Sensitivity Analysis for the 2D Navier–Stokes Equations with Applications to Continuous Data Assimilation

Abstract

We study a nonlinear-nudging modification of the Azouani-Olson-Titi continuous data assimilation (downscaling) algorithm for the 2D incompressible Navier-Stokes equations. We give a rigorous proof that the nonlinear-nudging system is globally well-posed, and moreover that its solutions converge to the true solution exponentially fast in time. Furthermore, we also prove that, once the error has decreased below a certain order one threshold, the convergence becomes double-exponentially fast in time, up until a precision determined by the sparsity of the observed data. In addition, we demonstrate the applicability of the analytical and sharpness of the results computationally.

Dynamically learning the parameters of a chaotic system using partial observations

“…All of these results, including the current investigation, are motivated by the continuous data assimilation approach pioneered by [5]. The approach was originally presented for the 2D Navier-Stokes equations but has since been generalized to settings beyond the Navier-Stokes system where only a subset of the prognostic variables are observable [2,3,11,9,10,12,13,17,19,20,25,23,26,33,35,34,36,37,38,42,43,44,45,46,50,51,53,55,56,60,62,63,68,72,81]. Further extensions of this approach to discrete-in-time observations [40,49,57], and the presence of stochastic noise in the observations [8,14] have been made with completely rigorous justification.…”

The Bleeps, the Sweeps, and the Creeps: Convergence Rates for Dynamic Observer Patterns via Data Assimilation for the 2D Navier-Stokes Equations

“…A.2 to A. 13. As we see from these results, all three of these methods enjoy exponential convergence to the reference solution in the L 2 norm.…”

“…Since its inception in [3,4], the AOT algorithm has been the subject of much recent study in both analytical studies [1,5,6,8,9,10,11,13,15,16,21,22,23,24,25,26,27,28,29,31,30,32,34,35,36,37,39,41,44,45,48,49,50,53] and computational studies [2,12,14,18,20,33,38,40,42,43].…”

Dynamically learning the parameters of a chaotic system using partial observations

“…All of these results, including the current investigation, are motivated by the continuous data assimilation approach pioneered by [5]. The approach was originally presented for the 2D Navier-Stokes equations but has since been generalized to settings beyond the Navier-Stokes system where only a subset of the prognostic variables are observable [2,3,11,9,10,12,13,17,19,20,25,23,26,33,35,34,36,37,38,42,43,44,45,46,50,51,53,55,56,60,62,63,68,72,81]. Further extensions of this approach to discrete-in-time observations [40,49,57], and the presence of stochastic noise in the observations [8,14] have been made with completely rigorous justification.…”