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

Elizabeth Carlson, Aseel Farhat, Vincent R. Martinez, Collin Victor

TL;DR

The paper investigates the relationship between direct-replacement (synchronization) and nudging data assimilation filters for the 2D NSE under idealized observations, proving that the nudging filter converges to the synchronization filter in the limit $\mu\to\infty$ and that the zero-nudging limit recovers the NSE. It develops a rigorous low/high-mode framework and a Lipschitz map to transfer stability from low to high modes, establishing convergence on finite horizons and, for large $N$, globally in time. The zero-nudging limit is also analyzed in parallel, linking nudging to the Bjerknes-style approach. Numerical experiments with both deterministic and noisy observations validate the theory and motivate an adaptive nudging strategy that improves accuracy by balancing convergence speed against noise amplification.

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.

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

TL;DR

The paper investigates the relationship between direct-replacement (synchronization) and nudging data assimilation filters for the 2D NSE under idealized observations, proving that the nudging filter converges to the synchronization filter in the limit and that the zero-nudging limit recovers the NSE. It develops a rigorous low/high-mode framework and a Lipschitz map to transfer stability from low to high modes, establishing convergence on finite horizons and, for large , globally in time. The zero-nudging limit is also analyzed in parallel, linking nudging to the Bjerknes-style approach. Numerical experiments with both deterministic and noisy observations validate the theory and motivate an adaptive nudging strategy that improves accuracy by balancing convergence speed against noise amplification.

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.
Paper Structure (8 sections, 12 theorems, 124 equations, 7 figures, 1 algorithm)

This paper contains 8 sections, 12 theorems, 124 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. The Infinite Nudging Limit of the Nudging Algorithm
  4. Computational Results
  5. Numerical Methods
  6. Convergence of Nudging to Synchronization -- Deterministic Observations
  7. Convergence of Nudging to Synchronization - Stochastic Observations
  8. The Zero-Nudging Limit: Convergence to the Bjerknes Filter

Key Result

Theorem 1.1

Given $f\in L^\infty(0,\infty;H)$ and $u_0\in V$, let $u$ denote the unique solution to the initial value problem corresponding to eq:nse:ff. Suppose that the nudging and direct-replacement algorithms are initialized identically. For any $N>0$ and $T>0$, one has for all $v_0\in V$ such that $P_Nv_0=P_Nu_0$. Moreover, there exists $N_*$ such that for all $N\geq N_*$

Figures (7)

  • Figure 1: Energy spectrum of the initial data with $\nu = 0.0001$, $G = 500,000$, and $\Delta t = 0.001$. The vertical red line is the 2/3 dealiasing cutoff as $\frac{2}{3}\frac{N}{2} = 341.\overline{3}.$
  • Figure 2: Error over time between reference and nudging solutions for different $\mu$ values when lowest 2 modes are observed. Plotted errors probe infinite-$\mu$ limit and display low mode error (left) and high mode error (right) in $L^2$ norm. Note that direct-replacement algorithm ($\mu=\infty$) achieves smallest total error among all values of $\mu$. Coloring corresponds to values $\mu = 10^k$, with $k$ indicated by the color bar; $k= \infty$ and $k= -\infty$ correspond to the direct-replacement and zero-nudging regime, respectively.
  • Figure 3: Error over time between reference and nudging solutions for different $\mu$ values when lowest 100 modes are observed. Nudging algorithm was initialized with low modes of the observed reference solution. Plotted errors probe infinite-$\mu$ limit and display low mode error (left) and high mode error (right) in $L^2$. Coloring corresponds to values $\mu = 10^k$, with $k$ indicated by the color bar; $k= \infty$ and $k= -\infty$ correspond to the direct-replacement and zero-nudging regime, respectively.
  • Figure 4: Error over time between reference and nudging solutions for different $\mu$ values assuming lowest 100 modes observed. Nudging algorithm was initialized with zero initial data. Plotted errors probe infinite-$\mu$ limit and display low mode error (left) and high mode error (right) in $L^2$ norm. Coloring corresponds to values $\mu = 10^k$, with $k$ indicated by the color bar; $k= \infty$ and $k= -\infty$ correspond to the direct-replacement and zero-nudging regime, respectively.
  • Figure 5: Error over time for different $\mu$ values assuming lowest 100 modes observed. Nudging algorithm was initialized with zero velocity. Plotted errors probe for zero-nudging limit and represent a splitting of $\norm{\tilde{u} - \tilde{v}}_{L^2}$, where $\tilde{u}$ is the solution of NSE with zero initial data, between low mode errors (left) and high mode errors (right). Coloring corresponds to values $\mu = 10^k$, with $k$ indicated by the color bar; $k= \infty$ and $k= -\infty$ correspond to the direct-replacement and zero-nudging regime, respectively.
  • ...and 2 more figures

Theorems & Definitions (26)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2: Theorem 3.1 & 3.3, OlsonTiti2003
  • Proposition 2.3: Theorem 1 & 6, AzouaniOlsonTiti2014
  • Theorem 3.1
  • Proposition 3.2
  • Corollary 3.3
  • proof : Proof of \ref{['cor:nudge:complete']}
  • Lemma 3.4
  • proof
  • ...and 16 more