SIAM Journal on Numerical Analysis· 2026Q1
Optimality of Quasi-Monte Carlo Methods and Suboptimality of the Sparse-Grid Gauss–Hermite Rule in Gaussian Sobolev Spaces
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- Q1SCImago
- 2026year
Short summary
Quasi-Monte Carlo (QMC) methods with a change of variables achieve optimal convergence rates in Gaussian Sobolev spaces, while the sparse-grid Gauss–Hermite quadrature rule is proven to be suboptimal, converging at a rate of only n^(-(1-1/p)) instead of the optimal n^(-1/p).
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Key points
- QMC methods with a change of variables achieve optimal convergence rates in Gaussian Sobolev spaces.
- Sparse-grid Gauss–Hermite quadrature is suboptimal, with a convergence rate of n^(-(1-1/p)) instead of the optimal n^(-1/p).
- The suboptimality of the sparse-grid rule stems from the structure of its nodes, not its weights.
- The optimal rate is n^(-1/p) for p-smooth functions with n function evaluations.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract. Optimality of several quasi-Monte Carlo methods and suboptimality of the sparse-grid quadrature based on the univariate Gauss–Hermite rule are proved in the Sobolev spaces of mixed dominating smoothness of order [Formula: see text], where the optimality is in the sense of worst-case convergence rate. For sparse-grid Gauss–Hermite quadrature, lower and upper bounds are established, with rates coinciding up to a logarithmic factor. This common algebraic rate is found to be only [Formula: see text] with [Formula: see text] function evaluations, although the optimal rate is known to be [Formula: see text]. The lower bound is obtained by exploiting the structure of the Gauss–Hermite nodes and is independent of the quadrature weights; consequently, no modification of the weights can improve the rate [Formula: see text]. In contrast, several quasi-Monte Carlo methods with a change of variables are shown to achieve the optimal rate, some up to, and one including, the logarithmic factor.
The authors' abstract, as published at the source. SIAM Journal on Numerical Analysis, 2026 · DOI ↗
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Field: Numerical Analysis
Numerical AnalysisMathematics