Spectral Filtering of Interpolant Observables for a Discrete-in-Time Downscaling Data Assimilation Algorithm

Abstract

We describe a spectrally-filtered discrete-in-time downscaling data assimilation algorithm and prove, in the context of the two-dimensional Navier-Stokes equations, that this algorithm works for a general class of interpolants, such as those based on local spatial averages as well as point measurements of the velocity. Our algorithm is based on the classical technique of inserting new observational data directly into the dynamical model as it is being evolved over time, rather than nudging, and extends previous results in which the observations were defined directly in terms of an orthogonal projection onto the large-scale (lower) Fourier modes. In particular, our analysis does not require the interpolant to be represented by an orthogonal projection, but requires only the interpolant to satisfy a natural approximation of the identity.and let Q λ = I − P λ be the orthogonal complement of P λ . Now, given λ > 0 and I h define J = P λ P σ I h and E = I − J.(1.6) Note, although no additional orthogonality or regularity properties other than those appearing in Definition 1.1 have been assumed on I h , the above spectral filtering yields an operator J which is nearly orthogonal and has a range contained in D(A). The downscaling data assimilation algorithm studied in this paper may now be stated as Definition 1.2. Let U be an exact solution of (1.2) which evolves according to dynamics given by the semi-process S. Let t n = t 0 + nδ be a sequence of times for which partial

Numerical efficacy study of data assimilation for the 2D magnetohydrodynamic equations

“…Indeed, an advantage of nudging is the flexibility of the interpolant. Practically speaking, Fourier modes are not expected to be directly measured in observed data, but could be obtained via a fast Fourier transform from data at nodes on a grid, albeit with an aliasing error (see [9] for more on spectral filtering of observables). The first algorithm we consider utilizes data collected on all the variables (except the pressure).…”

Continuous Data Assimilation with a Moving Cluster of Data Points for a Reaction Diffusion Equation: A Computational Study

“….. K > 0 here is an integer dependent on N , the minimum length scale λ (see Section 5) and the number of nodes in the probe, m h . For N = 2 12 and ν = 7.5e − 6, the frequency-locked velocities were observed to correspond to the value K = 64. It was found that the probe took significantly longer to converge or it did not converge at all at these velocities.…”

“…In addition to this, the resulting matrix is tri-diagonal, so solving this system has a time complexity of only O(N ) using the Thomas algorithm. In every simulation in this work, we use time-step ∆t = 1e − 3 for stability, and spatial resolution N = 2 12 = 4096 (i.e., ∆x = 2 −12 ≈ 2.441e − 4), which are sufficient to resolve the spatial scales in our simulations (i.e., below the cubic aliasing cut-off number N/4) to machine precision over the range of ν-values and µ − values we consider. Note that since we handle the data assimilation term explicitly, a CFL condition arises, requiring ∆t ≤ 2/µ, which is satisfied in all of our simulations.…”

“…A large amount of recent literature has built upon the AOT algorithm; see, e.g., [1,2,6,7,8,9,11,12,18,19,20,21,22,23,24,25,26,27,28,31,32,33,34,37,39,40,41,43,44]. Computational experiments on the AOT algorithm and its variants were carried out in the cases of the 2D Navier-Stokes equations [27], the 2D Bénard convection equations [2], and the 1D Kuramoto-Sivashinsky equations [39,37].…”

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.…”