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Full-Text Articles in Physical Sciences and Mathematics
Localized Bases For Kernel Spaces On The Unit Sphere, E. Fuselier, T. Hangelbroek, F. J. Narcowich, J. D. Ward, G. B. Wright
Localized Bases For Kernel Spaces On The Unit Sphere, E. Fuselier, T. Hangelbroek, F. J. Narcowich, J. D. Ward, G. B. Wright
Mathematics Faculty Publications and Presentations
Approximation/interpolation from spaces of positive definite or conditionally positive definite kernels is an increasingly popular tool for the analysis and synthesis of scattered data and is central to many meshless methods. For a set of N scattered sites, the standard basis for such a space utilizes N globally supported kernels; computing with it is prohibitively expensive for large N. Easily computable, well-localized bases with “small-footprint” basis elements—i.e., elements using only a small number of kernels—have been unavailable. Working on S2, with focus on the restricted surface spline kernels (e.g., the thin-plate splines restricted to the sphere), we …
Stability And Error Estimates For Vector Field Interpolation And Decomposition On The Sphere With Rbfs, Edward J. Fuselier, Grady Wright
Stability And Error Estimates For Vector Field Interpolation And Decomposition On The Sphere With Rbfs, Edward J. Fuselier, Grady Wright
Mathematics Faculty Publications and Presentations
A new numerical technique based on radial basis functions (RBFs) is presented for fitting a vector field tangent to the sphere, S2, from samples of the field at "scattered" locations on S2. The method naturally provides a way to decompose the reconstructed field into its individual Helmholtz–Hodge components, i.e., into divergence-free and curl-free parts, which is useful in many applications from the atmospheric and oceanic sciences (e.g., in diagnosing the horizontal wind and ocean currents). Several approximation results for the method will be derived. In particular, Sobolevtype error estimates are obtained for both the interpolant and …
Error And Stability Estimates For Surface-Divergence Free Rbf Interpolants On The Sphere, Edward J. Fuselier, Francis J. Narcowich, Joseph D. Ward, Grady Wright
Error And Stability Estimates For Surface-Divergence Free Rbf Interpolants On The Sphere, Edward J. Fuselier, Francis J. Narcowich, Joseph D. Ward, Grady Wright
Mathematics Faculty Publications and Presentations
Recently, a new class of surface-divergence free radial basis function interpolants has been developed for surfaces in R3. In this paper, several approximation results for this class of interpolants will be derived in the case of the sphere, S2. In particular, Sobolev-type error estimates are obtained, as well as optimal stability estimates for the associated interpolation matrices. In addition, a Bernstein estimate and an inverse theorem are also derived. Numerical validation of the theoretical results is also given.