Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  adapt a previously introduced continuous in time data assimilation (downscaling) algorithm for the two-dimensional Navier--Stokes equations to the more realistic case when the measurements are obtained discretely in time and may be contaminated by systematic errors. Our algorithm is designed to work with a general class of observables, such as low Fourier modes and local spatial averages over finite volume elements. Under suitable conditions on the relaxation (nudging) parameter, the spatial mesh resolution, and the time step beten successive measurements,  obtain an asymptotic in time estimate of the difference beten the approximating solution and the unknown reference solution corresponding to the measurements, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors. A stationary statistical analysis of our discrete data assimilation algorithm is also provided.
['Ciprian Foias', 'Cecilia F. Mondaini', 'Edriss S. Titi']
I am writing a research paper on a idea A Discrete Data Assimilation Scheme for the Solutions of the Two-Dimensional Navier--Stokes Equations and Their Statistics
[{"from": "human", "value": "I am writing a research paper on a idea A Discrete Data Assimilation Scheme for the Solutions of the Two-Dimensional Navier--Stokes Equations and Their Statistics"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  adapt a previously introduced continuous in time data assimilation (downscaling) algorithm for the two-dimensional Navier--Stokes equations to the more realistic case when the measurements are obtained discretely in time and may be contaminated by systematic errors. Our algorithm is designed to work with a general class of observables, such as low Fourier modes and local spatial averages over finite volume elements. Under suitable conditions on the relaxation (nudging) parameter, the spatial mesh resolution, and the time step beten successive measurements,  obtain an asymptotic in time estimate of the difference beten the approximating solution and the unknown reference solution corresponding to the measurements, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors. A stationary statistical analysis of our discrete data assimilation algorithm is also provided."}]