An adaptive hierarchical ensemble Kalman filter with reduced basis models

TL;DR

This work develops adaptive, online reduced-basis enhancements to multi-level and multi-fidelity ensemble Kalman filters for state estimation of nonlinear parabolic PDEs. By alternating model inflation and deflation, and by retraining reduced spaces from limited high-fidelity solves together with past surrogate data, the authors construct surrogate models that are both accurate and computationally cheap. The proposed aRB-ML-EnKF and aRB-MF-EnKF demonstrate competitive reconstruction accuracy with substantially reduced surrogate dimensionality and cost on quasi-geostrophic dynamics, highlighting the benefits of online adaptivity and information reuse. The framework offers a principled way to link surrogate-space accuracy to state uncertainty and shows promise for scalable, real-time data assimilation in complex spatio-temporal systems.

Abstract

The use of model order reduction techniques in combination with ensemble-based methods for estimating the state of systems described by nonlinear partial differential equations has been of great interest in recent years in the data assimilation community. Methods such as the multi-fidelity ensemble Kalman filter (MF-EnKF) and the multi-level ensemble Kalman filter (ML-EnKF) are recognized as state-of-the-art techniques. However, in many cases, the construction of low-fidelity models in an offline stage, before solving the data assimilation problem, prevents them from being both accurate and computationally efficient. In our work, we investigate the use of adaptive reduced basis techniques in which the approximation space is modified online based on the information that is extracted from a limited number of full order solutions and that is carried by the past models. This allows to simultaneously ensure good accuracy and low cost for the employed models and thus improve the performance of the multi-fidelity and multi-level methods.

Figures (8)

• Figure 1: Reference solutions at time $t_0=1000$. On the left: vorticity $\omega(t_0)$ and measurement locations (in white). On the right: stream function $\psi(t_0)$ and corresponding velocity field $\mathbf{u}(t_0)$ (in red).
• Figure 2: This plot compares different data assimilation algorithms. The left column shows the results of filters employing multi-level updates, while the right column shows those using multi-fidelity updates. The top row reports the relative errors versus assimilation time, the center row displays how the reduced basis size changes over the assimilation time, and the bottom row shows the wall clock time per single data assimilation step versus assimilation time.
• Figure 3: This plot compares different sampling priors. The left column shows the results of filters employing multi-level updates, while the right column shows those using multi-fidelity updates. The sub-plots in the top row illustrate relative errors versus assimilation time, those in the center row display the relationship between reduced basis size and assimilation time, and those at bottom the wall clock time per single data assimilation step versus assimilation time.
• Figure 4: This plot compares different tolerances for the surrogate model. The left column shows the results of filters employing multi-level updates, while the right column shows those using multi-fidelity updates. The sub-plots in the top row illustrate relative errors versus assimilation time, those in the center row display the relationship between reduced basis size and assimilation time, and those at bottom the wall clock time per single data assimilation step versus assimilation time.
• Figure 5: This plot compares different data assimilation algorithms: the standard EnKF, the two-level aRB-ML-EnKF, and three RB-EnKF with different accuracy levels and ensemble sizes. In the left column are reported the results of filters employing the inverse Laplacian prior, while in the right column those using the invariant measure prior. The top row reports the relative errors versus assimilation time, the center row displays how the reduced basis size changes over the assimilation time, and the bottom row shows the wall clock time per single data assimilation step versus assimilation time.

Theorems & Definitions (2)

• Remark 1
• Remark 2