Parameter Recovery for the 2 Dimensional Navier--Stokes Equations via Continuous Data Assimilation

Abstract

We study a continuous data assimilation algorithm proposed by Azouani, Olson, and Titi (AOT) in the context of an unknown Reynolds number. We determine the large-time error between the true solution of the 2D Navier-Stokes equations and the assimilated solution due to discrepancy between an approximate Reynolds number and the physical Reynolds number. Additionally, we develop an algorithm that can be run in tandem with the AOT algorithm to recover both the true solution and the Reynolds number (or equivalently the true viscosity) using only spatially discrete velocity measurements. The algorithm we propose involves changing the viscosity mid-simulation. Therefore, we also examine the sensitivity of the equations with respect to the Reynolds number. We prove that a sequence of difference quotients with respect to the Reynolds number converges to the unique solution of the sensitivity equations for both the 2D Navier-Stokes equations and the assimilated equations. We also note that this appears to be the first such rigorous proof of existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier-Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the Reynolds number.

Concurrent multi-parameter learning demonstrated on the Kuramoto-Sivashinsky equation

“…This is often a difficult task, as such parameters are difficult to measure and/or noise in the measurement process precludes identifying the parameters exactly. Motivated by [9], we develop a robust algorithm that estimates multiple parameters in a partial differential equation (PDE) when those parameters can be written as coefficients of the various terms in the PDE. As the current algorithm relies on the feedback control data assimilation mechanism introduced in [1], we simultaneously identify not only the parameters of the system, but also obtain the dynamical state of the system based on reduced observations of the true state.…”

“…All of these extensions have broadened the applicability of the AOT algorithm but rely on the accurate representation of the underlying model parameters. Recently, data assimilation for an imperfect model has been considered in different contexts [15,9,18]. It is rigorously established for two distinct settings that when a parameter of the system is unknown, that error in the assimilation will still converge up to the error in a single unknown parameter (see [9,18]).…”

“…Recently, data assimilation for an imperfect model has been considered in different contexts [15,9,18]. It is rigorously established for two distinct settings that when a parameter of the system is unknown, that error in the assimilation will still converge up to the error in a single unknown parameter (see [9,18]). An upcoming article [10] takes these concepts a step further, deriving analogous update formulae for the three parameters of the Lorenz equations [34] and carrying out a thorough numerical investigation to explore the limitations of these formulae.…”

Data Assimilation in Large Prandtl Rayleigh--Bénard Convection from Thermal Measurements

“…Moreover our studies are carried out to a similar high degree of numerical precision as found in [FJJT18]. We refer the reader to [ANLT16,FET17,DLMB18,CHL18] for various other studies in the context of turbulent flows such as how one can leverage the nudging scheme to infer unknown parameters of the flow. In [BM17,IMT18,MT18] analytical studies on the various modes of synchronization of the algorithm (1.2) and on certain variants on its numerical discretization were carried out.…”

Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations

“…This seemingly minor change had profound impacts, and the authors of [5] were able to prove that using only sparse observations, the CDA algorithm applied to the 2D Navier-Stokes equations converges to the correct solution exponentially fast in time, independent of the choice initial data. This stimulated a large amount of recent research on the CDA algorithm; see, e.g., [3,6,7,10,11,13,17,18,19,20,21,22,27,31,30,35,40,41] and the references therein. The recent paper [15] showed that CDA can be effectively used for weather prediction, showing that it can indeed be a powerful tool on practical large scale problems.…”