On “Upper Error Bounds for Quadrature Formulas on Function Classes” by K.K. Frolov

Ullrich, Mario

doi:10.1007/978-3-319-33507-0_31

Cited by 23 publications

(31 citation statements)

References 11 publications

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“…The advantage over previous related works lies in the potential for implementation. Other known algorithms which achieve the optimal error bounds, such as those based on Frolov's method (see, e.g., [6,9,11,28,29,30]), are very difficult to implement especially in high dimensions. For our algorithm, a simple probabilistic approach can be used to obtain suitable generating vectors, see Remark 10 below.…”

Section: Introductionmentioning

confidence: 99%

Lattice rules with random n achieve nearly the optimal O(n−α−1<

Kritzer

Kuo

Nuyens

et al. 2019

Journal of Approximation Theory

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Section: Introductionmentioning

confidence: 99%

Lattice rules with random n achieve nearly the optimal O(n−α−1<

Kritzer

Kuo

Nuyens

et al. 2019

Journal of Approximation Theory

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“…We now proceed to the case d ≥ 3. It is well known that the Frolov point sets are very good for numerical integration of smoothness classes of functions of several variables (see [15], [29], [31], [42], [14], [43]). Theorem 6.2 below, which was proved in [39], shows that the Frolov point sets have good fixed volume discrepancy.…”

Section: Fixed Volume Discrepancymentioning

confidence: 99%

Connections between numerical integration, discrepancy, dispersion, and universal discretization

Temlyakov¹

2019

The SMAI Journal of Computational Mathematics

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The main goal of this paper is to provide a brief survey of recent results which connect together results from different areas of research. It is well known that numerical integration of functions with mixed smoothness is closely related to the discrepancy theory. We discuss this connection in detail and provide a general view of this connection. It was established recently that the new concept of fixed volume discrepancy is very useful in proving the upper bounds for the dispersion. Also, it was understood recently that point sets with small dispersion are very good for the universal discretization of the uniform norm of trigonometric polynomials.

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“…To gain this, we use a reduction of the problem to theÅ s p,θ setting via tailored transformations in connection with the Frolov cubature formulae [12,36], which recently attracted significant interest [19,21,[44][45][46]. It has been already proved by Bykovskii [3] …”

Section: State Of the Art And Relevant Literaturementioning

confidence: 99%

Change of Variable in Spaces of Mixed Smoothness and Numerical Integration of Multivariate Functions on the Unit Cube

Ullrich

Ullrich²

2017

Constr Approx

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In a recent article, two of the present authors studied Frolov's cubature formulae and their optimality in Besov-Triebel-Lizorkin spaces of functions with dominating mixed smoothness supported in the unit cube. In this paper, we give a general result that the asymptotic order of the minimal worst-case integration error is not affected by boundary conditions in the above mentioned spaces. In fact, we propose two tailored modifications of Frolov's cubature formulae suitable for functions supported on the cube (not in the cube) that yield the same order of convergence up to a constant. This constant involves the norms of a "change of variable" and a "pointwise multiplication" mapping, respectively, between the function spaces of interest. We complement, extend, and improve classical results on the boundedness of change of variable mappings in Besov-Sobolev spaces of mixed smoothness. The second modification, suitable for classes of periodic functions, is based on a pointwise multiplication and is therefore most likely more suitable for applications than the (traditional) "change of variable" approach. These new theoretical insights are expected to be useful for the design of new (and robust) cubature rules for multivariate functions on the cube.

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On “Upper Error Bounds for Quadrature Formulas on Function Classes” by K.K. Frolov

Cited by 23 publications

References 11 publications

Lattice rules with random n achieve nearly the optimal O(n−α−1<

Lattice rules with random n achieve nearly the optimal O(n−α−1<

Connections between numerical integration, discrepancy, dispersion, and universal discretization

Change of Variable in Spaces of Mixed Smoothness and Numerical Integration of Multivariate Functions on the Unit Cube

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