Lp-error estimates for radial basis function interpolation on the sphere

Hubbert, Simon; Morton, Tanya M.

doi:10.1016/j.jat.2004.04.006

Cited by 44 publications

(55 citation statements)

References 16 publications

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“…[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.…”

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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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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“…For any zonal continuous kernel φ : S 2 × S 2 → R, the function in (2.5) is continuous and thus, in particular, is in L 2 ([−1, 1]), and can be expanded into a Legendre series 6) with the Legendre coefficients a defined by…”

Section: Preliminariesmentioning

confidence: 99%

“…: 6) where the estimate in the last step follows from (4.5). Now we apply Theorem 4.3 to the finite-dimensional space V and the norming set Z = {δ y j : j = 1, 2, .…”

Section: Proofsmentioning

confidence: 99%

Local Radial Basis Function Approximation on the Sphere

Hesse

Gia

2008

Bulletin of the Australian Mathematical Society

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In this paper we derive local error estimates for radial basis function interpolation on the unit sphere S 2 ⊂ R 3 . More precisely, we consider radial basis function interpolation based on data on a (global or local) point set X ⊂ S 2 for functions in the Sobolev space H s (S 2 ) with norm · s , where s > 1. The zonal positive definite continuous kernel φ, which defines the radial basis function, is chosen such that its native space can be identified with H s (S 2 ). Under these assumptions we derive a local estimate for the uniform error on a spherical cap S(z; r ): the radial basis function interpolant X f of f ∈ H s (S 2 ) satisfies sup x∈S(z;r ) | f (x) − X f (x)| ≤ ch (s−1)/2 f s , where h = h X,S(z;r ) is the local mesh norm of the point set X with respect to the spherical cap S(z; r ). Our proof is intrinsic to the sphere, and makes use of the Videnskii inequality. A numerical test illustrates the theoretical result.2000 Mathematics subject classification: 41A05, 41A15, 41A25, 65D05, 65D07.

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“…The mesh norm is also of practical importance since it appears in many proofs of error bounds for standard RBF interpolation on the sphere (e.g., [17,15]). Indeed, in the context of infinitely smooth radial kernels, it is shown in [17] that, provided the underlying function being interpolated is sufficiently smooth, the standard RBF interpolation method converges (in the max.…”

Section: Node Distributionsmentioning

confidence: 99%

Divergence-Free RBFs on Surfaces

Narcowich

Ward

Wright

2007

J Fourier Anal Appl

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This article presents a new tool for fitting a divergence-free vector field tangent to a two-dimensional orientable surface P ∈ R 3 to samples of such a field taken at scattered sites on P. This method, which involves a kernel constructed from radial basis functions, has applications to problems in geophysics, and has the advantage of avoiding problems with poles. Numerical examples testing the method on the sphere are included.

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Lp-error estimates for radial basis function interpolation on the sphere

Cited by 44 publications

References 16 publications

Transport schemes on a sphere using radial basis functions

Transport schemes on a sphere using radial basis functions

Local Radial Basis Function Approximation on the Sphere

Divergence-Free RBFs on Surfaces

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