On weak tractability of the Clenshaw–Curtis Smolyak algorithm

Hinrichs, Aicke; Novak, Erich; Ullrich, Mario

doi:10.1016/j.jat.2014.03.012

Cited by 17 publications

(16 citation statements)

References 31 publications

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“…this is Open Problem 2 from [89]. We know from Vybíral [120] and [62] that the curse is present for somewhat larger spaces and that a weak tractability holds for smaller classes; this can be proved with the Smolyak algorithm, see [63].…”

Section: Some Recent Resultsmentioning

confidence: 99%

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Some Results on the Complexity of Numerical Integration

Novak

2016

Springer Proceedings in Mathematics &Amp; Statistics

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We present some results on the complexity of numerical integration. We start with the seminal paper of Bakhvalov (1959) and end with new results on the curse of dimensionality and on the complexity of oscillatory integrals. This survey paper consists of four parts:1. Classical results till 1971 2. Randomized algorithms 3. Tensor product problems, tractability and weighted norms 4. Some recent results: C k functions and oscillatory integrals Classical Results till 1971I start with a warning: We do not discuss the complexity of path integration and infinite-dimensional integration on R N or other domains although there are exciting new results in that area, see [8,14,21,22,23,41,43,44,53,69,77,90,96,121,123]. For parametric integrals see [16,17], for quantum computers, see [48,49,80,115].We mainly study the problem of numerical integration, i.e., of approximating the integral

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Section: Some Recent Resultsmentioning

confidence: 99%

“…For recent results on the order of convergence see Sickel and T. Ullrich [100,101] and Dinh Dũng and T. Ullrich [30]. The recent paper [63]…”

Section: ⊓ ⊔mentioning

confidence: 99%

Some Results on the Complexity of Numerical Integration

Novak

2016

Springer Proceedings in Mathematics &Amp; Statistics

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“…The by now classical research topic of numerically integrating d-variate functions with mixed smoothness properties goes back to the work of Korobov [25], Hlawka [22], and Bakhvalov [2] in the 1960s and was continued later by numerous authors including Frolov [12], Temlyakov [43,45,46,49], Dubinin [7,8], Skriganov [39], Triebel [53], Hinrichs et al [16,17,21], Hinrichs, Novak, M. Ullrich, Woźniakowski [18,19], Hinrichs, Novak, M. Ullrich [20], Dũng, T. Ullrich [9], T. Ullrich [56], Dick and Pillichshammer [6], and Markhasin [28,29,30] to mention just a few. In contrast to the quadrature of univariate functions, where equidistant point grids lead to optimal formulas, the multivariate problem is much more involved.…”

Section: Introductionmentioning

confidence: 99%

The Role of Frolov's Cubature Formula for Functions with Bounded Mixed Derivative

Ullrich¹,

Ullrich²

2016

SIAM J. Numer. Anal.

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We prove upper bounds on the order of convergence of Frolov's cubature formula for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov B s p,θ and Triebel-Lizorkin spaces F s p,θ and our results treat the whole range of admissible parameters (s ≥ 1/p). In particular, we obtain upper bounds for the difficult the case of small smoothness which is given for Triebel-Lizorkin spaces F s p,θ in case 1 < θ < p < ∞ with 1/p < s ≤ 1/θ. The presented upper bounds on the worst-case error show a completely different behavior compared to "large" smoothness s > 1/θ. In the latter case the presented upper bounds are optimal, i.e., they can not be improved by any other cubature formula. The optimality for "small" smoothness is open.

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“…Nevertheless, the proof does not cover all D d with a size λ d (D d ) ≥ α d and it would be interesting to know whether the curse also holds for these more general domains. If one assumes that all directional derivatives of all orders are bounded by one and D d = [0, 1] d then one can prove the weak tractability of the integration problem using the Clenshaw-Curtis Smolyak algorithm, see [13].…”

Section: Arbitrary Linear Informationmentioning

confidence: 99%

Algorithms and complexity for functions on general domains

Novak

2020

Journal of Complexity

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Error bounds and complexity bounds in numerical analysis and information-based complexity are often proved for functions that are defined on very simple domains, such as a cube, a torus, or a sphere. We study optimal error bounds for the approximation or integration of functions defined on D d ⊂ R d and only assume that D d is a bounded Lipschitz domain. Some results are even more general. We study three different concepts to measure the complexity: order of convergence, asymptotic constant, and explicit uniform bounds, i.e., bounds that hold for all n (number of pieces of information) and all (normalized) domains.It is known for many problems that the order of convergence of optimal algorithms does not depend on the domain D d ⊂ R d . We present examples for which the following statements are true:1. Also the asymptotic constant does not depend on the shape of D d or the imposed boundary values, it only depends on the volume of the domain.2. There are explicit and uniform lower (or upper, respectively) bounds for the error that are only slightly smaller (or larger, respectively) than the asymptotic error bound. *Much is known about the order of convergence of the numbers e n , in particular for simple domains D d , such as the cube or the torus. General bounded Lipschitz domains are studied in [7,10,11,12,26,27,29,38,39,40,42,43]. In many cases, the optimal order of the e n is of the form e n ≍ n −α (log n) β , where α and β do not depend on the domain D d . It is interesting to know also the exact asymptotic constant C := lim n→∞ e n n α (log n) −β , if it exists. The value of C is known only in rare cases (unless d = 1, we do not discuss the univariate case in detail), and usually only for very special domains,

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On weak tractability of the Clenshaw–Curtis Smolyak algorithm

Cited by 17 publications

References 31 publications

Some Results on the Complexity of Numerical Integration

Some Results on the Complexity of Numerical Integration

The Role of Frolov's Cubature Formula for Functions with Bounded Mixed Derivative

Algorithms and complexity for functions on general domains

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