Tractability of multivariate approximation defined over Hilbert spaces with exponential weights

Irrgeher, Christian; Kritzer, Peter; Pillichshammer, Friedrich; Woźniakowski, Henryk

doi:10.1016/j.jat.2016.02.020

Cited by 46 publications

(55 citation statements)

References 20 publications

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“…This approach goes back to the seminal paper [55]. Another approach is the concept of increasing smoothness with respect to properly ordered variables, see, e.g., [11,25,31,37,38,49,54]. The precise definition of Hilbert spaces of functions depending on infinitely many variables of increasing smoothness can be found in [25,Section 3].…”

Section: Remark 22mentioning

confidence: 99%

Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration

Gnewuch

Wnuk

2020

Journal of Approximation Theory

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Smolyak's method, also known as hyperbolic cross approximation or sparse grid method, is a powerful tool to tackle multivariate tensor product problems solely with the help of efficient algorithms for the corresponding univariate problem.In this paper we study the randomized setting, i.e., we randomize Smolyak's method. We provide upper and lower error bounds for randomized Smolyak algorithms with explicitly given dependence on the number of variables and the number of information evaluations used. The error criteria we consider are the worst-case root mean square error (the typical error criterion for randomized algorithms, often referred to as "randomized error") and the root mean square worst-case error (often referred to as "worst-case error").Randomized Smolyak algorithms can be used as building blocks for efficient methods such as multilevel algorithms, multivariate decomposition methods or dimensionwise quadrature methods to tackle successfully high-dimensional or even infinitedimensional problems. As an example, we provide a very general and sharp result on the convergence rate of N -th minimal errors of infinite-dimensional integration on weighted reproducing kernel Hilbert spaces. Moreover, we are able to characterize the spaces for which randomized algorithms for infinte-dimensional integration are superior to deterministic ones. We illustrate our findings for the special instance of weighted Korobov spaces. We indicate how these results can be extended, e.g., to spaces of functions whose smooth dependence on successive variables increases ("spaces of increasing smoothness") and to the problem of L 2 -approximation (function recovery).

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Section: Remark 22mentioning

confidence: 99%

Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration

Gnewuch

Wnuk

2020

Journal of Approximation Theory

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“…See, e.g., [22] and the references therein, for the following example in the context of tractability analysis of high-dimensional problems. For further information about Korobov spaces see, e.g., [ Example 3.1.…”

Section: 3mentioning

confidence: 99%

“…with equivalent norms. See, e.g., [22] and the references therein for this type of Fourier weights. Put r 0 := 0 as previously.…”

Section: 3mentioning

confidence: 99%

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Embeddings for infinite-dimensional integration and L2-approximation with increasing smoothness

Gnewuch

Hefter

Hinrichs

et al. 2019

Journal of Complexity

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We study integration and L 2 -approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh, Haar, and Sobolev spaces. For the proofs we derive embedding theorems between spaces of increasing smoothness and appropriate weighted function spaces of fixed smoothness.

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“…The case of L 2 -approximation for ρ(h) depending exponentially on h was dealt with in the recent paper [2]. The results there can be also obtained as special cases of a more general approach presented in the paper [4].…”

Section: Introductionmentioning

confidence: 99%

L∞-Approximation in Korobov spaces with exponential weights

Kritzer

Pillichshammer

Woźniakowski

2017

Journal of Complexity

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We study multivariate L ∞ -approximation for a weighted Korobov space of periodic functions for which the Fourier coefficients decay exponentially fast. The weights are defined, in particular, in terms of two sequences a = {a j } and b = {b j } of positive real numbers bounded away from zero. We study the minimal worstcase error e L∞−app,Λ (n, s) of all algorithms that use n information evaluations from a class Λ in the s-variate case. We consider two classes Λ in this paper: the class Λ all of all linear functionals and the class Λ std of only function evaluations.We study exponential convergence of the minimal worst-case error, which means that e L∞−app,Λ (n, s) converges to zero exponentially fast with increasing n. Furthermore, we consider how the error depends on the dimension s. To this end, we define the notions of κ-EC-weak, EC-polynomial and EC-strong polynomial tractability, where EC stands for "exponential convergence". In particular, EC-polynomial tractability means that we need a polynomial number of information evaluations in s and 1+log ε −1 to compute an ε-approximation. We derive necessary and sufficient conditions on the sequences a and b for obtaining exponential error convergence, and also for obtaining the various notions of tractability. The results are the same for both classes Λ.L 2 -approximation for functions from the same function space has been considered in [2]. It is surprising that most results for L ∞ -approximation coincide with their counterparts for L 2 -approximation. This allows us to deduce also results for L p -approximation for p ∈ [2, ∞].

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Tractability of multivariate approximation defined over Hilbert spaces with exponential weights

Cited by 46 publications

References 20 publications

Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration

Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration

Embeddings for infinite-dimensional integration and L2-approximation with increasing smoothness

L∞-Approximation in Korobov spaces with exponential weights

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