Continuous data assimilation with stochastically noisy data

Bessaih, Hakima; Olson, Eric J.; Titi, Edriss S.

doi:10.1088/0951-7715/28/3/729

Cited by 99 publications

(98 citation statements)

References 46 publications

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“…These issues are the focus of the current paper. Our results extend the work of Hayden, Olson and Titi [17] on discrete-in-time data assimilation from the case where the low-resolution observations are given by projection onto the low Fourier modes to both the first and second type of general interpolant observables that appear in Azouani, Olson and Titi [3], see also Bessaih, Olson and Titi [4]. To make this extension, we apply a spectral filter based on the Stokes operator to the interpolant observables and call the new method spectrally-filtered discrete-in-time downscaling data assimilation.…”

Section: Introductionsupporting

confidence: 83%

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

Çelik¹,

Olson²,

Titi³

2019

SIAM J. Appl. Dyn. Syst.

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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

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Section: Introductionsupporting

confidence: 83%

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

Çelik¹,

Olson²,

Titi³

2019

SIAM J. Appl. Dyn. Syst.

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“…The number of velocity degrees of freedom for constant functions on the coarse mesh is just 5,772. The simulation is run on [0,5] (so the actual corresponding times for the DNS would be [5,10]). Figure 4 at the top, we observe exponential decay of v n h − u n L 2 in time, as predicted by our theory.…”

Section: Numerical Experiments 3: 2d Channel Flow Past a Cylindermentioning

confidence: 99%

Global in time stability and accuracy of IMEX-FEM data assimilation schemes for Navier–Stokes equations

Larios

Rebholz

Zerfas

2019

Computer Methods in Applied Mechanics and Engineering

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We study numerical schemes for incompressible Navier-Stokes equations using IMEX temporal discretizations, finite element spacial discretizations, and equipped with continuous data assimilation (a technique recently developed by Azouani, Olson, and Titi in 2014). We analyze stability and accuracy of the proposed methods, and are able to prove well-posedness, long time stability, and long time accuracy estimates, under restrictions of the time step size and data assimilation parameter. We give results for several numerical tests that illustrate the theory, and show that, for good results, the choice of discretization parameter and element choices can be critical. IntroductionData assimilation (DA) refers to a wide class of schemes for incorporating observational data in simulations, in order to increase the accuracy of solutions and to obtain better estimates of initial conditions. It is the subject of a large body of work (see, e.g., [13,31,33], and the references therein). DA algorithms are widely used in weather modeling, climate science, and hydrological and environmental forecasting [31]. Classically, these techniques are based on linear quadratic estimation, also known as the Kalman Filter. The Kalman Filter is described in detail in several textbooks, including [13,31,33,10], and the references therein.Recently, a promising new approach to data assimilation was pioneered by Azouani, Olson, and Titi [3,4] (see also [9,25,39] for early ideas in this direction). This new approach, which we call AOT Data Assimilation or continuous data assimilation, adds a feedback control term at the PDE level that nudges the computed solution towards the reference solution corresponding to the observed data. A similar approach is taken by Blömker, Law, Stuart, and Zygalakis in [7] in the context of stochastic differential equations. The AOT algorithm is based on feedback control at the PDE (partial differential equation) level, described below. The first works in this area assumed noise-free observations, but [5] adapted the method to the case of noisy data, and [19] adapted to the case in which measurements are obtained discretely in time and may be contaminated by systematic errors. Computational experiments on the AOT algorithm and its variants were carried * An important property of the b operator is that b(u, v, v) = 0 for u, v ∈ X.We will utilize the following bounds on b.Lemma 2.1. There exists a constant M > 0 dependent only on Ω satisfyingfor all u, v, w ∈ X for which the norms on the right hand sides are finite.Remark 2.2. Here and throughout, sharper estimates are possible if we restrict to 2D. However, for simplicity and generality, we do not make this restriction.Proof. These well known bounds follow from Hölder's inequality, Sobolev inequalities, and the Poincaré inequality. Discretization preliminariesDenote by τ h a regular, conforming triangulation of the domain Ω, and let X h ⊂ X, Q h ⊂ Q be an inf-sup stable pair of discrete velocity -pressure spaces. For simplicity, we will take X h = X ∩ P k and Q h = Q ∩...

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“…The convergence of this synchronization algorithm for the 2D NSE, in higher order (Gevery class) norm and in L ∞ norm, was later studied in [7] for smoother forcing. An extension of the approach in [4] to the case when the observational data contains stochastic noise was analyzed in [6]. A study of the algorithm for the 2D NSE when the measurements are obtained discretely in time and are contaminated by the measurement device error is presented in [22] (see also [29]).…”

Section: Introductionmentioning

confidence: 99%

Assimilation of Nearly Turbulent Rayleigh–Bénard Flow Through Vorticity or Local Circulation Measurements: A Computational Study

et al. 2018

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We introduce a continuous (downscaling) data assimilation algorithm for the 2D Bénard convection problem using vorticity or local circulation measurements only. In this algorithm, a nudging term is added to the vorticity equation to constrain the model. Our numerical results indicate that the approximate solution of the algorithm is converging to the unknown reference solution (vorticity and temperature) corresponding to the measurements of the 2D Bénard convection problem when only spatial coarse-grain measurements of vorticity are assimilated. Moreover, this convergence is realized using data which is much more coarse than the resolution needed to satisfy rigorous analytical estimates.

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Continuous data assimilation with stochastically noisy data

Cited by 99 publications

References 46 publications

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

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

Global in time stability and accuracy of IMEX-FEM data assimilation schemes for Navier–Stokes equations

Assimilation of Nearly Turbulent Rayleigh–Bénard Flow Through Vorticity or Local Circulation Measurements: A Computational Study

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