The power of standard information for multivariate approximation in the randomized setting

Wasilkowski, Grzegorz W.; Woźniakowski, Henryk

doi:10.1090/s0025-5718-06-01944-2

Cited by 38 publications

(59 citation statements)

References 33 publications

(51 reference statements)

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“…It is unknown if similar relations hold for = std . We need to discuss a recent result from [17] that provides the relation between the nth minimal errors e wor (n; L 2,ρ , all ) and e ran (n; L 2,ρ , std ). This result states that, modulo a power of the double logarithm of n, the minimal errors behave in a similar way.…”

Section: Known Results For Weighted L 2 Approximationmentioning

confidence: 99%

“…In the randomized setting, we allow the sampling points x j and/or the functions a j in (1.1) to be randomly selected. The following result has been established in [17]:…”

Section: Introductionmentioning

confidence: 95%

“…Such problems have been usually studied in the worst case and randomized settings with the error measured in a weighted L 2 norm. It is known that the results in the randomized and worst case settings are practically the same for a weighted L 2 norm, see [9,16,17]. There are also papers where the worst case or randomized setting is studied for specific Banach spaces that are not Hilbert.…”

Section: Introductionmentioning

confidence: 99%

“…The aim of the present paper is to study the power of standard information for L ∞ approximation problem-a problem that is typically more challenging than L 2,ρ approximation problems. Based on [5,6,17], we show that for spaces F satisfying certain assumptions, see (A2) and (A3) in Sect. 4, we have…”

Section: Introductionmentioning

confidence: 99%

“…The proofs of this paper follow the techniques from a number of papers including [5,6,17]. The results of this paper are constructive and provide efficient randomized algorithms for a general class of reproducing kernel Hilbert spaces.…”

Section: Introductionmentioning

confidence: 99%

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On the power of standard information for L ∞ approximation in the randomized setting

2009

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We study approximation of multivariate functions from a general separable reproducing kernel Hilbert space in the randomized setting with the error measured in the L ∞ norm. We consider algorithms that use standard information consisting of function values or general linear information consisting of arbitrary linear functionals. The power of standard or linear information is defined as, roughly speaking, the optimal rate of convergence of algorithms using n function values or linear functionals. We prove under certain assumptions that the power of standard information in the randomized setting is at least equal to the power of linear information in the worst case setting, and that the powers of linear and standard information in the randomized setting differ at most by 1/2. These assumptions are satisfied for spaces with weighted Korobov and Wiener reproducing kernels. For the Wiener case, the parameters in these assumptions are prohibitively large, and therefore we also present less restrictive assumptions and obtain other bounds on the power of standard information. Finally, we study tractability, which means that we want to guarantee that the errors depend at most polynomially on the number of variables and tend to zero polynomially in n −1 when n function values are used.

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Section: Known Results For Weighted L 2 Approximationmentioning

confidence: 99%

“…In the randomized setting, we allow the sampling points x j and/or the functions a j in (1.1) to be randomly selected. The following result has been established in [17]:…”

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the power of standard information for L ∞ approximation in the randomized setting

2009

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Multivariate Approximation in Downward Closed Polynomial Spaces

Cohen

Migliorati

2018

Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

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The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of dimensionality. As a typical example, standard polynomial spaces, such as those of total degree type, are often uneffective to reach a prescribed accuracy unless a prohibitive number of evaluations is invested. In recent years it has been shown that, for certain relevant applications, there are substantial advantages in using certain sparse polynomial spaces having anisotropic features with respect to the different variables. These applications include in particular the numerical approximation of high-dimensional parametric and stochastic partial differential equations. We start by surveying several results in this direction, with an emphasis on the numerical algorithms that are available for the construction of the approximation, in particular through interpolation or discrete least-squares fitting. All such algorithms rely on the assumption that the set of multi-indices associated with the polynomial space is downward closed. In the present paper we introduce some tools for the study of approximation in multivariate spaces under this assumption, and use them in the derivation of error bounds, sometimes independent of the dimension d, and in the development of adaptive strategies.

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