WEIGHTED THIN PLATE SPLINES

2005 ◽  
Vol 03 (03) ◽  
pp. 297-324 ◽  
Author(s):  
A. BOUHAMIDI

A widely known method in multivariate interpolation and approximation theory consists of the use of thin plate splines. In this paper, we investigate some results and properties relative to a wide variety of variational splines in some space of functions arising from a nonnegative weight function. This model includes thin plate splines, splines in tension and discusses smoothing and interpolating splines. Pointwise error estimates are given for both problems.

2018 ◽  
Vol 39 (3) ◽  
pp. 1085-1109 ◽  
Author(s):  
R H Nochetto ◽  
D Ntogkas ◽  
W Zhang

Abstract In this paper we continue the analysis of the two-scale method for the Monge–Ampère equation for dimension d ≥ 2 introduced in the study by Nochetto et al. (2017, Two-scale method for the Monge–Ampère equation: convergence to the viscosity solution. Math. Comput., in press). We prove continuous dependence of discrete solutions on data that in turn hinges on a discrete version of the Alexandroff estimate. They are both instrumental to prove pointwise error estimates for classical solutions with Hölder and Sobolev regularity. We also derive convergence rates for viscosity solutions with bounded Hessians which may be piecewise smooth or degenerate.


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