Computing Reviews
Today's Issue Hot Topics Search Browse Recommended My Account Log In
Review Help
Search
Convexity and solvability for compactly supported radial basis functions with different shapes
Zhu S., Wathen A. Journal of Scientific Computing63 (3):862-884,2015.Type:Article
Date Reviewed: Aug 28 2015

Radial functions depend only on the distance from the center of the domain, for example, Gaussians. Interpolations using radial functions of the same shape lead to a non-singular coefficient matrix. Unfortunately, when using adaptive mesh, the shape of radial functions is not the same. This in turn leads to an ill-conditioned system. The authors give sufficient conditions to guarantee a diagonally dominant interpolation matrix when using radial functions with different shapes.

A radial function for which the first derivative is negative and the second derivative is non-negative is called convex. The authors focus on Wendland functions, a set of compactly supported radial basis functions (RBFs) (see Wendland [1]).

They show that the interpolation matrix is non-singular under a local geometric property of the neighborhood of each center. The interpolation matrices guarantee incomplete lower upper (LU) factorization. The authors give MATLAB code to construct the interpolation matrix with differently scaled RBFs. They give several examples in 2D and 3D surface reconstruction (see also Morse et al. [2]). The algorithm was demonstrated to be efficient for large-scale problems but not very accurate. Error estimates were not given.

Reviewer:  Beny Neta Review #: CR143728 (1511-0968)
1) Wendland, H. Scattered data approximation. Cambridge University Press, Cambridge, UK, 2005.
2) Morse, B. S.; Yoon, T. S.; Rheingans, P.; Chen, D. T.; Subramanian, K. R. Interpolating implicit surfaces from scattered data using compactly supported radial basis functions. In Proc. of the International Conference on Shape Modeling and Applications. IEEE, 2001, 1–10.
Bookmark and Share
  Featured Reviewer  
 
Interpolation (G.1.1 )
 
 
Numerical Algorithms (G.1.0 ... )
 
Would you recommend this review?
yes
no
Other reviews under "Interpolation": Date
Systolic computation of interpolating polynomials
Cappello P., Koç Ç., Gallopoulos E. Computing 45(2): 95-117, 2000. Type: Article
Jun 1 1991
Interpolation of data on the surface of a sphere
Renka R. (ed) ACM Transactions on Mathematical Software 10(4): 417-436, 1984. Type: Article
Nov 1 1985
Incremental linear interpolation
Field D. ACM Transactions on Graphics (TOG) 4(1): 1-11, 1985. Type: Article
Jun 1 1986
more...

E-Mail This Printer-Friendly
Send Your Comments
Contact Us
Reproduction in whole or in part without permission is prohibited.   Copyright 1999-2024 ThinkLoud®
Terms of Use
| Privacy Policy