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

TL;DR

This paper introduces a continuous data assimilation method to accurately recover the parameter alpha in the simplified Bardina model using Fourier mode observations, ensuring convergence of both the parameter and solution.

Contribution

The study presents a novel recursive parameter update algorithm with rigorous convergence analysis for the simplified Bardina model.

Findings

The parameter alpha can be accurately recovered using finitely many Fourier modes.

The approximate solution converges to the true solution under certain conditions.

The method guarantees the convergence of the estimated parameter to its true value.

Abstract

In this study, we develop a continuous data assimilation algorithm to recover the parameter $\alpha$ 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.