On the inadequacy of nudging data assimilation algorithms for non-dissipative systems: A computational examination of the Korteweg de-Vries and Lorenz equations

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 the Korteweg de-Vries (KdV) equation and a partially dissipative variant of the Lorenz 1963 system. Our analysis reveals that the KdV equation lacks 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. We numerically verify that the AOT algorithm successfully recovers these counterexamples for the damped and driven KdV equation, as studied in [1], which is dissipative. Additionally, we demonstrate numerically that the AOT algorithm is not effective in accurately recovering solutions for a partially dissipative variant of the Lorenz 1963 system.