Radial basis interpolation on homogeneous manifolds: convergence rates

Abstract

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.

Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates

“…Since M is a smooth and compact manifold, we get a smooth atlas A = {(Ψ j , U j )} containing finitely many charts for M. By refining our charts as necessary, we may assume that the images of these charts is equal to a ball in R k centered around the origin. Since our manifold is smooth and the images of our charts are convex, the arguments in the proof of [22,Proposition 3.3] follow to find positive constants c 1,j and c 2,j such that…”

“…Although kernel approximants were initially considered with the domain being a Euclidean space or a sphere [43], the ideas have since been generalized enough to handle functions defined on other mathematical objects. Kernels have been studied that are positive definite on various Riemannian manifolds [18,27,22], and there are also kernels that exploit the group structure of their underlying manifold, with domains such Lie groups, projective spaces, and motion groups [12,14,47]. These kernels are obtained in various ways in the literature.…”

Error Bounds for a Least Squares Meshless Finite Difference Method on Closed Manifolds

“…If an orthonormal basis is known for a Sobolev space, for example for L 2 (M) = H 0 (M), then reproducing kernels can be constructed via infinite series, see e.g. [33,36] and [57,Section 17.4]. We discuss this in somewhat more detail in Section 2.3.…”

“…Meshless numerical methods are particularly attractive for solving operator equations on manifolds, in particular on the sphere [22,23], because of the difficulties of creating and maintaining suitable meshes or grids in this setting. Error bounds for meshless methods for the approximation of functions and specifically solutions of operator equations on manifolds have been studied for example in [24,26,27,32,33,35,36,37,43]. However, these results do not apply to localized finite difference type methods considered below.…”

Colocación con funciones de base radial para la solución de modelos matemáticos en fenoménos de transporte

“…Una función de base radial (Radial basis function RBF), es una función real cuyo valor depende de la distancia entre dos puntos o nodos, un nodo fuente que actúa como centro y un nodo de campo o punto donde se evalúa la función. La norma usada para la distancia es frecuentemente la norma euclidiana aunque otras pueden ser admisibles en aplicaciones especificas (Aftab, Moinuddin, y Shaikh, 2014) o en problemas sobre manifolds (Hubbert, L.Gia, y Morton, 2015;Levesley y Ragozin, 2007).…”