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Constructive Approximation

, Volume 45, Issue 3, pp 497–519 | Cite as

Discrete Least-Squares Approximations over Optimized Downward Closed Polynomial Spaces in Arbitrary Dimension

  • Albert CohenEmail author
  • Giovanni Migliorati
  • Fabio Nobile
Article

Abstract

We analyze the accuracy of the discrete least-squares approximation of a function \(u\) in multivariate polynomial spaces \(\mathbb {P}_\varLambda :=\mathrm{span} \{y\mapsto y^\nu : \nu \in \varLambda \}\) with \(\varLambda \subset \mathbb {N}_0^d\) over the domain \(\varGamma :=[-1,1]^d\), based on the sampling of this function at points \(y^1,\ldots ,y^m \in \varGamma \). The samples are independently drawn according to a given probability density \(\rho \) belonging to the class of multivariate beta densities, which includes the uniform and Chebyshev densities as particular cases. Motivated by recent results on high-dimensional parametric and stochastic PDEs, we restrict our attention to polynomial spaces associated with downward closed sets \(\varLambda \) of prescribed cardinality n, and we optimize the choice of the space for the given sample. This implies, in particular, that the selected polynomial space depends on the sample. We are interested in comparing the error of this least-squares approximation measured in \(L^2(\varGamma ,d\rho )\) with the best achievable polynomial approximation error when using downward closed sets of cardinality n. We establish conditions between the dimension n and the size m of the sample, under which these two errors are proved to be comparable. Our main finding is that the dimension d enters only moderately in the resulting trade-off between m and n, in terms of a logarithmic factor \(\ln (d)\), and is even absent when the optimization is restricted to a relevant subclass of downward closed sets, named anchored sets. In principle, this allows one to use these methods in arbitrarily high or even infinite dimension. Our analysis builds upon (Chkifa et al. in ESAIM Math Model Numer Anal 49(3):815–837, 2015), which considered fixed and nonoptimized downward closed multi-index sets. Potential applications of the proposed results are found in the development and analysis of efficient numerical methods for computing the solution to high-dimensional parametric or stochastic PDEs, but are not limited to this area.

Keywords

Multivariate polynomial approximation Discrete least squares Convergence rate Best n-term approximation Downward closed set 

Mathematics Subject Classification

41A10 41A25 41A50 41A63 65M70 

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Albert Cohen
    • 1
    Email author
  • Giovanni Migliorati
    • 1
  • Fabio Nobile
    • 2
  1. 1.Laboratoire Jacques-Louis Lions, CNRS, UMR 7598, UPMC Univ Paris 06Sorbonne UniversitésParisFrance
  2. 2.MATH-CSQIÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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