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

DOI10.1007/978-3-319-33507-0_31zbMath1356.65089arXiv1404.5457OpenAlexW3098368813MaRDI QIDQ2957059

Mario Ullrich

Publication date: 20 January 2017

Published in: Springer Proceedings in Mathematics & Statistics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1404.5457

zbMATH Keywords

Sobolev spacenumerical integrationtutorial paperFrolov cubature

Mathematics Subject Classification ID

Multidimensional problems (41A63) Approximate quadratures (41A55) Numerical quadrature and cubature formulas (65D32) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis (65-01)

Related Items (17)

Complexity of oscillatory integrals on the real line ⋮ On generation and enumeration of orthogonal Chebyshev-Frolov lattices ⋮ Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube ⋮ On the orthogonality of the Chebyshev-Frolov lattice and applications ⋮ Connections between numerical integration, discrepancy, dispersion, and universal discretization ⋮ A Monte Carlo Method for Integration of Multivariate Smooth Functions ⋮ Some Results on the Complexity of Numerical Integration ⋮ Lattice enumeration via linear programming ⋮ Consistency of randomized integration methods ⋮ A note on the dispersion of admissible lattices ⋮ Lattice rules with random \(n\) achieve nearly the optimal \(\mathcal{O}(n^{-\alpha-1/2})\) error independently of the dimension ⋮ Digital net properties of a polynomial analogue of Frolov's construction ⋮ Lattice-based integration algorithms: Kronecker sequences and rank-1 lattices ⋮ Optimal order quadrature error bounds for infinite-dimensional higher-order digital sequences ⋮ Numerical performance of optimized Frolov lattices in tensor product reproducing kernel Sobolev spaces ⋮ Smooth fixed volume discrepancy, dispersion, and related problems ⋮ An Explicit Construction of Optimal Order Quasi--Monte Carlo Rules for Smooth Integrands

Cites Work

• Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions
• On Chebyshev's polynomials and certain combinatorial identities
• Cubature formulas, discrepancy, and nonlinear approximation
• Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube
• The Role of Frolov's Cubature Formula for Functions with Bounded Mixed Derivative
• Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square
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