On the nudging approach to continuous data assimilation in the limit of infinite error feedback gain

TL;DR

This paper explores the relationship between synchronization and nudging filters in continuous data assimilation for 2D Navier-Stokes equations, showing that infinite nudging leads to synchronization, supported by theoretical analysis and numerical experiments.

Contribution

It establishes the convergence of the nudging filter to the synchronization filter as the nudging parameter approaches infinity, filling a gap in the understanding of these algorithms.

Findings

Nudging filter converges to the synchronization filter at infinite nudging parameter.

Numerical experiments confirm theoretical convergence results.

An adaptive nudging strategy improves performance over constant nudging.

Abstract

This article studies the intimate relationship between two filtering algorithms for continuous data assimilation, the synchronization filter and the nudging filter, in the paradigmatic context of the two-dimensional (2D) Navier-Stokes equations (NSE) for incompressible fluids. In this setting, the nudging filter can formally be viewed as an affine perturbation of the 2D NSE. Thus, in the degenerate limit of zero nudging parameter, the nudging filter converges to the solution of the 2D NSE. However, when the nudging parameter of the nudging filter is large, the perturbation becomes singular. It is shown that in the singular limit of infinite nudging parameter, the nudging filter converges to the synchronization filter. In establishing this result, the article fills a notable gap in the literature surrounding these algorithms. Numerical experiments are then presented that confirm the theoretical results and probes the issue of selecting a nudging strategy in the presence of observational noise. In this direction, an adaptive nudging strategy is proposed that leverages the insight gained from the relationship between the synchronization filter and the nudging filter that produces measurable improvement over the constant nudging strategy.