Scattered Data Interpolation on Spheres: Error Estimates and Locally Supported Basis Functions

Narcowich, Francis J.; Ward, Joseph D.

doi:10.1137/s0036141001395054

Cited by 118 publications

(104 citation statements)

References 16 publications

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“…There is quite some literature on how to find good approximations to data on the sphere by using interpolation by zonal kernels, see e.g. [7,6].…”

Section: Harmonic Functions In the Unit Ballmentioning

confidence: 99%

“…Following [7,6] we define Sobolev spaces H τ on the sphere by (2) for orthonormal expansions (1). If considered on S 2 they contain continuous functions and allow continuous point evaluations if τ > 3/2.…”

Section: Error Boundsmentioning

confidence: 99%

“…To proceed towards a citation of Proposition 2.1 of [7], we consider a Sobolev space H τ with τ > 3/2 and work with functions g from H τ and kernels L on S 2 whose native space is contained in H τ . Interpolation of data of g is done in scattered point sets X ⊂ S 2 with fill distance…”

Section: Error Boundsmentioning

confidence: 99%

“…We do not need harmonicity on the sphere, because we only look at the approximation quality there, but the smoothness of R enters crucially into what follows. We can use the standard literature on kernel-based methods on spheres [7,6] to derive error bounds on S for functions with Sobolev smoothness, and this provides convergence rates which are dominated by the smoothness of R and the boundary data, as is to be expected.…”

Section: Introductionmentioning

confidence: 99%

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Solving the 3D Laplace equation by meshless collocation via harmonic kernels

Hon

Schaback

2011

Adv Comput Math

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This paper solves the Laplace equation ∆u = 0 on domains Ω ⊂ R 3 by meshless collocation on scattered points of the boundary ∂Ω. In contrast to the Method of Fundamental Solutions, there are no singularities and no artificial boundaries, since we use new singularity-free positive definite kernels which are harmonic in both arguments. In contrast to many other techniques, e.g. the Boundary Point Method or the Method of Fundamental Solutions, we provide a solid mathematical foundation which includes error bounds and works for general star-shaped domains. The convergence rates depend only on the smoothness of the domain and the boundary data. Some numerical examples are included.

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“…There is quite some literature on how to find good approximations to data on the sphere by using interpolation by zonal kernels, see e.g. [7,6].…”

Section: Harmonic Functions In the Unit Ballmentioning

confidence: 99%

Section: Error Boundsmentioning

confidence: 99%

Section: Error Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Solving the 3D Laplace equation by meshless collocation via harmonic kernels

Hon

Schaback

2011

Adv Comput Math

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“…(13) If iter > 0, then set ζ 0 = ζ 1 . (14) Set ζ 1 = p · r. (15) Update the counter, iter = iter +1. (16) If iter = 1, then define p = p else p = p + (ζ 1 /ζ 0 )p. (17) Update the residual vector…”

Section: An Overlapping Additive Schwarz Algorithmmentioning

confidence: 99%

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

2009

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Abstract. We present an overlapping domain decomposition technique for solving elliptic partial differential equations on the sphere. The approximate solution is constructed using shifts of a strictly positive definite kernel on the sphere. The condition number of the Schwarz operator depends on the way we decompose the scattered set into smaller subsets. The method is illustrated by numerical experiments on relatively large scattered point sets taken from MAGSAT satellite data.

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The capability of approximation for neural networks interpolant on the sphere

Cao

Shao

2010

Math. Meth. Appl. Sci.

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Communicated by Y. XuCompared with planar hyperplane, fitting data on the sphere has been an important and active issue in geoscience, metrology, brain imaging, and so on. In this paper, using a functional approach, we rigorously prove that for given distinct samples on the unit sphere there exists a feed-forward neural network with single hidden layer which can interpolate the samples, and simultaneously near best approximate the target function in continuous function space. Also, by using the relation between spherical positive definite radial basis functions and the basis function on the Euclidean space R d+1 , a similar result in a spherical Sobolev space is established.

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Scattered Data Interpolation on Spheres: Error Estimates and Locally Supported Basis Functions

Cited by 118 publications

References 16 publications

Solving the 3D Laplace equation by meshless collocation via harmonic kernels

Solving the 3D Laplace equation by meshless collocation via harmonic kernels

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

The capability of approximation for neural networks interpolant on the sphere

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