eprintid: 3123
rev_number: 6
eprint_status: archive
userid: 69
dir: disk0/00/00/31/23
datestamp: 2016-02-26 12:39:16
lastmod: 2016-02-26 12:39:16
status_changed: 2016-02-26 12:39:16
type: article
metadata_visibility: show
creators_name: Gnecco, Giorgio
creators_id: giorgio.gnecco@imtlucca.it
title: On the Curse of Dimensionality in the Ritz Method
ispublished: pub
subjects: QA75
divisions: CSA
full_text_status: none
keywords: Ritz method; Curse of dimensionality; Infinite-dimensional optimization; Approximation schemes; Extended Ritz method
note: First online: 03 September 2015
abstract: It is shown that the classical Ritz method of the calculus of variations suffers from the “curse of dimensionality,” i.e., an exponential growth, as a function of the number of variables, of the dimension a linear subspace needs in order to achieve a desired relative improvement in the accuracy of approximation of the optimal solution value. The proof is constructive and is obtained by exhibiting a family of infinite-dimensional optimization problems for which this happens, namely those with quadratic functional and spherical constraint. The results provide a theoretical motivation for the search of alternative solution methods, such as the so-called “extended Ritz method,” to deal with the curse of dimensionality.
date: 2016
date_type: published
publication: Journal of Optimization Theory and Applications
volume: 168
number: 2
publisher: Springer
pagerange: 488-509
id_number: 10.1007/s10957-015-0804-y
refereed: TRUE
issn: 0022-3239
official_url: http://link.springer.com/article/10.1007%2Fs10957-015-0804-y
citation:   Gnecco, Giorgio  On the Curse of Dimensionality in the Ritz Method.  Journal of Optimization Theory and Applications, 168 (2).  pp. 488-509.  ISSN 0022-3239      (2016)