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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.21838 |
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| _version_ | 1866917188864049152 |
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| author | Xu, Xiaoda |
| author_facet | Xu, Xiaoda |
| contents | This paper studies the expected $L_p$-discrepancy ($2 \leq p < \infty$) for stratified sampling schemes under importance sampling. We introduce a parametric family of equivolume partitions $Ω_{θ,\sim}$ and leverage recent exact formulas for the expected $L_2$-discrepancy \cite{xian2025improved}. Our main contribution is a weighted discrepancy reduction lemma that relates weighted $L_p$-discrepancy to standard $L_p$-discrepancy with explicit constants depending on the weight function. For $p=2$, we obtain explicit bounds using the exact discrepancy formulas. For $p>2$, we derive probabilistic bounds via dyadic chaining techniques. The results yield uniform error estimates for multivariate integration in Sobolev spaces $\mathcal{H}^1(K)$ and $F^*_{d,q}$, demonstrating improved performance over classical jittered sampling in importance sampling scenarios. Numerical experiments validate our theoretical findings and illustrate the practical advantages of parametric stratified sampling. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21838 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted $L_p$-Discrepancy Bounds for Parametric Stratified Sampling and Applications to High-Dimensional Integration Xu, Xiaoda Numerical Analysis This paper studies the expected $L_p$-discrepancy ($2 \leq p < \infty$) for stratified sampling schemes under importance sampling. We introduce a parametric family of equivolume partitions $Ω_{θ,\sim}$ and leverage recent exact formulas for the expected $L_2$-discrepancy \cite{xian2025improved}. Our main contribution is a weighted discrepancy reduction lemma that relates weighted $L_p$-discrepancy to standard $L_p$-discrepancy with explicit constants depending on the weight function. For $p=2$, we obtain explicit bounds using the exact discrepancy formulas. For $p>2$, we derive probabilistic bounds via dyadic chaining techniques. The results yield uniform error estimates for multivariate integration in Sobolev spaces $\mathcal{H}^1(K)$ and $F^*_{d,q}$, demonstrating improved performance over classical jittered sampling in importance sampling scenarios. Numerical experiments validate our theoretical findings and illustrate the practical advantages of parametric stratified sampling. |
| title | Weighted $L_p$-Discrepancy Bounds for Parametric Stratified Sampling and Applications to High-Dimensional Integration |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2512.21838 |