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A Relaxed Direct-insertion Downscaling Method For Discrete-in-time Data Assimilation

Emine Celik, Eric Olson

TL;DR

This work addresses the challenge that increasing observation frequency can degrade synchronization in spectrally-filtered discrete-in-time data assimilation for dissipative systems. It introduces a relaxation parameter $κ$ tied to the observation gap via $κ = μ δ$, stabilizing updates and enabling convergence across a wide range of $δ$, with a clear connection to continuous nudging in the limit. Numerically, the optimal $κ$ scales linearly with $δ$ for $δ \le 0.5$ (approximately $κ_{\min} ≈ μ δ$ with $μ ≈ 0.966$) across an ensemble for the 2D Navier–Stokes equations. Theoretically, the paper proves that the $κ$-relaxed discrete-in-time method converges to the continuous nudging solution on any finite interval as $δ \to 0$, and there exist parameter choices ensuring $\|u(T)-U(T)\|$ can be made arbitrarily small for large enough $T$ and small enough $δ$, establishing a rigorous bridge between discrete and continuous data assimilation approaches.

Abstract

This paper improves the spectrally-filtered direct-insertion downscaling method for discrete-in-time data assimilation by introducing a relaxation parameter that overcomes a constraint on the observation frequency. Numerical simulations demonstrate that taking the relaxation parameter proportional to the time between observations allows one to vary the observation frequency over a wide range while maintaining convergence of the approximating solution to the reference solution. Under the same assumptions we analytically prove that taking the observation frequency to infinity results in the continuous-in-time nudging method.

A Relaxed Direct-insertion Downscaling Method For Discrete-in-time Data Assimilation

TL;DR

This work addresses the challenge that increasing observation frequency can degrade synchronization in spectrally-filtered discrete-in-time data assimilation for dissipative systems. It introduces a relaxation parameter tied to the observation gap via , stabilizing updates and enabling convergence across a wide range of , with a clear connection to continuous nudging in the limit. Numerically, the optimal scales linearly with for (approximately with ) across an ensemble for the 2D Navier–Stokes equations. Theoretically, the paper proves that the -relaxed discrete-in-time method converges to the continuous nudging solution on any finite interval as , and there exist parameter choices ensuring can be made arbitrarily small for large enough and small enough , establishing a rigorous bridge between discrete and continuous data assimilation approaches.

Abstract

This paper improves the spectrally-filtered direct-insertion downscaling method for discrete-in-time data assimilation by introducing a relaxation parameter that overcomes a constraint on the observation frequency. Numerical simulations demonstrate that taking the relaxation parameter proportional to the time between observations allows one to vary the observation frequency over a wide range while maintaining convergence of the approximating solution to the reference solution. Under the same assumptions we analytically prove that taking the observation frequency to infinity results in the continuous-in-time nudging method.
Paper Structure (5 sections, 16 theorems, 196 equations, 5 figures)

This paper contains 5 sections, 16 theorems, 196 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Numerical Results
  3. Theoretical Results
  4. Concluding Remarks
  5. Appendix

Key Result

Theorem 1.1

Let $U$ be a solution to nse1 with $f\in V$ and initial condition $U_0\in {\cal D}(A)$. There exist bounds $\rho_\alpha$ and $\widetilde{\rho}_\alpha$ depending only on $\delta_{\rm max}$, $\nu$, $\|f\|$ and $U_0$ such that and Furthermore, nse1 possesses a unique global attractor ${\cal A}$ bounded in ${\cal D}(A)$. Moreover, if $U$ lies on that global attractor, then the constants $\rho_\alpha

Figures (5)

  • Figure 1: The error $|U(t)-u(t)|$ for a reference solution $U(t)$ where the approximating solution $u(t)$ was computed for different observation intervals $\delta$. Large $\delta$ is shown on the left; small on the right.
  • Figure 2: The same calculation as Figure \ref{['epaths']} redone for a different reference solution $U(t)$ lying on the global attractor.
  • Figure 3: The ensemble average and geometric mean compared to box plots (no outliers removed) of the error $|U(T)-u(T)|$ at time $T=t_0+1024$ for $500$ simulations varying $\delta$.
  • Figure 4: The geometric mean of $\|U(T)-u(T)\|_{L^2}$ at $T=t_0+1024$ for varying values of $\delta$ and $\kappa$.
  • Figure 5: Box plots (no outliers removed) depicting the values of $\kappa$ that minimize $\|U(T)-u(T)\|_{L^2}$ when $T=t_0+1024$ for each trajectory in the ensemble. Note $\kappa=\mu\delta$ was fitted for $\delta\le 0.5$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • proof : Proof of Theorem \ref{['partialresult']}.
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Proposition 3.3
  • ...and 17 more