Greedy Algorithms for Reduced Bases in Banach Spaces

By Ronald A. DeVore, Guergana Petrova, Przemyslaw Wojtaszczyk. Constructive approximation, 2012

191 citations recorded by the source corpus.

Given a Banach space X and one of its compact sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{F}$\end{document}, we consider the problem of finding a good n-dimensional space Xn⊂X which can be used to approximate the elements of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{F}$\end{document}. The best possible error we can achieve for such an approximation is given by the Kolmogorov width \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d_{n}(\mathcal{F})_{X}$\end{document}. However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in Buffa et al. (Modél. Math. Anal. Numér. 46:595–603, 2012) in the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X=\mathcal{H}$\end{document} is a Hilbert space. The results of Buffa et al. (Modél. Math. Anal. Numér. 46:595–603, 2012) were significantly improved upon in Binev et al. (SIAM J. Math. Anal. 43:1457–1472, 2011). The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/50e80e11402fd0ed11c0a12dc1401018f8e0c1e2

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Citations in the corpus

Sparse-grid, reduced-basis Bayesian inversion (2015, background): “However, this argument of proof [23] for Banach spaces X and Y, only a convergence rate estimate for the RB error of the form CN, α ≥ 1/2, which is inferior to dN ≤ CN.”
A parameterized‐background data‐weak approach to variational data assimilation: formulation, analysis, and application to acoustics (2015, methodology): “The algorithm has been proven to generate an optimal sequence of spaces with respect to the Kolmogorov width of Mbk in [5, 25, 26].”
Neural network approximation (2020, background): “The reader interested in the more general Banach space setting can consult (DeVore et al. 2013).”
Low-rank methods for high-dimensional approximation and model order reduction (2015, unknown): “as presented in section 5.1 (replacing kw(y) − u(y)kV by kB(y)w(y) − f(y)kC(y)). This is the so-called oﬄine phase of Reduced Basis methods, and convergence results for this algorithm can be found in [10, 12, 32], where the error ku−urk∞,K = supy∈Ξ K ku(y)− PVru(y)kV is compared with the best rank-r”
Approximation of high-dimensional parametric PDEs * (2015, methodology): “These results were improved upon in [32] and extended to the case of a general Banach space V .”
A Stochastic Collocation Algorithm with Multifidelity Models (2014, background): “to the manifold u(Γ) that, as N increases, decays commensurately with the N -width [1, 5].”