Sobolev Error Estimates and a Bernstein Inequality for Scattered Data Interpolation via Radial Basis Functions

Narcowich, Francis J.; Ward, Joseph D.; Wendland, Holger

doi:10.1007/s00365-005-0624-7

Cited by 110 publications

(101 citation statements)

References 16 publications

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“…does not decline faster than h X,T , however, it can be shown that the error rate is of the same order as if a kernel with the correct degree of smoothness was used [57]. Hence, for quasi-uniform sets X, i.e.…”

Section: Interpolation With Misspecified Kernelsmentioning

confidence: 88%

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Interpolation of spatial data – A stochastic or a deterministic problem?

2013

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Interpolation of spatial data is a very general mathematical problem with various applications. In geostatistics, it is assumed that the underlying structure of the data is a stochastic process which leads to an interpolation procedure known as kriging. This method is mathematically equivalent to kernel interpolation, a method used in numerical analysis for the same problem, but derived under completely different modelling assumptions. In this article we present the two approaches and discuss their modelling assumptions, notions of optimality and the different concepts to quantify the interpolation accuracy. Their relation is much closer than has been appreciated so far, and even results on convergence rates of kernel interpolants can be translated to the geostatistical framework. We sketch the different answers obtained in the two fields concerning the issue of kernel misspecification, present some methods for kernel selection, and discuss the scope of these methods with a data example from the computer experiments literature.

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Section: Interpolation With Misspecified Kernelsmentioning

confidence: 88%

“…We state two results from the kernel interpolation literature (see [83,Sec. 11.6] and [57,Sec. 4]) and formulate their consequences for kriging:…”

Section: Error Estimatesmentioning

confidence: 99%

Interpolation of spatial data – A stochastic or a deterministic problem?

2013

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“…As standard results [11,16,19] on kernels suggest, the expectable optimal form of γ(Yr, m, µ) is γ(Yr, m, µ) ≃ q µ−m r where qr is the separation distance qr := 1 2 min y =z, y,z∈Yr y − z 2 of Yr, which is proportional to the fill distance hr if the trial centers are not too wildly scattered. In such a case, the inequality (21) will be satisfied if the ratio hs/hr of fill distances stays bounded above by a sufficiently small constant determined by the other ingredients like smoothness and scaling of the kernels.…”

Section: Theoremmentioning

confidence: 94%

Recovery of functions from weak data using unsymmetric meshless kernel-based methods

Schaback¹

2008

Applied Numerical Mathematics

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“…The intial escape was first proven in the case of scalar RBFs by Narcowich, Ward and Wendland, and their technique has since been applied to various situations involving RBFs [11,12,26,27,30]. A common theme in all these cases is to use functions that are band-limited, that is, functions whose Fourier transforms are compactly supported.…”

Section: Interpolation and Approximation Via Vector Spherical Polynommentioning

confidence: 99%

“…Many of the arguments that follow have been applied in several other situations. Sobolev error estimates have been derived in a similar way for scalar and matrix-valued RBFs and scalar SBFs [11,26,27,30]. In fact, the methods we employ here have been recently used to derive similar results for the divergence-free tangent kernel Ψ div [12,Section 4].…”

Section: Interpolation Error For Smooth and Rough Target Functionsmentioning

confidence: 99%

Stability and Error Estimates for Vector Field Interpolation and Decomposition on the Sphere with RBFs

Fuselier¹,

Wright²

2009

SIAM J. Numer. Anal.

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A new numerical technique based on radial basis functions (RBFs) is presented for fitting a vector field tangent to the sphere, S 2 , from samples of the field at "scattered" locations on S 2 . The method naturally provides a way to decompose the reconstructed field into its individual Helmholtz-Hodge components, i.e. into divergence-free and curl-free parts, which is useful in many applications from the atmospheric and oceanic sciences (e.g., in diagnosing the horizontal wind and ocean currents). Several approximation results for the method will be derived. In particular, Sobolev-type error estimates are obtained for both the interpolant and its decomposition. Optimal stability estimates for the associated interpolation matrices are also presented. Finally, numerical validation of the theoretical results is given for vector fields with similar characteristics to those of atmospheric wind fields.

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Sobolev Error Estimates and a Bernstein Inequality for Scattered Data Interpolation via Radial Basis Functions

Cited by 110 publications

References 16 publications

Interpolation of spatial data – A stochastic or a deterministic problem?

Interpolation of spatial data – A stochastic or a deterministic problem?

Recovery of functions from weak data using unsymmetric meshless kernel-based methods

Stability and Error Estimates for Vector Field Interpolation and Decomposition on the Sphere with RBFs

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