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

DOI10.1137/15M1014814zbMath1336.65023arXiv1503.08846MaRDI QIDQ2798206

Mario Ullrich, Tino Ullrich

Publication date: 4 April 2016

Published in: SIAM Journal on Numerical Analysis (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1503.08846

zbMATH Keywords

convergenceSobolev spacenumerical integrationworst-case errorFrolov cubature formulaBesov-Triebel-Lizorkin space

Mathematics Subject Classification ID

Sobolev spaces and other spaces of ``smooth functions, embedding theorems, trace theorems (46E35) Multidimensional problems (41A63) Approximate quadratures (41A55) Numerical quadrature and cubature formulas (65D32)

Related Items (26)

Approximation of functions with small mixed smoothness in the uniform norm ⋮ Optimal cubature formulas on classes of periodic functions in several variables ⋮ Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions ⋮ 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 ⋮ Optimal randomized quadrature for weighted Sobolev and Besov classes with the Jacobi weight on the ball ⋮ A universal algorithm for multivariate integration ⋮ On the orthogonality of the Chebyshev-Frolov lattice and applications ⋮ A Monte Carlo Method for Integration of Multivariate Smooth Functions ⋮ Complexity of Monte Carlo integration for Besov classes on the unit sphere ⋮ Some Results on the Complexity of Numerical Integration ⋮ On “Upper Error Bounds for Quadrature Formulas on Function Classes” by K.K. Frolov ⋮ A Universal Median Quasi-Monte Carlo 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 ⋮ Bounds on Kolmogorov widths and sampling recovery for classes with small mixed smoothness ⋮ 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 ⋮ Gelfand numbers related to structured sparsity and Besov space embeddings with small mixed smoothness ⋮ Hyperbolic wavelet analysis of classical isotropic and anisotropic Besov-Sobolev spaces ⋮ Numerical performance of optimized Frolov lattices in tensor product reproducing kernel Sobolev spaces ⋮ An Explicit Construction of Optimal Order Quasi--Monte Carlo Rules for Smooth Integrands ⋮ A higher order Faber spline basis for sampling discretization of functions

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