This is mainly a review paper, concerned with some applications of the concept of Nonlinear Approximation to adaptive convex minimization. At first, we recall the basic ideas and we compare linear to nonlinear approximation for three relevant families of bases used in practice: Fourier bases, finite element bases, wavelet bases. Next, we show how nonlinear approximation can be used to design rigorously justified and optimally efficient adaptive methods to solve abstract minimization problems in Banach spaces, using either wavelet or finite element bases. In particular, a wavelet adaptive steepest-descent algorithm is presented and investigated.
In questo articolo, a prevalente carattere di rassegna, si considerano varie applicazioni del concetto di Approssimazione Nonlineare alla minimizzazione convessa adattativa. Dapprima, si ricordano alcuni concetti di base e si confrontano l'approssimazione lineare e quella nonlineare nel caso di tre basi funzionali notevoli: la base di Fourier, le basi degli elementi finiti e le basi di ondine. Successivamente, indichiamo come l'approssimazione nonlineare possa essere usata nella definizione di metodi adattativi per la risoluzione di problemi di minimizzazione astratta in spazi di Banach. Gli algoritmi risultanti, che impiegano sia basi di ondine sia basi di elementi finiti, risultano rigorosamente giustificabili e con proprietà di ottimalità dal punto di vista dell'efficienza. In questo ambito, si descrive con un qualche dettaglio un algoritmo di «steepest-descent» per discretizzazioni in ondine.
@article{BUMI_2003_8_6B_2_263_0,
author = {Claudio Canuto},
title = {Adaptive convex optimization in Banach spaces: a multilevel approach},
journal = {Bollettino dell'Unione Matematica Italiana},
volume = {6-A},
year = {2003},
pages = {263-287},
zbl = {1177.42028},
mrnumber = {1988205},
language = {en},
url = {http://dml.mathdoc.fr/item/BUMI_2003_8_6B_2_263_0}
}
Canuto, Claudio. Adaptive convex optimization in Banach spaces: a multilevel approach. Bollettino dell'Unione Matematica Italiana, Tome 6-A (2003) pp. 263-287. http://gdmltest.u-ga.fr/item/BUMI_2003_8_6B_2_263_0/
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