Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation

DźNg, Dinh

doi:10.1007/s10208-015-9274-8

Cited by 32 publications

(32 citation statements)

References 40 publications

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“…As a next step, we obtain as a consequence of Lemma the following result. Its proof is similar to the one in [, Theorem 2.1(ii)] (see also [, Lemma 2.5]). The main tool is an application of the discrete Hardy inequality, see [, (2.28)–(2.29)].…”

Section: Besov Spaces Of Mixed Smoothnessmentioning

confidence: 70%

Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

Dũng

Ullrich²

2014

Math. Nachr.

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We prove lower bounds for the error of optimal cubature formulae for d-variate functions from Besov spaces of mixed smoothness B α p,θ (G d ) in the case 0 < p, θ ≤ ∞ and α > 1/p, whereWe prove upper bounds for QMC methods of integration on the Fibonacci lattice for bivariate periodic functions from B α p,θ (T 2 ) in the case 1 ≤ p ≤ ∞, 0 < θ ≤ ∞, α > 1/p. A non-periodic modification of this classical formula yields upper bounds for B α p,θ (I 2 ) if 1/p < α < 1 + 1/p. In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from B α p,θ (G 2 ) and indicate that a corresponding result is most likely also true in case d > 2. This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formula on Smolyak grids is never optimal for the general setting.

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Section: Besov Spaces Of Mixed Smoothnessmentioning

confidence: 70%

Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

Dũng

Ullrich²

2014

Math. Nachr.

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“…There are still many open cases in this framework which actually lack the suitable lower bounds. Let us refer to the works by Temlyakov [41,42] and the more recent papers Sickel, Ullrich [35,36,46], Dinh Dũng [10,11], [2], as well as [12] and the references therein for upper bounds in case p ≥ q and the question-marked region. We emphasize that our technique allows to reproduce all those results, including the upper bound in [41], within a few lines of proof.…”

Section: New Matching Bounds For (Non-)linear Sampling Recoverymentioning

confidence: 99%

Optimal sampling recovery of mixed order Sobolev embeddings via discrete Littlewood–Paley type characterizations

Byrenheid¹,

Ullrich²

2017

Anal Math

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In this paper we consider the L q -approximation of multivariate periodic functions f with L p -bounded mixed derivative (difference). The (possibly non-linear) reconstruction algorithm is supposed to recover the function from function values, sampled on a discrete set of n sampling nodes. The general performance is measured in terms of (non-)linear sampling widths ̺ n . We conduct a systematic analysis of Smolyak type interpolation algorithms in the framework of Besov-Lizorkin-Triebel spaces of dominating mixed smoothness based on specifically tailored discrete Littlewood-Paley type characterizations. As a consequence, we provide sharp upper bounds for the asymptotic order of the (non-)linear sampling widths in various situations and close some gaps in the existing literature. For example, in case 2 ≤ p < q < ∞ and r > 1/p the linear sampling widthsshow the asymptotic behavior of the corresponding Gelfand n-widths, whereas in case 1 < p < q ≤ 2 and r > 1/p the linear sampling widths match the corresponding linear widths. In the mentioned cases linear Smolyak interpolation based on univariate classical trigonometric interpolation turns out to be optimal.

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“…For an overview we refer to [4] and the references therein. Additionally, let us mention the work of Temlyakov [40,39], Griebel et al [3,10,11], Dinh [7,9,4] , Sickel [31,32,33,34,4], Ullrich [33,41,34,4].…”

Section: Introductionmentioning

confidence: 99%

Tight error bounds for rank-1 lattice sampling in spaces of hybrid mixed smoothness

et al. 2017

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We consider the approximate recovery of multivariate periodic functions from a discrete set of function values taken on a rank-s integration lattice. The main result is the fact that any (non-)linear reconstruction algorithm taking function values on a rank-s lattice of size M has a dimension-independent lower bound of 2 −(α+1)/2 M −α/2 when considering the optimal worst-case error with respect to function spaces of (hybrid) mixed smoothness α > 0 on the d-torus. We complement this lower bound with upper bounds that coincide up to logarithmic terms. These upper bounds are obtained by a detailed analysis of a rank-1 lattice sampling strategy, where the rank-1 lattices are constructed by a componentby-component (CBC) method. This improves on earlier results obtained in [25] and [27]. The lattice (group) structure allows for an efficient approximation of the underlying function from its sampled values using a single one-dimensional fast Fourier transform. This is one reason why these algorithms keep attracting significant interest. We compare our results to recent (almost) optimal methods based upon samples on sparse grids. This paper deals with the reconstruction of multivariate periodic functions from a discrete set of M function values along rank-1 lattices. Such lattices are widely used for the efficient numerical integration of multivariate periodic functions since the 1950ies [1,21,29,35,6] and represent a well-distributed set of points in [0, 1) d . A rank-1 lattice with M ∈ N points and generating vector z ∈ Z d is given byIn this paper we will show that restricting the set of available discrete information to samples from a rank-s lattice, cf.[35], seriously affects the rate of convergence of a corresponding worst-case error with respect to classes of functions with (hybrid) mixed smoothness α > 0.To be more precise, for any (possibly nonlinear) reconstruction procedure from sampled values along rank-s lattices we can find a function in the periodic Sobolev spaces H α mix such that the L 2 (T d ) mean square error is at least 2 −(α+1)/2 M −α/2 . In contrast to that, it has been proved recently, that the sampling recovery from (energy) sparse grids leads to much better convergence rates, namely M −α in the main term, see [33,41,4].Subsequently, we study particular reconstructing algorithms, which are based on the naive approach of approximating the potentially "largest" Fourier coefficients (integrals) with the same rank-1 lattice rule. Despite the lacking asymptotical optimality, recovery from so-called reconstructing rank-1 lattices, cf. [15,18], has some striking advantages.First, the matrix of the underlying linear system of equations has orthogonal columns due to the group structure [2] and the reconstructing property of the used rank-1 lattices. Consequently, the computation is stable, cf. [17,15].Second, the CBC strategy [14, Tab. 3.1] provides a search method for a reconstructing rank-1 lattice which allows for the computation of the approximate Fourier coefficients belonging to frequencies lying on ...

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Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation

Cited by 32 publications

References 40 publications

Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

Optimal sampling recovery of mixed order Sobolev embeddings via discrete Littlewood–Paley type characterizations

Tight error bounds for rank-1 lattice sampling in spaces of hybrid mixed smoothness

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