Abstract

Surface modeling is closely related to interpolation and approximation by using level set methods, radial basis functions methods, and moving least squares methods. Although radial basis functions with global support have a very good approximation effect, this is often accompanied by an ill-conditioned algebraic system. The exceedingly large condition number of the discrete matrix makes the numerical calculation time consuming. The paper introduces a truncated exponential function, which is radial on arbitrary n-dimensional space R n and has compact support. The truncated exponential radial function is proven strictly positive definite on R n while internal parameter l satisfies l ≥ ⌊ n 2 ⌋ + 1 . The error estimates for scattered data interpolation are obtained via the native space approach. To confirm the efficiency of the truncated exponential radial function approximation, the single level interpolation and multilevel interpolation are used for surface modeling, respectively.

Highlights

  • Radial basis functions can be used to construct trial spaces that have high precision in arbitrary dimensions with arbitrary smoothness

  • Since the truncated radial basis function Φ is only in C0 (Ω × Ω), hkX,Ω is vanishing in the above error estimate of Theorem 5

  • Numerical experiments suggest that truncated exponential radial basis function (TERBF) interpolation is essentially faster than the scattered data interpolation with globally supported radial basis functions

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Summary

Introduction

Radial basis functions can be used to construct trial spaces that have high precision in arbitrary dimensions with arbitrary smoothness. The applications of RBFs (or so-called meshfree methods) can be found in many different areas of science and engineering, including geometric modeling with surfaces [1].The globally supported radial basis functions such as Gaussians or generalized (inverse) multiquadrics have excellent approximation properties. They often produce dense discrete systems, which tend to have poor conditioning and lead to a high computational cost. The radial basis functions with compact supports can lead to a very well conditioned sparse system. The goal of this work is to design a truncated exponential function that has compact support and is strictly positive definite and radial on arbitrary n-dimensional space Rn and to show the advantages of the truncated exponential radial function approximation for surface modeling

Radial Basis Functions
Multiply Monotonicity
Native Spaces
Truncated Exponential Function
Errors in Native Spaces
Single-Level Approximation
Multilevel Approximation
Conclusions
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