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Recovering the Parameter $α$ in the Simplified Bardina Model through Continuous Data Assimilation

Débora A. F. Albanez, Maicon José Benvenutti, Jing Tian

TL;DR

This paper develops a continuous data assimilation scheme to recover the lengthscale parameter $\alpha$ in the three-dimensional simplified Bardina turbulence model using observations of a finite set of Fourier modes. By embedding a recursive update for a surrogate parameter $\beta$ within a nudged assimilated system, it proves that $\beta_n$ converges to $\alpha$ and the assimilated state $w_n$ converges to the true solution $u$ under explicit, verifiable conditions. The main contributions are a constructive update rule, detailed auxiliary estimates, and a rigorous convergence proof showing exponential decay of both the parameter and state errors. The results provide a theoretical foundation for parameter identification in alpha-regularization turbulence models and set the stage for future computational implementations and data-driven applications.

Abstract

In this study, we develop a continuous data assimilation algorithm to recover the parameter $α$ in the simplified Bardina model. Our method utilizes the observations of finitely many Fourier modes by using a nudging framework that involves recursive parameter updates. We provide a rigorous convergence analysis, showing that the approximate parameter approaches the true value under suitable conditions, while the approximate solution also converges to the true solution.

Recovering the Parameter $α$ in the Simplified Bardina Model through Continuous Data Assimilation

TL;DR

This paper develops a continuous data assimilation scheme to recover the lengthscale parameter in the three-dimensional simplified Bardina turbulence model using observations of a finite set of Fourier modes. By embedding a recursive update for a surrogate parameter within a nudged assimilated system, it proves that converges to and the assimilated state converges to the true solution under explicit, verifiable conditions. The main contributions are a constructive update rule, detailed auxiliary estimates, and a rigorous convergence proof showing exponential decay of both the parameter and state errors. The results provide a theoretical foundation for parameter identification in alpha-regularization turbulence models and set the stage for future computational implementations and data-driven applications.

Abstract

In this study, we develop a continuous data assimilation algorithm to recover the parameter in the simplified Bardina model. Our method utilizes the observations of finitely many Fourier modes by using a nudging framework that involves recursive parameter updates. We provide a rigorous convergence analysis, showing that the approximate parameter approaches the true value under suitable conditions, while the approximate solution also converges to the true solution.

Paper Structure

This paper contains 7 sections, 7 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Functional analytic framework and preliminaries
  3. Overview of the parameter-$\alpha$ recovery algorithm
  4. Main Results
  5. Auxiliary Estimates
  6. Proof of the Main Result
  7. Conclusions

Key Result

Theorem 1

Consider $u$ solution of Bardina1Leray with initial condition $u(0) \in \dot{V}_{2}$ and $w_{0} \in \dot{V}_{2}$. Let $0<\varepsilon<\alpha^{2}_{0}$ and $\beta_{1} \in [\alpha_{0},\alpha_{1}]$, with $\alpha_0,\alpha_1$ satisfying estimates00. Suppose that for each $n \in \mathbb{N}$, there exist $\t with $t_{n+1} >\hat{t}_{n}\geq t_{n}$, where $t_{n}$ is the final time from the previous iteration

Theorems & Definitions (15)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Proposition 1
  • proof
  • ...and 5 more