doi:10.1016/j.camwa.2006.06.005 
Copyright © 2007 Elsevier Ltd All rights reserved.
Adaptive residual subsampling methods for radial basis function interpolation and collocation problems
aDepartment of Mathematical Sciences, University of Delaware, Newark, DE, 19716, United States
Received 31 January 2006;
revised 1 June 2006;
accepted 15 June 2006.
Available online 6 April 2007.
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Abstract
We construct a new adaptive algorithm for radial basis functions (RBFs) method applied to interpolation, boundary-value, and initial-boundary-value problems with localized features. Nodes can be added and removed based on residuals evaluated at a finer point set. We also adapt the shape parameters of RBFs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix. The performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains.
Keywords: Adaptive; Radial basis functions; Interpolation; Collocation; Residual subsampling
Fig. 1. Left: Runge function with final RBF node distribution. Right: Error convergence of residual subsampling method compared to standard Chebyshev interpolation (dashes).
Fig. 2. An example of geometric refinement for some domain Ω. Initially, coarse collection consisting of 4×4 nonoverlapping boxes is used to cover the domain and then the domain is refined up to first generation. Accepted RBF centers are shown as dots, and interior residual check points are crosses.
Fig. 3. Left: tanh(60x−.01). Right: Error convergence of residual subsampling method compared to standard Chebyshev interpolation (dashes).
Fig. 4. Left: |x+.04|, the box-based method adaptively refines centers around corners. Right: Error convergence of residual subsampling method compared to standard Chebyshev interpolation (dashes).
(a) 
g=102.
(b) g=4.
Fig. 5. Local and global features of modified Franke function.
Fig. 6. Centers are located neatly around regions with steep gradients.
Fig. 7. Left: Numerical solution of stationary viscous Burgers’ equation with Robin condition. Right: Solid lines are the absolute error with respect to the exact solution and dots are pointwise residuals of the PDE at check points.
Fig. 8. Numerical solution of stationary Allen–Cahn with parameter ν=10−4.
Fig. 9. Numerical solution of Poisson’s equation in starfish-like domain.
Fig. 10. Numerical solution of Poisson’s equation in L-shaped domain.
Fig. 11. Solution of the Allen–Cahn equation (11) using a model of alternating time stepping and adaptation by residual subsampling of the approximate solution. Each line in the figure represents a time at which adaptation has occurred. The nodes clearly adapt themselves to emerging steep gradients.
Fig. 12. Solution of the Burgers’ equation using a model of alternating time stepping and adaptation by residual subsampling of the approximate solution. Each line in the figure represents a time at which adaptation has occurred.
Fig. 13. Numerical solution of time-dependent Burgers’ equation using direct implementation of residual subsampling method in space–time.
Table 1.
Iterative progress of the refinement algorithm for interpolation of tanh(60x−0.1)
Nr,Nc = Number of centers to be added/removed respectively.
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