Kernel-based methods provide flexible and accurate algorithms for
the reconstruction of functions from meshless samples. A major question
in the use of such methods is the influence of the samples� locations
on the behavior of the approximation, and feasible optimal strategies
are not known for general problems. Nevertheless, efficient and greedy
point-selection strategies are known. This paper gives a proof of
the convergence rate of the data-independent $P$-greedy algorithm,
based on the application of the convergence theory for greedy algorithms
in reduced basis methods. The resulting rate of convergence is shown
to be quasi-optimal in the case of kernels generating Sobolev spaces.
As a consequence, this convergence rate proves that, for kernels
of Sobolev spaces, the points selected by the algorithm are asymptotically
uniformly distributed, as conjectured in the paper where the algorithm
has been introduced
%0 Journal Article
%1 santin2017convergence
%A Santin, G.
%A Haasdonk, B.
%D 2017
%J Dolomites Research Notes on Approximation
%K from:mhartmann ians imported vorlaeufig
%P 68--78
%T Convergence rate of the data-independent P-greedy algorithm in
kernel-based approximation
%U http://www.emis.de/journals/DRNA/9-2.html
%V 10
%X Kernel-based methods provide flexible and accurate algorithms for
the reconstruction of functions from meshless samples. A major question
in the use of such methods is the influence of the samples� locations
on the behavior of the approximation, and feasible optimal strategies
are not known for general problems. Nevertheless, efficient and greedy
point-selection strategies are known. This paper gives a proof of
the convergence rate of the data-independent $P$-greedy algorithm,
based on the application of the convergence theory for greedy algorithms
in reduced basis methods. The resulting rate of convergence is shown
to be quasi-optimal in the case of kernels generating Sobolev spaces.
As a consequence, this convergence rate proves that, for kernels
of Sobolev spaces, the points selected by the algorithm are asymptotically
uniformly distributed, as conjectured in the paper where the algorithm
has been introduced
@article{santin2017convergence,
abstract = {Kernel-based methods provide flexible and accurate algorithms for
the reconstruction of functions from meshless samples. A major question
in the use of such methods is the influence of the samples� locations
on the behavior of the approximation, and feasible optimal strategies
are not known for general problems. Nevertheless, efficient and greedy
point-selection strategies are known. This paper gives a proof of
the convergence rate of the data-independent $P$-greedy algorithm,
based on the application of the convergence theory for greedy algorithms
in reduced basis methods. The resulting rate of convergence is shown
to be quasi-optimal in the case of kernels generating Sobolev spaces.
As a consequence, this convergence rate proves that, for kernels
of Sobolev spaces, the points selected by the algorithm are asymptotically
uniformly distributed, as conjectured in the paper where the algorithm
has been introduced},
added-at = {2018-07-20T10:55:06.000+0200},
author = {Santin, G. and Haasdonk, B.},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2ca770b9c5d18c1ebc84a598fa8aad0cb/mathematik},
file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/SH16b_www_p-greedy.pdf:PDF},
interhash = {059244121289561b790e31929cc535d6},
intrahash = {ca770b9c5d18c1ebc84a598fa8aad0cb},
journal = {Dolomites Research Notes on Approximation},
keywords = {from:mhartmann ians imported vorlaeufig},
owner = {haasdonk},
pages = {68--78},
timestamp = {2019-12-18T14:37:55.000+0100},
title = {Convergence rate of the data-independent {P}-greedy algorithm in
kernel-based approximation},
url = {http://www.emis.de/journals/DRNA/9-2.html},
volume = 10,
year = 2017
}
