Related papers: Continuous data assimilation for 2D stochastic Nav…
We study a continuous data assimilation (CDA) algorithm for a velocity-vorticity formulation of the 2D Navier-Stokes equations in two cases: nudging applied to the velocity and vorticity, and nudging applied to the velocity only. We prove…
Continuous data assimilation (CDA) is a method that continuously integrates observational data into a dynamical system to improve model accuracy in real-time. The AOT algorithm is one of the most widely used methods in CDA due to its…
Consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times. Discrete data assimilation inserts these observational measurements of the reference dynamical system…
Continuous data assimilation (CDA) nudges observational data into governing equations to recover the underlying flow and improve predictions. Existing rigorous CDA analyses focus primarily on incompressible flows, yet no physical flow is…
We introduce a continuous data assimilation (downscaling) algorithm for the two-dimensional Navier-Stokes equations employing coarse mesh measurements of only one component of the velocity field. This algorithm can be implemented with a…
The Gray--Scott model governs the interaction of two chemical species via a system of reaction-diffusion equations. Despite its simple form, it produces extremely rich patterns such as spots, stripes, waves, and labyrinths. That makes it…
An algorithm for continuous data assimilation for the two- dimensional B\'enard convection problem is introduced and analyzed. It is inspired by the data assimilation algorithm developed for the Navier-Stokes equations, which allows for the…
We consider nonlinear solvers for the incompressible, steady (or at a fixed time step for unsteady) Navier-Stokes equations in the setting where partial measurement data of the solution is available. The measurement data is…
We introduce a localized version of the nudging data assimilation algorithm for the periodic 2D Navier-Stokes equations in which observations are confined (i.e., localized) to a window that moves across the entire domain along a…
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…
This work investigates the effectiveness of the Back-and-Forth Nudging (BFN) data assimilation algorithm, specifically its performance when employing the Azouani-Olson-Titi (AOT) continuous data assimilation downscaling nudging algorithm,…
We propose, analyze, and test a novel continuous data assimilation reduced order model (DA-ROM) for simulating incompressible flows. While ROMs have a long history of success on certain problems with recurring dominant structures, they tend…
We consider a continuous data assimilation method for the barotropic Navier--Stokes system. The observed solution is supposed to be bounded on the whole time period of observation, while the synchronized solution, usually provided by a…
Continuous data assimilation addresses time-dependent problems with unknown initial conditions by incorporating observations of the solution into a nudging term. For the prototypical heat equation with variable conductivity and the Neumann…
We rigorously prove the well-posedness of the formal sensitivity equations with respect to the Reynolds number corresponding to the 2D incompressible Navier-Stokes equations. Moreover, we do so by showing a sequence of difference quotients…
This paper considers a nudging-based scheme for data assimilation for the two-dimensional (2D) Navier-Stokes equations (NSE) with periodic boundary conditions and studies the synchronization of the signal produced by this algorithm with the…
Data assimilation methodologies are designed to incorporate noisy observations of a physical system into an underlying model in order to infer the properties of the state of the system. Filters refer to a class of data assimilation…
The error analysis of a proper orthogonal decomposition (POD) data assimilation (DA) scheme for the Navier-Stokes equations is carried out. A grad-div stabilization term is added to the formulation of the POD method. Error bounds with…
This paper considers improving the Picard and Newton iterative solvers for the Navier-Stokes equations in the setting where data measurements or solution observations are available. We construct adapted iterations that use continuous data…
The paper presents a new state estimation algorithm for a bilinear equation representing the Fourier- Galerkin (FG) approximation of the Navier-Stokes (NS) equations on a torus in R2. This state equation is subject to uncertain but bounded…