Within kernel-based interpolation and its many applications, the handling of the scaling or the shape parameter is a well-documented but unsolved problem. We consider native spaces whose kernels allow us to change the kernel scale of a d-variate interpolation problem locally, depending on the requirements of the application. The trick is to define a scale function c on the domain Ω Rd to transform an interpolation problem from data locations xj in Rd to data locations (xj, c(xj)) and to use a fixed-scale kernel on Rd+1 for interpolation there. The (d+1)-variate solution is then evaluated at (x, c(x)) for x Rd to give a d-variate interpolant with a varying scale. A large number of examples show how this can be done in practice to get results that are better than the fixed-scale technique, with respect to both condition number and error. The background theory coincides with fixed-scale interpolation on the submanifold of Rd+1 given by the points (x, c(x)) of the graph of the scale function c.

Bozzini, M., Lenarduzzi, L., Rossini, M., Schaback, R. (2015). Interpolation with variably scaled kernels. IMA JOURNAL OF NUMERICAL ANALYSIS, 35(1), 199-219 [10.1093/imanum/drt071].

Interpolation with variably scaled kernels

BOZZINI, MARIA TUGOMIRA;ROSSINI, MILVIA FRANCESCA
;
2015

Abstract

Within kernel-based interpolation and its many applications, the handling of the scaling or the shape parameter is a well-documented but unsolved problem. We consider native spaces whose kernels allow us to change the kernel scale of a d-variate interpolation problem locally, depending on the requirements of the application. The trick is to define a scale function c on the domain Ω Rd to transform an interpolation problem from data locations xj in Rd to data locations (xj, c(xj)) and to use a fixed-scale kernel on Rd+1 for interpolation there. The (d+1)-variate solution is then evaluated at (x, c(x)) for x Rd to give a d-variate interpolant with a varying scale. A large number of examples show how this can be done in practice to get results that are better than the fixed-scale technique, with respect to both condition number and error. The background theory coincides with fixed-scale interpolation on the submanifold of Rd+1 given by the points (x, c(x)) of the graph of the scale function c.
Articolo in rivista - Articolo scientifico
positive-definite radial basis functions; scaling; shape parameter
English
2015
35
1
199
219
none
Bozzini, M., Lenarduzzi, L., Rossini, M., Schaback, R. (2015). Interpolation with variably scaled kernels. IMA JOURNAL OF NUMERICAL ANALYSIS, 35(1), 199-219 [10.1093/imanum/drt071].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/53809
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