Exponentially convergent data assimilation algorithm for Navier-Stokes equations

TL;DR

This paper introduces an exponentially convergent data assimilation algorithm for Navier-Stokes equations, providing a robust state estimation method that handles noise and incomplete data, with applications to chaotic fluid flow reconstruction.

Contribution

The paper develops a novel, exponentially convergent data assimilation algorithm for Navier-Stokes equations using algebraic Riccati equations for gain design.

Findings

Algorithm achieves exponential convergence of estimation error.

Successfully reconstructs chaotic fluid flow from noisy, incomplete data.

Provides a practical method for robust state estimation in fluid dynamics.

Abstract

The paper presents a new state estimation algorithm for a bilinear equation representing the Fourier- Galerkin (FG) approximation of the Navier-Stokes (NS) equations on a torus in R2. This state equation is subject to uncertain but bounded noise in the input (Kolmogorov forcing) and initial conditions, and its output is incomplete and contains bounded noise. The algorithm designs a time-dependent gain such that the estimation error converges to zero exponentially. The sufficient condition for the existence of the gain are formulated in the form of algebraic Riccati equations. To demonstrate the results we apply the proposed algorithm to the reconstruction a chaotic fluid flow from incomplete and noisy data.