Continuous Data Assimilation Using General Interpolant Observables

TL;DR

This paper introduces a new continuous data assimilation algorithm for dissipative systems like the 2D Navier-Stokes equations, leveraging finite-dimensional parameters and general interpolation operators to ensure convergence of the approximated solution to the true system over time.

Contribution

The paper develops a novel data assimilation method that uses general measurement data and interpolation operators, providing conditions for guaranteed convergence in dissipative systems.

Findings

The algorithm guarantees convergence under specific spatial resolution conditions.

Applicable to various measurement data types via interpolation operators.

Effective in signal synchronization for dissipative systems.

Abstract

We present a new continuous data assimilation algorithm based on ideas that have been developed for designing finite-dimensional feedback controls for dissipative dynamical systems, in particular, in the context of the incompressible two-dimensional Navier--Stokes equations. These ideas are motivated by the fact that dissipative dynamical systems possess finite numbers of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages which govern their long-term behavior. Therefore, our algorithm allows the use of any type of measurement data for which a general type of approximation interpolation operator exists. Our main result provides conditions, on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, obtained by our algorithm from the measurement data, converges to the unknown reference solution over time. Our algorithm is also applicable in the context of signal synchronization in which one can recover, asymptotically in time, the solution (signal) of the underlying dissipative system that is corresponding to a continuously transmitted partial data.