Closed form representations and properties of the generalised Wendland functions

Chernih, Andrew; Hubbert, Simon

doi:10.1016/j.jat.2013.09.005

Cited by 30 publications

(19 citation statements)

References 8 publications

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“…For two given non negative functions g 1 (x) and g 2 (x), with g 1 (x) g 2 (x) we mean that there exist two constants c and C such that 0 < c < C < ∞ and cg 2 (x) ≤ g 1 (x) ≤ Cg 2 (x) for each x. The next result follows from Zastavnyi (2006), Chernih and Hubbert (2014), and from standard properties of Fourier transforms. Their proofs are thus omitted.…”

mentioning

confidence: 94%

“…We now define GW correlation functions ϕ µ,κ as introduced by Gneiting (2002b), Zastavnyi (2006) and Chernih and Hubbert (2014). For κ > 0, we define…”

mentioning

confidence: 99%

“…Note that the case κ = 0 is not included in Theorem 1. We consider it in the following result, whose proof follows the lines of Zastavnyi (2006) and Chernih and Hubbert (2014) for the case κ > 0.…”

mentioning

confidence: 99%

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Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics

et al. 2019

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We study estimation and prediction of Gaussian random fields with covariance models belonging to the generalized Wendland (GW) class, under fixed domain asymptotics. As for the Matérn case, this class allows for a continuous parameterization of smoothness of the underlying Gaussian random field, being additionally compactly supported. The paper is divided into three parts: first, we characterize the equivalence of two Gaussian measures with GW covariance function, and we provide sufficient conditions for the equivalence of two Gaussian measures with Matérn and GW covariance functions. In the second part, we establish strong consistency and asymptotic distribution of the maximum likelihood estimator of the microergodic parameter associated to GW covariance model, under fixed domain asymptotics. The third part elucidates the consequences of our results in terms of (misspecified) best linear unbiased predictor, under fixed domain asymptotics. Our findings are illustrated through a simulation study: the former compares the finite sample behavior of the maximum likelihood estimation of the microergodic parameter with the given asymptotic distribution. The latter compares the finitesample behavior of the prediction and its associated mean square error when using two equivalent Gaussian measures with Matérn and GW covariance models, using covariance tapering as benchmark.1 imsart-aos ver.

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mentioning

confidence: 94%

“…We now define GW correlation functions ϕ µ,κ as introduced by Gneiting (2002b), Zastavnyi (2006) and Chernih and Hubbert (2014). For κ > 0, we define…”

mentioning

confidence: 99%

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Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics

et al. 2019

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“…, it is indeed possible to find such a function; see [27]. The most prominent examples of such compactly supported RBFs were given by Wendland [28].…”

Section: Definition 1 (Rbf)mentioning

confidence: 99%

“…The Fourier transform of the compactly supported RBF shall satisfy

c_{1} {((), 1 + ∥ ω ∥_{2}^{2})}^{- σ} \leq \overset{normalΦ}{false} (ω) \leq c_{2} {((), 1 + ∥ ω ∥_{2}^{2})}^{- σ}

for 0 < c 1 ≤ c 2 . For

σ > \frac{d + 1}{2}

, it is indeed possible to find such a function; see .…”

Section: Introductionmentioning

confidence: 99%

Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs

Farrell

Pestana

2015

Numerical Linear Algebra App

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Symmetric collocation methods with radial basis functions allow approximation of the solution of a partial differential equation, even if the right-hand side is only known at scattered data points, without needing to generate a grid. However, the benefit of a guaranteed symmetric positive definite block system comes at a high computational cost. This cost can be alleviated somewhat by considering compactly supported radial basis functions and a multiscale technique. But the condition number and sparsity will still deteriorate with the number of data points. Therefore, we study certain block diagonal and triangular preconditioners. We investigate ideal preconditioners and determine the spectra of the preconditioned matrices before proposing more practical preconditioners based on a restricted additive Schwarz method with coarse grid correction (ARASM). Numerical results verify the effectiveness of the preconditioners.

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Spatio-Spectral Assessment of Some Isotropic Polynomial Covariance Functions on the Sphere

Piretzidis

Kotsakis

Mertikas

et al. 2023

International Association of Geodesy Symposia

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In gravity field modeling, covariance functions are mainly associated with least squares collocation. Prior to the implementation of least squares collocation, the characteristics of the selected analytical covariance function need to be well understood. In this contribution, we study four polynomial covariance functions, i.e., the spherical, Askey, C2-Wendland and C4-Wendland models. All of them are defined on the sphere and correspond to isotropic, positive definite and compactly supported functions. We examine them in the spatial and spectral domains, and assess their characteristics, such as the correlation length, the curvature parameter, the spectral maximum and the spectral decay rate. We also provide analytical expressions and numerical estimates for these parameters.

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Closed form representations and properties of the generalised Wendland functions

Cited by 30 publications

References 8 publications

Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics

Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics

Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs

Spatio-Spectral Assessment of Some Isotropic Polynomial Covariance Functions on the Sphere

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