On the optimal order of integration in Hermite spaces with finite\n smoothness

2016/08/22 by Josef Dick, Dick, Josef, Christian Irrgeher +5 · 3 citations
Economics, Econometrics and Finance · Mathematics · #65D30 #65D32 #65Y20 #FOS: Mathematics #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1608.06061

openalex publication_date 2016/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the numerical approximation of integrals over \ℝ^s with\nrespect to the standard Gaussian measure for integrands which lie in certain\nHermite spaces of functions. The decay rate of the associated sequence is\nspecified by a single integer parameter which determines the smoothness classes\nand the inner product can be expressed via L₂ norms of the derivatives of\nthe function.\n We map higher order digital nets from the unit cube to a suitable subcube of\n\ℝ^s via a linear transformation and show that such rules achieve,\napart from powers of \log N, the optimal rate of convergence of the\nintegration error.\n

Cited by

• Second order interlaced polynomial lattice rules for integration over ℝ^s
• Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss--Hermite rule in Gaussian Sobolev spaces
• Quasi-Monte Carlo integration over ℝ^s with boundary-damping importance sampling

Related

• Optimal numerical integration and approximation of functions on ℝ^d equipped with Gaussian measure
• Integration in Hermite spaces of analytic functions
• Approximation in Hermite spaces of smooth functions
• Tractability of L₂-approximation and integration in weighted Hermite spaces of finite smoothness
• Sub-optimality of Gauss--Hermite quadrature and optimality of the trapezoidal rule for functions with finite smoothness