Error estimates for scattered data interpolation on spheres

Jetter, Kurt; Stöckler, Joachim; Ward, Joseph D.

doi:10.1090/s0025-5718-99-01080-7

Cited by 121 publications

(122 citation statements)

References 14 publications

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“…(The growth function is also related to the norming constant used in the error bounds introduced in [7].) The following statement is easily obtainable by the duality of linear functionals, see e.g.…”

Section: Error Bounds In Terms Of a Growth Functionmentioning

confidence: 99%

Error bounds for kernel-based numerical differentiation

Davydov

Schaback

2015

Numer. Math.

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The literature on meshless methods observed that kernel-based numerical differentiation formulae are robust and provide high accuracy at low cost. This paper analyzes the error of such formulas, using the new technique of growth functions. It allows to bypass certain technical assumptions that were needed to prove the standard error bounds on interpolants and their derivatives. Since differentiation formulas based on polynomials also have error bounds in terms of growth functions, we have a convenient way to compare kernel-based and polynomial-based formulas. It follows that kernel-based formulas are comparable in accuracy to the best possible polynomial-based formulas. A variety of examples is provided.

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Section: Error Bounds In Terms Of a Growth Functionmentioning

confidence: 99%

Error bounds for kernel-based numerical differentiation

Davydov

Schaback

2015

Numer. Math.

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“…This quantity is referred to as the mesh-norm [24,25] and, geometrically, it represents the radius of the largest cap that covers the area between any subset of nodes on the sphere. The ME node sets have the property that h decays approximately uniformly like the inverse of the square root of the number of nodes N, i.e.…”

Section: Node Distribution and Convergence Of Rbf Interpolantsmentioning

confidence: 99%

“…[25,26]). Indeed, in the context of infinitely smooth RBFs, it is shown in [25] that, provided the underlying function being interpolated is sufficiently smooth, RBF interpolants converge (in the L ∞ norm) like h −1/2 e −c/4h , i.e at an exponential rate, for some constant c > 0 that depends on the RBF. For the ME node sets, convergence will thus proceed like N 1/4 e −c √ N /4 .…”

Section: Node Distribution and Convergence Of Rbf Interpolantsmentioning

confidence: 99%

Transport schemes on a sphere using radial basis functions

Flyer

Wright

2007

Journal of Computational Physics

133

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This is an author-produced, peer-reviewed version of this article. AbstractThe aim of this work is to introduce the physics community to the high performance of radial basis functions (RBFs) compared to other spectral methods for modeling transport (pure advection) and to provide the first known application of the RBF methodology to hyperbolic partial differential equations on a sphere. First, it is shown that even when the advective operator is posed in spherical coordinates (thus having singularities at the poles), the RBF formulation of it is completely singularity-free. Then, two classical test cases are conducted: 1) linear advection, where the initial condition is simply transported around the sphere and 2) deformational flow (idealized cyclogenesis), where an angular velocity is applied to the initial condition, spinning it up around an axis of rotation. The results show that RBFs allow for a much lower spatial resolution (i.e. lower number of nodes) while being able to take unusually large time-steps to achieve the same accuracy as compared to other commonly used spectral methods on a sphere such as spherical harmonics, double Fourier series, and spectral element methods. Furthermore, RBFs are algorithmically much simpler to program.

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“…[7,8,9,12,16,17]) that deals with direct theorems for scattered data interpolation by radial basis functions uses the native space as the space of smooth functions. Even in the case of "radial" functions on the sphere the concept of native spaces is carried over ( [5,4]). The easiest way to introduce native spaces on R d is by a Fourier transform.…”

Section: Direct Theoremsmentioning

confidence: 99%

Inverse and saturation theorems for radial basis function interpolation

Schaback¹,

Wendland²

2001

Math. Comp.

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Abstract. While direct theorems for interpolation with radial basis functions are intensively investigated, little is known about inverse theorems so far. This paper deals with both inverse and saturation theorems. For an inverse theorem we especially show that a function that can be approximated sufficiently fast must belong to the native space of the basis function in use. In case of thin plate spline interpolation we also give certain saturation theorems.

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Error estimates for scattered data interpolation on spheres

Cited by 121 publications

References 14 publications

Error bounds for kernel-based numerical differentiation

Error bounds for kernel-based numerical differentiation

Transport schemes on a sphere using radial basis functions

Inverse and saturation theorems for radial basis function interpolation

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