A Simple Approach to the Variational Theory for Interpolation on Spheres

Levesley, Jeremy; Light, Will; Ragozin, David L.; Sun, Xingping

doi:10.1007/978-3-0348-8696-3_9

Cited by 18 publications

(35 citation statements)

References 12 publications

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“…The analysis is then carried out for functions in a more restricted setting, and a doubled rate of convergence is obtained (see Theorem 6.13 and the subsequent corollaries for the final results), using a relatively simple analysis. (In brief, a modified kernel and modified native space close to those used in [10] are defined, with respect to which the approximation is an orthogonal projection. Schaback 's trick [14] then yields the doubled convergence rate for sufficiently smooth functions.…”

Section: Introductionmentioning

confidence: 99%

Approximation on the sphere using radial basis functions plus polynomials

Sloan

Sommariva

2007

Adv Comput Math

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In this paper we analyse a hybrid approximation of functions on the sphere S^2 ⊂ R^3 by radial basis functions combined with polynomials, with the radial basis functions assumed to be generated by a (strictly) positive definite kernel. The approximation is determined by interpolation at scattered data points, supplemented by side conditions on the coefficients to ensure a square linear system. The analysis is first carried out in the native space associated with the kernel (with no explicit polynomial component, and no side conditions). A more refined error estimate is obtained for functions in a still smaller space. Numerical calculations support the utility of this hybrid approximation

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Section: Introductionmentioning

confidence: 99%

Approximation on the sphere using radial basis functions plus polynomials

Sloan

Sommariva

2007

Adv Comput Math

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“…The proof techniques in [8] are completely different from ours, since they map the sphere S d with charts onto subsets of R d and make use of results for the Euclidean case. In [9] and [5], global error estimates for radial basis function interpolation on S 2 and S d , respectively, were proved in terms of the global mesh norm. A closer inspection of the proof, however, shows that the same proof does also imply local error estimates on a spherical cap in terms of the local mesh norm with respect to a small neighbourhood of the spherical cap.…”

Section: Introductionmentioning

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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“…There are a number of different approximation methods currently available on spheres, including wavelets [5], piecewise polynomial splines [1], and the subject of this paper, radial functions (sometimes called zonal splines or variational splines) [5,8]. Error estimates and convergence rates for radial approximation on spheres, of an optimal nature, are recent in vintage [6,7], and rely on some technically demanding mathematics.…”

Section: Introductionmentioning

confidence: 99%

Radial basis interpolation on homogeneous manifolds: convergence rates

Levesley

Ragozin

2007

Adv Comput Math

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Pointwise error estimates for approximation on compact homogeneous manifolds using radial kernels are presented. For a C 2r positive definite kernel κ the pointwise error at x for interpolation by translates of κ goes to 0 like ρ r , where ρ is the density of the interpolating set on a fixed neighbourhood of x. Tangent space techniques are used to lift the problem from the manifold to Euclidean space, where methods for proving such error estimates are well established.

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A Simple Approach to the Variational Theory for Interpolation on Spheres

Cited by 18 publications

References 12 publications

Approximation on the sphere using radial basis functions plus polynomials

Approximation on the sphere using radial basis functions plus polynomials

Local Radial Basis Function Approximation on the Sphere

Radial basis interpolation on homogeneous manifolds: convergence rates

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