Error Estimates for Interpolation by Compactly Supported Radial Basis Functions of Minimal Degree

Wendland, Holger

doi:10.1006/jath.1997.3137

Cited by 315 publications

(162 citation statements)

References 12 publications

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“…We describe several strategies for specifying the modulating function K. The first is to use a tapering function, the result is referred to as FSA-Taper, which sets the correlation of distant spatio-temporal pairs to zero. In the univariate spatial case, a number of compactly supported covariance functions have been used for covariance tapering, for example the spherical covariance function, the family of Wendland covariance functions, and the bisquare function, to name a few (Wendland (1995(Wendland ( , 1998Gneiting (2002); Cressie and Johannesson (2008)). In the spatio-temporal context, we consider tapering functions as Schur products of a purely spatial and a purely temporal tapering function.…”

Section: Covariance Approximation For Large Computation Of Spatiotempmentioning

confidence: 99%

Full-Scale Approximations of Spatio-Temporal Covariance Models for Large Datasets

Zhang¹,

Sang²,

Huang³

2014

STAT SINICA

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. (2015). Full-scale approximations of spatio-temporal covariance models for large datasets. Statistica Sinica, 25 99-114. Full-scale approximations of spatio-temporal covariance models for large datasets AbstractVarious continuously-indexed spatio-temporal process models have been constructed to characterize spatiotemporal dependence structures, but the computational complexity for model fitting and predictions grows in a cubic order with the size of dataset and application of such models is not feasible for large datasets. This article extends the full-scale approximation (FSA) approach by Sang and Huang (2012) to the spatiotemporal context to reduce computational complexity. A reversible jump Markov chain Monte Carlo (RJMCMC) algorithm is proposed to select knots automatically from a discrete set of spatio-temporal points. Our approach is applicable to nonseparable and nonstationary spatio-temporal covariance models. We illustrate the effectiveness of our method through simulation experiments and application to an ozone measurement dataset. Disciplines Engineering | Science and Technology Studies FULL-SCALE APPROXIMATIONS OF SPATIO-TEMPORAL COVARIANCE MODELS FOR LARGE DATASETSBohai Zhang, Huiyan Sang and Jianhua Z. Huang Texas A&M UniversityAbstract: Various continuously-indexed spatio-temporal process models have been constructed to characterize spatio-temporal dependence structures, but the computational complexity for model fitting and predictions grows in a cubic order with the size of dataset and application of such models is not feasible for large datasets. This article extends the full-scale approximation (FSA) approach by Sang and Huang (2012) to the spatio-temporal context to reduce computational complexity. A reversible jump Markov chain Monte Carlo (RJMCMC) algorithm is proposed to select knots automatically from a discrete set of spatio-temporal points. Our approach is applicable to nonseparable and nonstationary spatio-temporal covariance models. We illustrate the effectiveness of our method through simulation experiments and application to an ozone measurement dataset.

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Section: Covariance Approximation For Large Computation Of Spatiotempmentioning

confidence: 99%

Full-Scale Approximations of Spatio-Temporal Covariance Models for Large Datasets

Zhang¹,

Sang²,

Huang³

2014

STAT SINICA

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“…It is shown in [28,Corollary 2.3] that Ψ n+1,m ∈ C 2m (R n+1 ). For any given N and any set of N pairwise distinct points {x 1 , .…”

Section: Spherical Basis Functionsmentioning

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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“…In order to illustrate how a suitable choice of a trust radius can be found experimentally we consider the following Example 4.7. Let ψ be the multiple of Wendland's function φ 1,3 (see [15]), given by…”

Section: Proposition 43 Leads Immediately To the Followingmentioning

confidence: 99%

A symmetric collocation method with fast evaluation

Johnson

2008

IMA Journal of Numerical Analysis

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Abstract. Symmetric collocation, which can be used to numerically solve linear partial differential equations, is a natural generalization of the well-established scattered data interpolation method known as radial basis function (rbf) interpolation. As with rbf interpolation, a major shortcoming of symmetric collocation is the high cost, in terms of floating point operations, of evaluating the obtained function. When solving a linear partial differential equation, one usually has some freedom in choosing the collocation points. We explain how this freedom can be exploited to allow the fast evaluation of the obtained function provided the basic function is chosen as a tensor product of compactly supported piecewise polynomials. Our proposed fast evaluation method, which is exact in exact arithmetic, is initially designed and analyzed in the univariate case. The multivariate case is then reduced, recursively, to multiple univariate evaluations. Along with the theoretical development of the method, we report the results of selected numerical experiments which help to clarify expectations.

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Error Estimates for Interpolation by Compactly Supported Radial Basis Functions of Minimal Degree

Cited by 315 publications

References 12 publications

Full-Scale Approximations of Spatio-Temporal Covariance Models for Large Datasets

Full-Scale Approximations of Spatio-Temporal Covariance Models for Large Datasets

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

A symmetric collocation method with fast evaluation

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