A low-rank ensemble Kalman filter for elliptic observations

Abstract

We propose a regularization method for ensemble Kalman filtering (EnKF) with elliptic observation operators. Commonly used EnKF regularization methods suppress state correlations at long distances. For observations described by elliptic partial differential equations, such as the pressure Poisson equation (PPE) in incompressible fluid flows, distance localization should be used cautiously, as we cannot disentangle slowly decaying physical interactions from spurious long-range correlations. This is particularly true for the PPE, in which distant vortex elements couple nonlinearly to induce pressure. Instead, these inverse problems have a low effective dimension: low-dimensional projections of the observations strongly inform a low-dimensional subspace of the state space. We derive a low-rank factorization of the Kalman gain based on the spectrum of the Jacobian of the observation operator. The identified eigenvectors generalize the source and target modes of the multipole expansion, independently of the underlying spatial distribution of the problem. Given rapid spectral decay, inference can be performed in the low-dimensional subspace spanned by the dominant eigenvectors. This low-rank EnKF is assessed on dynamical systems with Poisson observation operators, where we seek to estimate the positions and strengths of point singularities over time from potential or pressure observations. We also comment on the broader applicability of this approach to elliptic inverse problems outside the context of filtering.

Bayesian inference of vorticity in unbounded flow from limited pressure measurements

“…In the latter context – for example, in an ensemble Kalman filter (da Silva & Colonius 2018; Darakananda & Eldredge 2019; Le Provost & Eldredge 2021; Le Provost et al. 2022) – the inference comprises the analysis part of every step, when data are assimilated into the prediction. Some of the challenges and uncertainty of the static inference identified in this paper are overcome with advancing time as the sensors observe the evolving configuration of the flow and the forecast model predicts the state's evolution.…”

“…(2018), Le Provost & Eldredge (2021) and Le Provost et al. (2022), in which pressure measurements were assimilated into the estimate of the state via an ensemble Kalman filter (Evensen 1994). Each step of such sequential data assimilation consists of the same Bayesian inference (or analysis) procedure: we start with an initial guess for the probability distribution (the prior) and seek an improved guess (the posterior).…”

Cluster-based Bayesian approach for noisy and sparse data: application to flow-state estimation

“…In this context, hydrodynamic and aerodynamic systems-owing to their highly nonlinear dynamics-are particularly challenging applications for data-driven approaches [2,3]. For twodimensional flows, simplified physical models were successfully applied to estimate flow states from sparse sensor data [4,5]. Furthermore, a series of recent studies addressed the problem of flow-state reconstruction from a data-driven perspective, demonstrating the capability of datadriven approaches for estimating complex, three-dimensional and turbulent flow states [6][7][8][9][10][11][12][13][14][15][16][17].…”

Physics-informed dynamic mode decomposition

“…Recent successful examples of ML algorithms that have been modified to respect physical principles include neural networks [2][3][4][5][6][7][8][9][10], kernel methods [11,12], deep generative models [13], data assimilation [14,15] and sparse regression [16][17][18][19][20]. These examples demonstrate that incorporating partially known physical principles into ML architectures can increase the accuracy, robustness and generalizability of the resulting models, while simultaneously decreasing the required training data.…”

“…Physical principles—such as conservation laws, symmetries and invariances—can be incorporated into ML algorithms in the form of inductive biases, thereby ensuring that the learned models are constrained to the correct physics. Recent successful examples of ML algorithms that have been modified to respect physical principles include neural networks [2–10], kernel methods [11,12], deep generative models [13], data assimilation [14,15] and sparse regression [16–20]. These examples demonstrate that incorporating partially known physical principles into ML architectures can increase the accuracy, robustness and generalizability of the resulting models, while simultaneously decreasing the required training data.…”