Second order interlaced polynomial lattice rules for integration over $\mathbb{R}^s$
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914068453916672 |
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| author | Cui, Tiangang Dick, Josef Pillichshammer, Friedrich |
| author_facet | Cui, Tiangang Dick, Josef Pillichshammer, Friedrich |
| contents | We study numerical integration of functions $f: \mathbb{R}^{s} \to \mathbb{R}$ with respect to a probability measure. By applying the corresponding inverse cumulative distribution function, the problem is transformed into integrating an induced function over the unit cube $(0,1)^{s}$. We introduce a new orthonormal system: \emph{order~2 localized Walsh functions}. These basis functions retain the approximation power of classical Walsh functions for twice-differentiable integrands while inheriting the spatial localization of Haar wavelets. Localization is crucial because the transformed integrand is typically unbounded at the boundary. We show that the worst-case quasi-Monte Carlo integration error decays like $\mathcal{O}(N^{-1/λ})$ for every $λ\in (1/2,1]$. As an application, we consider elliptic partial differential equations with a finite number of log-normal random coefficients and show that our error estimates remain valid for their stochastic Galerkin discretizations by applying a suitable importance sampling density. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26624 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Second order interlaced polynomial lattice rules for integration over $\mathbb{R}^s$ Cui, Tiangang Dick, Josef Pillichshammer, Friedrich Numerical Analysis We study numerical integration of functions $f: \mathbb{R}^{s} \to \mathbb{R}$ with respect to a probability measure. By applying the corresponding inverse cumulative distribution function, the problem is transformed into integrating an induced function over the unit cube $(0,1)^{s}$. We introduce a new orthonormal system: \emph{order~2 localized Walsh functions}. These basis functions retain the approximation power of classical Walsh functions for twice-differentiable integrands while inheriting the spatial localization of Haar wavelets. Localization is crucial because the transformed integrand is typically unbounded at the boundary. We show that the worst-case quasi-Monte Carlo integration error decays like $\mathcal{O}(N^{-1/λ})$ for every $λ\in (1/2,1]$. As an application, we consider elliptic partial differential equations with a finite number of log-normal random coefficients and show that our error estimates remain valid for their stochastic Galerkin discretizations by applying a suitable importance sampling density. |
| title | Second order interlaced polynomial lattice rules for integration over $\mathbb{R}^s$ |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2509.26624 |