On the inadequacy of nudging data assimilation algorithms for non-dissipative systems

TL;DR

This paper demonstrates that the Azouani-Olson-Titi nudging algorithm is inadequate for non-dissipative systems like Euler and KdV equations, as they lack finitely determining modes, leading to indistinguishable solutions from sparse data.

Contribution

The study reveals fundamental limitations of the AOT data assimilation algorithm for non-dissipative systems, supported by theoretical analysis and numerical experiments.

Findings

AOT fails to distinguish solutions in non-dissipative systems due to infinite solutions with same data.

AOT successfully recovers solutions in dissipative KdV systems.

AOT is ineffective for partially dissipative Lorenz 1963 systems.

Abstract

In this work, we study the applicability of the Azouani-Olson-Titi (AOT) nudging algorithm for continuous data assimilation to evolutionary dynamical systems that are not dissipative. Specifically, we apply the AOT algorithm to a partially dissipative variant of the Lorenz 1963 system, the Korteweg-de Vries equation (KdV) in 1D, and the 2D incompressible Euler equations. Our analysis reveals that both the Euler and KdV equations lack the finitely many determining modes property, leading to the construction of infinitely many solutions with exactly the same sparse observational data, which data assimilation methods cannot distinguish between. Simultaneously, we numerically verify that the AOT algorithm successfully recovers these counterexamples for the damped and driven KdV equation, which is dissipative. Additionally, to further support our argument, we present numerical evidence showing that the AOT algorithm is ineffective at accurately recovering solutions for a partially dissipative variant of the Lorenz 1963 system.