Error Estimates and Convergence Rates for Variational Hermite Interpolation

Abstract

This paper considers the variational problem of Hermite interpolation and its error bounds. The optimal Hermite interpolant, which minimises the semi-norm of the reproducing kernel Hilbert space C h determined by given r-CPD m function h, is just the h-spline Hermite interpolant. The results on error estimation and convergence rate of the h-spline interpolant generalise those of W.

Kernel techniques: From machine learning to meshless methods

“…If the recovery problem (3.1) is generalized to (3.3), there is a similar theory (Wu 1992, Luo andLevesley 1998) concerning optimal recovery, replacing the kernel matrix with entries K(x j , x k ) by a symmetric matrix with elements λ x j λ y k K(x, y), where we used an upper index x at λ x to indicate that the functional λ acts with respect to the variable x. The system (3.5) becomes…”

Error Bounds for Solving Pseudodifferential Equations on Spheres by Collocation with Zonal Kernels

“…For our purposes, the approach taken in [14] will prove most useful. In addition, the work presented here also draws on some ideas from other recent sources on interpolation in R m [18,23], solving differential equations in R m using collocation [4,7,8], and spherical interpolation and approximation [5,10,17]. The system of Eqs.…”

Multiquadric quasi‐interpolation methods for solving partial differential algebraic equations

“…When the Gaussian elimination with pivoting is applied to solve the linear equations, Tables I and II compare the performances of different methods with regular points. It can be seen that ICN-QIE is more accurate than ICN-QID, ICN-Kansa and ICN-HCM in this case, where ICN-Kansa method refers to the implicit Crank-Nicolson with Kansa's method by multiquadrics as a RBF; ICN-HCM method refers to the implicit Crank-Nicolson with the HCM by multiquadrics as a RBF ( for details see [23][24][25][26][27][28]).…”

“…Multiquadric functions which are a class of RBFs were proposed by Hardy [8]. Using multiquadric functions for solving partial differential equations (PDEs), there are many so-called meshless collocation methods have been investigated and developed successfully, see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. However, a big obstacle for the meshless collocation method is that the companion matrix is generally ill-conditioned, nonsymmetric and full dense matrix, which constrains the applicability of the method to solve large-scale problems.…”