Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data

Larios, Adam; Pei, Yuan

doi:10.3934/eect.2020031

Cited by 18 publications

(6 citation statements)

References 80 publications

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“…Dissipation is then crucially used to prove that the system converges not only to the observed projection of the true state, but in fact to the full true state. It was noted in [16,33,54] for different systems that if the parameters of the model are not that of the true state, then the system will converge to a finite amount of unrecoverable error. This remaining error is a direct consequence of the parameter error, and hence provides an avenue to estimating the true value of the parameter.…”

Section: Introductionmentioning

confidence: 99%

Dynamically learning the parameters of a chaotic system using partial observations

Carlson¹,

Hudson²,

Larios³

et al. 2022

DCDS

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<p style='text-indent:20px;'>Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, we supply a rigorous analytical proof that guarantees the convergence of this algorithm to the true parameter values when the system in question is the classic three-dimensional Lorenz system. Such a result appears to be the first of its kind for dynamical parameter estimation of nonlinear systems. Computationally, we demonstrate the efficacy of this algorithm on the Lorenz system by recovering any proper subset of the three non-dimensional parameters of the system, so long as a corresponding subset of the state is observable. We moreover probe the limitations of the algorithm by identifying dynamical regimes under which certain parameters cannot be effectively inferred having only observed certain state variables.</p>

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Section: Introductionmentioning

confidence: 99%

Dynamically learning the parameters of a chaotic system using partial observations

Carlson¹,

Hudson²,

Larios³

et al. 2022

DCDS

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“…In this paper, inspired by the methodology of data assimilation, especially variational data assimilation in continuous time (for relevant works we refer e.g. to [13,19,26,29,30,44,46,52]), we seek to minimise the misfit functional (u, p, y) → (1 − λ) Q(•, •, u, ∇u, p) − q + λ y over all admissible triplets (u, p, y) which satisfy (1.1), for a fixed weight λ ∈ (0, 1). The role of this weight is to obtain essentially a Pareto family of extremals, one for each value λ, even though in this paper we do not pursue further this viewpoint of vector-valued minimisation (the interested reader may e.g.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Vectorial variational problems in $L^\infty$ constrained by the Navier-Stokes equations

Clark¹,

Katzourakis²,

Muha³

2021

Preprint

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We study a minimisation problem in L p and L ∞ for certain cost functionals, where the class of admissible mappings is constrained by the Navier-Stokes equations. Problems of this type are motivated by variational data assimilation for atmospheric flows arising in weather forecasting. Herein we establish the existence of PDE-constrained minimisers for all p, and also that L p minimisers converge to L ∞ minimisers as p → ∞. We further show that L p minimisers solve an Euler-Lagrange system. Finally, all special L ∞ minimisers constructed via approximation by L p minimisers are shown to solve a divergence PDE system involving measure coefficients, which is a divergenceform counterpart of the corresponding non-divergence Aronsson-Euler system. Contents 1. Introduction and main results 1 2. Preliminaries and the main estimates 9 3. Minimisers of L p problems and convergence as p → ∞ 12 4. The equations for L p PDE-constrained minimisers 13 5. The equations for L ∞ PDE-constrained minimisers 15 Acknowledgement 19 References 19

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“…In such methods, nudging of the computed solution is done by penalizing its difference to measurement data, that is, I H ( v ) is penalized to be close to I H ( u ). Since the initial work of Azouani et al in 2014 [4], there has been a large amount of work done for these types of methods, including for many different equations arising from physics such as Navier–Stokes equations (NSE) and Boussinesq, for different variations of DA implementation, and noisy data: CDA for NSE is studied in [7, 18, 42], with noisy data [6], noisy data discrete in time [20], and fully discrete [37]; CDA for NS‐ α is considered in [1] for higher Reynolds number flow; CDA is applied to Benard convection in [2, 16, 19] with various methods of nudging; CDA for Brinkman Forchheimer‐extended Darcy is studied in [43] and for surface quasi‐geostrophic equation in [34]; general interpolating operators for CDA are considered in [21]; CDA for reduced order modeling of the NSE is studied in [50]; and nonlinear nudging is proposed and analyzed in [36]. Although most of these works are in 2D, some research has been done in 3D including good numerical results for CDA applied to the 3D NSE in [40, 41] and analytical results for 3D Leray‐ α [17] and NS‐ α [1].…”

Section: Introductionmentioning

confidence: 99%

Simple and efficient continuous data assimilation of evolution equations via algebraic nudging

Rebholz

Zerfas

2021

Numerical Methods Partial

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We introduce, analyze, and test an interpolation operator designed for use with continuous data assimilation (DA) of evolution equations that are discretized spatially with the finite element method. The interpolant is constructed as an approximation of the L 2 projection operator onto piecewise constant functions on a coarse mesh, but which allows nudging to be done completely at the linear algebraic level, independent of the rest of the discretization, with a diagonal matrix that is simple to construct; it can even completely remove the need for explicit construction of a coarse mesh. We prove the interpolation operator has sufficient stability and accuracy properties, and we apply it to algorithms for both fluid transport DA and incompressible Navier-Stokes DA. For both applications we prove the DA solutions with arbitrary initial conditions converge to the true solution (up to discretization error) exponentially fast in time, and are thus long-time accurate. Results of several numerical tests are given, which both illustrate the theory and demonstrate its usefulness on practical problems. KEYWORDS continuous data assimilation, finite element method, Navier-Stokes equations 1 INTRODUCTION Data assimilation (DA) algorithms are widely used in weather prediction, climate modeling, and many other applications [35]. The term DA generally refers to schemes that incorporate observational data in simulations in order to increase accuracy and/or obtain better initial conditions. There is a large amount of literature on the general topic of DA [13, 35, 38], and several different techniques exist, such as the

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Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data

Cited by 18 publications

References 80 publications

Dynamically learning the parameters of a chaotic system using partial observations

Dynamically learning the parameters of a chaotic system using partial observations

Vectorial variational problems in $L^\infty$ constrained by the Navier-Stokes equations

Simple and efficient continuous data assimilation of evolution equations via algebraic nudging

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