On the Optimal Order of Integration in Hermite Spaces with Finite Smoothness

DOI10.1137/16M1087461MaRDI QIDQ4635513

Josef Dick, Gunther Leobacher, Christian Irrgeher, Friedrich Pillichshammer

Publication date: 23 April 2018

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

Full work available at URL: https://arxiv.org/abs/1608.06061

zbMATH Keywords

numerical integrationHermite polynomialsworst-case errorhigher order digital nets

Mathematics Subject Classification ID

Numerical quadrature and cubature formulas (65D32) Complexity and performance of numerical algorithms (65Y20) Numerical integration (65D30)

Related Items (9)

Scaled lattice rules for integration on ℝ^{𝕕} achieving higher-order convergence with error analysis in terms of orthogonal projections onto periodic spaces ⋮ Countable tensor products of Hermite spaces and spaces of Gaussian kernels ⋮ MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM ⋮ Suboptimality of Gauss–Hermite Quadrature and Optimality of the Trapezoidal Rule for Functions with Finite Smoothness ⋮ Numerical weighted integration of functions having mixed smoothness ⋮ Tractability of \(L_2\)-approximation and integration in weighted Hermite spaces of finite smoothness ⋮ Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs ⋮ Truncation in average and worst case settings for special classes of \(\infty \)-variate functions ⋮ Integration in reproducing kernel Hilbert spaces of Gaussian kernels

Cites Work

• Tractability of multivariate problems. Volume I: Linear information
• Optimal order quadrature error bounds for infinite-dimensional higher-order digital sequences
• High-dimensional integration on \(\mathbb{R}^d\), weighted Hermite spaces, and orthogonal transforms
• Integration in Hermite spaces of analytic functions
• Embeddings of weighted Hilbert spaces and applications to multivariate and infinite-dimensional integration
• Explicit Constructions of Quasi-Monte Carlo Rules for the Numerical Integration of High-Dimensional Periodic Functions
• Walsh Spaces Containing Smooth Functions and Quasi–Monte Carlo Rules of Arbitrary High Order
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