On greedy algorithm approximating Kolmogorov widths in Banach spaces

The greedy algorithm to produce n-dimensional subspaces Xn to approximate a compact set F contained in a Hilbert space was introduced in the context of reduced basis method in [12,13]. The same algorithm works for a general Banach space and in this context was studied in [4]. In this paper we study...

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Bibliographic Details
Published inJournal of mathematical analysis and applications Vol. 424; no. 1; pp. 685 - 695
Main Author Wojtaszczyk, P.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.04.2015
Subjects
Greedy algorithm
Kolmogorov widths
Non-linear approximation
Reduced basis method
Greedy algorithm
Kolmogorov widths
Non-linear approximation
Reduced basis method
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ISSN0022-247X
1096-0813
DOI10.1016/j.jmaa.2014.11.054

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Summary:The greedy algorithm to produce n-dimensional subspaces Xn to approximate a compact set F contained in a Hilbert space was introduced in the context of reduced basis method in [12,13]. The same algorithm works for a general Banach space and in this context was studied in [4]. In this paper we study the case F⊂Lp. If Kolmogorov diameters dn(F) of F decay as n−α we give an almost optimal estimate for the decay of σn:=dist(F,Xn). We also give some direct estimates of the form σn≤Cndn(F).
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2014.11.054