Novel results for the anisotropic sparse grid quadrature

Haji-Ali, Abdul-Lateef; Harbrecht, Helmut; Peters, Michael; Siebenmorgen, Markus

doi:10.1016/j.jco.2018.02.003

Cited by 27 publications

(45 citation statements)

References 28 publications

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“…These are much better than N −1/2 , N −1 , and N −3/2 , respectively, which are predicted by Theorem 3. Indeed, the rates are at least twice as much as predicted, which issues from the fact that the factor 1/2 in the exponent on the right hand side of (4) issues from the crude estimate N (q) ≤ #X w (q, m) 2 , see [3] for details.…”

Section: Numerical Resultsmentioning

confidence: 90%

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Multiscale and High-Dimensional Problems

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Section: Numerical Resultsmentioning

confidence: 90%

“…More advanced examples from the uncertainty quantification of boundary value problems with random input parameters can be found in [3].…”

Section: Numerical Resultsmentioning

confidence: 99%

Multiscale and High-Dimensional Problems

et al. 2018

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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%

“…Today Smolyak's method is widely used in scientific computing and there exists a huge number of scientific articles dealing with applications and modifications of it. A partial list of papers (which is, of course, very far from being complete) on deterministic Smolyak algorithms may contain, e.g., the articles [64,65] for general approximation problems, [16,6,58,3,13,46,17,18,51,26,31] for numerical integration, [28,5,57,59,53,62,11] for function recovery, and [50,70,45,68,69,14,15] for other applications. Additional references and further information may be found in the survey articles [4,29], the book chapters [47,Chapter 15], [60,Chapter 4], and the books [7,12].…”

Section: Introductionmentioning

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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“…to allow a bounded holomorphic extension into certain cylindrical subsets of C N : here, as in [20], we consider parametric integrands which are holomorphic in cartesian products of discs with increasing radii. The rate at which those radii increase is a measure of the sparsity of the function, and as was observed in [20,22,30] governs the (dimension-independent) rate of convergence of the quadrature. These assumptions on the integrand are condensed in the notion of ( , )-holomorphy for a positive sequence = ( ) ∈N ∈ ℓ (N) and some ∈ (0, 1), see Definition 3.1 and also cp.…”

Section: Introductionmentioning

confidence: 95%

Convergence rates of high dimensional Smolyak quadrature

Zech

Schwab

2020

ESAIM: M2AN

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We analyse convergence rates of Smolyak integration for parametric maps u: U → X taking values in a Banach space X, defined on the parameter domain U = [−1,1]N. For parametric maps which are sparse, as quantified by summability of their Taylor polynomial chaos coefficients, dimension-independent convergence rates superior to N-term approximation rates under the same sparsity are achievable. We propose a concrete Smolyak algorithm to a priori identify integrand-adapted sets of active multiindices (and thereby unisolvent sparse grids of quadrature points) via upper bounds for the integrands’ Taylor gpc coefficients. For so-called “(b,ε)-holomorphic” integrands u with b∈lp(∕) for some p ∈ (0, 1), we prove the dimension-independent convergence rate 2/p − 1 in terms of the number of quadrature points. The proposed Smolyak algorithm is proved to yield (essentially) the same rate in terms of the total computational cost for both nested and non-nested univariate quadrature points. Numerical experiments and a mathematical sparsity analysis accounting for cancellations in quadratures and in the combination formula demonstrate that the asymptotic rate 2/p − 1 is realized computationally for a moderate number of quadrature points under certain circumstances. By a refined analysis of model integrand classes we show that a generally large preasymptotic range otherwise precludes reaching the asymptotic rate 2/p − 1 for practically relevant numbers of quadrature points.

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Novel results for the anisotropic sparse grid quadrature

Cited by 27 publications

References 28 publications

Multiscale and High-Dimensional Problems

Multiscale and High-Dimensional Problems

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

Convergence rates of high dimensional Smolyak quadrature

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