisher–KPP and a 1D arterial-blood-flow model show that, for moderate ranks, DLR-Enkf achieves accuracy close to full-order EnKF at substantially reduced cost, with rank choice and potential adaptivity critically influencing covariance representation and parameter identifiability. The approach offers a practical, scalable pathway for online data assimilation in large-scale PDE systems and highlights avenues for further efficiency gains through rank adaptation and optimized hyper-reduction.

Abstract

We propose a Dynamical Low-Rank Ensemble Kalman Filter (DLR-ENKF) for efficient joint state-parameter estimation in high-dimensional dynamical systems. The method extends the DLR-ENKF formulation of arXiv:2509.11210 to the augmented state-parameter framework, tracking the filtering density within a dynamically evolving low-dimensional subspace. Key developments include 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. The results highlight the potential of dynamically evolving reduced bases to achieve accurate filtering and parameter estimation at reduced computational cost.

Figures (14)

• Figure 1: Modes of the diffusion coefficient $\nu(x)$
• Figure 2: True and identified diffusion coefficient $\nu(x, {\color{black} \theta})$ at final time (full observations)
• Figure 3: Full Order Model Senkf parameter identification with $P = 200$ particles (full observations)
• Figure 4: Dlr-Senkf ($R = 7$) parameter identification with $P = 200$ particles (full observations)
• Figure 5: Parameter $\theta_1$ for partial observations ($P = 200$); (a)--(c) Dlr-Senkf (d) Senkf

Theorems & Definitions (4)

• Remark 2.1
• Remark 3.1
• Remark 3.2
• Remark 3.3