Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss--Hermite rule in Gaussian Sobolev spaces
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| Format: | Preprint |
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2025
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| _version_ | 1866918402665218048 |
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| author | Kazashi, Yoshihito Suzuki, Yuya Goda, Takashi |
| author_facet | Kazashi, Yoshihito Suzuki, Yuya Goda, Takashi |
| contents | Optimality of several quasi-Monte Carlo methods and suboptimality of the sparse-grid quadrature based on the univariate Gauss--Hermite rule is proved in the Sobolev spaces of mixed dominating smoothness of order $α$, 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. The dominant rate is found to be only $N^{-α/2}$ with $N$ function evaluations, although the optimal rate is known to be $N^{-α}(\ln N)^{(d-1)/2}$. 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 $N^{-α/2}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18712 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss--Hermite rule in Gaussian Sobolev spaces Kazashi, Yoshihito Suzuki, Yuya Goda, Takashi Numerical Analysis Optimality of several quasi-Monte Carlo methods and suboptimality of the sparse-grid quadrature based on the univariate Gauss--Hermite rule is proved in the Sobolev spaces of mixed dominating smoothness of order $α$, 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. The dominant rate is found to be only $N^{-α/2}$ with $N$ function evaluations, although the optimal rate is known to be $N^{-α}(\ln N)^{(d-1)/2}$. 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 $N^{-α/2}$. 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. |
| title | Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss--Hermite rule in Gaussian Sobolev spaces |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2509.18712 |