Extreme $L_p$ discrepancy, numerical integration and the curse of dimensionality
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866913171728498688 |
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| author | Novak, Erich Pillichshammer, Friedrich |
| author_facet | Novak, Erich Pillichshammer, Friedrich |
| contents | The classical notion of extreme $L_p$ discrepancy is a quantitative measure for the irregularity of distribution of finite point sets in the $d$-dimensinal unit cube. In this paper we find a dual integration problem whose worst-case error is exactly the extreme $L_p$ discrepancy of the underlying integration nodes. Studying this integration problem we show that the extreme $L_p$ discrepancy suffers from the curse of dimensionality for all $p \in (1,\infty)$. It is known that the problem is tractable for $p=\infty$; the case $p=1$ stays open. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19760 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extreme $L_p$ discrepancy, numerical integration and the curse of dimensionality Novak, Erich Pillichshammer, Friedrich Numerical Analysis Number Theory 11K38, 65C05, 65Y20 The classical notion of extreme $L_p$ discrepancy is a quantitative measure for the irregularity of distribution of finite point sets in the $d$-dimensinal unit cube. In this paper we find a dual integration problem whose worst-case error is exactly the extreme $L_p$ discrepancy of the underlying integration nodes. Studying this integration problem we show that the extreme $L_p$ discrepancy suffers from the curse of dimensionality for all $p \in (1,\infty)$. It is known that the problem is tractable for $p=\infty$; the case $p=1$ stays open. |
| title | Extreme $L_p$ discrepancy, numerical integration and the curse of dimensionality |
| topic | Numerical Analysis Number Theory 11K38, 65C05, 65Y20 |
| url | https://arxiv.org/abs/2602.19760 |