In recent years, in the setting of radial basis function, the study
of approximation algorithms has particularly focused on the construction
of (stable) bases for the associated Hilbert spaces. One of the ways
of describing such spaces and their properties is the study of a
particular integral operator and its spectrum. We proposed in a recent
work the so-called WSVD basis, which is strictly connected to the
eigen-decomposition of this operator and allows to overcome some
problems related to the stability of the computation of the approximant
for a wide class of radial kernels. Although effective, this basis
is computationally expensive to compute. In this paper we discuss
a method to improve and compute in a fast way the basis using methods
related to Krylov subspaces. After reviewing the connections between
the two bases, we concentrate on the properties of the new one, describing
its behavior by numerical tests.
%0 Journal Article
%1 demarchi2015computation
%A De Marchi, Stefano
%A Santin, Gabriele
%D 2015
%I Springer Netherlands
%J BIT Numerical Mathematics
%K 15A23; 41A05; 42A82; 65Y20 Fast Interpolation; Matrix Positive computation; definite factorization; functions; vorlaeufig
%N 4
%P 949--966
%R 10.1007/s10543-014-0537-6
%T Fast computation of orthonormal basis for RBF spaces through Krylov
space methods
%U http://dx.doi.org/10.1007/s10543-014-0537-6
%V 55
%X In recent years, in the setting of radial basis function, the study
of approximation algorithms has particularly focused on the construction
of (stable) bases for the associated Hilbert spaces. One of the ways
of describing such spaces and their properties is the study of a
particular integral operator and its spectrum. We proposed in a recent
work the so-called WSVD basis, which is strictly connected to the
eigen-decomposition of this operator and allows to overcome some
problems related to the stability of the computation of the approximant
for a wide class of radial kernels. Although effective, this basis
is computationally expensive to compute. In this paper we discuss
a method to improve and compute in a fast way the basis using methods
related to Krylov subspaces. After reviewing the connections between
the two bases, we concentrate on the properties of the new one, describing
its behavior by numerical tests.
@article{demarchi2015computation,
abstract = {In recent years, in the setting of radial basis function, the study
of approximation algorithms has particularly focused on the construction
of (stable) bases for the associated Hilbert spaces. One of the ways
of describing such spaces and their properties is the study of a
particular integral operator and its spectrum. We proposed in a recent
work the so-called WSVD basis, which is strictly connected to the
eigen-decomposition of this operator and allows to overcome some
problems related to the stability of the computation of the approximant
for a wide class of radial kernels. Although effective, this basis
is computationally expensive to compute. In this paper we discuss
a method to improve and compute in a fast way the basis using methods
related to Krylov subspaces. After reviewing the connections between
the two bases, we concentrate on the properties of the new one, describing
its behavior by numerical tests.},
added-at = {2018-07-20T10:54:15.000+0200},
author = {De Marchi, Stefano and Santin, Gabriele},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2b881c5bd45bba967a418431f264925f3/mhartmann},
doi = {10.1007/s10543-014-0537-6},
file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/DeMSa2015_www_Fast_computation_onb_RBF.pdf:PDF},
interhash = {5fc476059e77e8e9226f6a15761cf6a5},
intrahash = {b881c5bd45bba967a418431f264925f3},
issn = {0006-3835},
journal = {BIT Numerical Mathematics},
keywords = {15A23; 41A05; 42A82; 65Y20 Fast Interpolation; Matrix Positive computation; definite factorization; functions; vorlaeufig},
language = {English},
number = 4,
owner = {santinge},
pages = {949--966},
publisher = {Springer Netherlands},
timestamp = {2018-07-20T08:54:15.000+0200},
title = {Fast computation of orthonormal basis for RBF spaces through Krylov
space methods},
url = {http://dx.doi.org/10.1007/s10543-014-0537-6},
volume = 55,
year = 2015
}