Perspectives on locally weighted ensemble Kalman methods

TL;DR

The paper presents locally weighted ensemble Kalman methods (lwEKI and lwEnSRF) as a framework to fuse the strengths of ensemble-based smoothing and particle-filter-like exploration with local, kernel-based conditioning. It introduces a local derivative operator $D_\kappa A$ and related mean-field analogues to enable pointwise, locally linear or quadratic approximations of nonlinear forward maps from ensemble evaluations. The key contribution is showing that locally weighted EK methods yield a preconditioned gradient flow that generalizes the classical EKI, preserves ensemble cohesion, and improves performance on nonlinear and multimodal problems; numerical experiments on Himmelblau, 2d/10d problems, and discontinuous maps illustrate improved exploration and inversion capabilities, at the cost of higher computation. The work highlights practical considerations and potential enhancements, including low-rank kernel structures, clustering, graph-based computation, adaptivity of kernel bandwidth, and potential Metropolis corrections to combine the benefits of particle filtering with ensemble Kalman methods for nonlinear/inverse problems.

Abstract

This manuscript derives locally weighted ensemble Kalman methods from the point of view of ensemble-based function approximation. This is done by using pointwise evaluations to build up a local linear or quadratic approximation of a function, tapering off the effect of distant particles via local weighting. This introduces a candidate method (the locally weighted Ensemble Kalman method for inversion) with the motivation of combining some of the strengths of the particle filter (ability to cope with nonlinear maps and non-Gaussian distributions) and the Ensemble Kalman filter (no filter degeneracy). We provide some numerical evidence for the accuracy of locally weighted ensemble methods, both in terms of approximation and inversion.

Figures (10)

• Figure 1: Illustration of Localisation (top) vs local weighting (bottom). Localisation forces decorrelation of components variables $x_i$ in a given dataset where these variables correspond to fixed geographical (or spatial) locations. Local weighting tracks the (changing) spatial configuration of dynamic particles, with ensemble-based effects (such as contraction to a joint mean) being mediated by the particles' respective distance (or a different kernel).
• Figure 2: Local frame-based approximation with (Gaussian) kernel bandwidths $r\in\{5,1,0.2\}$ (top to bottom) for the sine function, with an ensemble of size $J=100$ uniformly sampled on the interval $[-3,3]$.
• Figure 3: Local frame-based approximation with (Gaussian) kernel bandwidth $r=1$. Ensemble size $J=100$. Top left: Contour plot of the Himmelblau function. Green square marker: Reference point $x$. Black markers: Ensemble members (with marker size proportional to weight $\kappa^{(i)}(x)$). Orange star: locally weighted ensemble mean $\mu^\kappa(x)$. Top right: Contour plot of (exact) second-order Taylor approximation centered at $\mu^\kappa(x)$. Bottom left: Approximate first order approximation $A_x$. Bottom right: Approximate second-order approximation $A_x^{(2)}$.
• Figure 4: Local frame-based approximation with (Gaussian) kernel bandwidth $r=5$. Ensemble size $J=100$. Top left: Contour plot of the Himmelblau function. Green square marker: Reference point $x$. Black markers: Ensemble members (with marker size proportional to weight $\kappa^{(i)}(x)$), identical ensemble as in figure \ref{['fig:approx_himmelblaur1']}. Orange star: locally weighted ensemble mean $\mu^\kappa(x)$. Top right: Contour plot of (exact) second-order Taylor approximation centered at $\mu^\kappa(x)$. Note that this is different from the top right plot in figure \ref{['fig:approx_himmelblaur1']} because $\mu^\kappa(x)$ depends on the kernel and its bandwidth. Bottom left: Approximate first order approximation $A_x$. Bottom right: Approximate second-order approximation $A_x^{(2)}$.
• Figure 5: locally weighted EKI for the Himmelblau benchmark problem. Misfit functional $\Phi$ is shown as contour plot, with evolution of ensemble shown for four time steps (in order: top left, top right, bottom left, bottom right). Inversion of $Au = y$ amounts to minimisation of $\Phi$. Ensemble eventually collapses to all four global minima (not shown).

Theorems & Definitions (21)

• Definition 1
• Lemma 1: Reconstruction formula for finite frames, [casazza2013introduction]
• Lemma 2
• proof
• Definition 2
• Lemma 3
• proof
• Definition 3
• Lemma 4
• proof