Lattice Rules Meet Kernel Cubature
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866918045843193856 |
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| author | Kaarnioja, Vesa Klebanov, Ilja Schillings, Claudia Suzuki, Yuya |
| author_facet | Kaarnioja, Vesa Klebanov, Ilja Schillings, Claudia Suzuki, Yuya |
| contents | Rank-1 lattice rules are a class of equally weighted quasi-Monte Carlo methods that achieve essentially linear convergence rates for functions in a reproducing kernel Hilbert space (RKHS) characterized by square-integrable first-order mixed partial derivatives. In this work, we explore the impact of replacing the equal weights in lattice rules with optimized cubature weights derived using the reproducing kernel. We establish a theoretical result demonstrating a doubled convergence rate in the one-dimensional case and provide numerical investigations of convergence rates in higher dimensions. We also present numerical results for an uncertainty quantification problem involving an elliptic partial differential equation with a random coefficient. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09500 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lattice Rules Meet Kernel Cubature Kaarnioja, Vesa Klebanov, Ilja Schillings, Claudia Suzuki, Yuya Numerical Analysis Statistics Theory 65D32, 65D30, 46E22 Rank-1 lattice rules are a class of equally weighted quasi-Monte Carlo methods that achieve essentially linear convergence rates for functions in a reproducing kernel Hilbert space (RKHS) characterized by square-integrable first-order mixed partial derivatives. In this work, we explore the impact of replacing the equal weights in lattice rules with optimized cubature weights derived using the reproducing kernel. We establish a theoretical result demonstrating a doubled convergence rate in the one-dimensional case and provide numerical investigations of convergence rates in higher dimensions. We also present numerical results for an uncertainty quantification problem involving an elliptic partial differential equation with a random coefficient. |
| title | Lattice Rules Meet Kernel Cubature |
| topic | Numerical Analysis Statistics Theory 65D32, 65D30, 46E22 |
| url | https://arxiv.org/abs/2501.09500 |