A new criterion for tractability of multivariate problems

Papageorgiou, Anargyros; Petras, Iasonas

doi:10.1016/j.jco.2014.03.001

Cited by 28 publications

(13 citation statements)

References 3 publications

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“…The study of exponential convergence-tractability (or for short, EC-tractability) has previously been dealt with in [6,13,14]. Similar notions of tractability have been introduced by Papageorgiou and Petras [24] which they call ''polylog-tractability'' and ''ln κ -weak tractability''. Tractability in the context of exponential error convergence is also addressed in the third volume of the books of Novak and Woźniakowski [22].…”

Section: Introductionmentioning

confidence: 99%

Integration in Hermite spaces of analytic functions

Irrgeher

Kritzer

Leobacher

et al. 2015

Journal of Complexity

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a b s t r a c tWe study integration in a class of Hilbert spaces of analytic functions defined on the R s . The functions are characterized by the property that their Hermite coefficients decay exponentially fast. We use Gauss-Hermite integration rules and show that the errors of our algorithms decay exponentially fast. Furthermore, we study tractability in terms of s and log ε −1 and give necessary and sufficient conditions under which we achieve exponential convergence with EC-weak, EC-polynomial, and EC-strong polynomial tractability.

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Section: Introductionmentioning

confidence: 99%

Integration in Hermite spaces of analytic functions

Irrgeher

Kritzer

Leobacher

et al. 2015

Journal of Complexity

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“…If App is EXP, then we can discuss the tractability with exponential convergence (EC-tractabilty). Recently, there are many papers where EC-tractability is considered (see [1,5,13,19]).…”

Section: For Abs We Havementioning

confidence: 99%

Average case tractability of multivariate approximation with Gaussian kernels

Chen

Wang

2019

Journal of Approximation Theory

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We study the problem of approximating functions of d variables in the average case setting for the L 2 space L 2,d with the standard Gaussian weight equipped with a zero-mean Gaussian measure. The covariance kernel of this Gaussian measure takes the form of a Gaussian kernel with non-increasing positive shape parameters γ 2 j for j = 1, 2, . . . , d. The error of approximation is defined in the norm of L 2,d . We study the average case error of algorithms that use at most n arbitrary continuous linear functionals. The information complexity n(ε, d) is defined as the minimal number of linear functionals which are needed to find an algorithm whose average case error is at most ε. We study different notions of tractability or exponentially-convergent tractability (ECtractability) which the information complexity n(ε, d) describe how behaves as a function of d and ε −1 or as one of d and (1 + ln ε −1 ). We find necessary and sufficient conditions on various notions of tractability and EC-tractability in terms of shape parameters. In particular, for any positive s > 0 and t ∈ (0, 1) we obtain that the sufficient and necessary condition on γ 2 j for which lim d+ε −1 →∞ n(ε, d) ε −s + d t = 0 holds is lim j→∞ j 1−t γ 2 j ln + γ −2 j = 0, where ln + x = max(1, ln x).

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“…The existing notions of tractability mainly include strong polynomial tractability (SPT), polynomial tractability (PT), quasi-polynomial tractability (QPT), weak tractability (WT), (s, t)-weak tractability ((s, t)-WT), and uniform weak tractability (UWT). In recent years the study of algebraic and exponential tractability has attracted much interest, and a great number of interesting results have been obtained (see [31,32,33,43,9,44,38,39,6,5,21,36,45,15,2,26,37] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

On the power of standard information for tractability for $L_2$-approximation in the randomized setting

Lu¹,

Wang²

2021

Preprint

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We study approximation of multivariate functions from a separable Hilbert space in the randomized setting with the error measured in the weighted L 2 norm. We consider algorithms that use standard information Λ std consisting of function values or general linear information Λ all consisting of arbitrary linear functionals. We use the weighted least squares regression algorithm to obtain the upper estimates of the minimal randomized error using Λ std . We investigate the equivalences of various notions of algebraic and exponential tractability for Λ std and Λ all for the normalized or absolute error criterion. We show that in the randomized setting for the normalized or absolute error criterion, the power of Λ std is the same as that of Λ all for all notions of exponential and algebraic tractability without any condition. Specifically, we solve four Open Problems 98, 100-102 as posed by E.Novak

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A new criterion for tractability of multivariate problems

Cited by 28 publications

References 3 publications

Integration in Hermite spaces of analytic functions

Integration in Hermite spaces of analytic functions

Average case tractability of multivariate approximation with Gaussian kernels

On the power of standard information for tractability for $L_2$-approximation in the randomized setting

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