The $L_p$-discrepancy for finite $p>1$ suffers from the curse of dimensionality
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866917623423303680 |
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| author | Novak, Erich Pillichshammer, Friedrich |
| author_facet | Novak, Erich Pillichshammer, Friedrich |
| contents | The $L_p$-discrepancy is a classical quantitative measure for the irregularity of distribution of an $N$-element point set in the $d$-dimensional unit cube. Its inverse for dimension $d$ and error threshold $\varepsilon \in (0,1)$ is the number of points in $[0,1)^d$ that is required such that the minimal normalized $L_p$-discrepancy is less or equal $\varepsilon$. It is well known, that the inverse of $L_2$-discrepancy grows exponentially fast with the dimension $d$, i.e., we have the curse of dimensionality, whereas the inverse of $L_{\infty}$-discrepancy depends exactly linearly on $d$. The behavior of inverse of $L_p$-discrepancy for general $p \not\in \{2,\infty\}$ was an open problem since many years. Recently, the curse of dimensionality for the $L_p$-discrepancy was shown for an infinite sequence of values $p$ in $(1,2]$, but the general result seemed to be out of reach. In the present paper we show that the $L_p$-discrepancy suffers from the curse of dimensionality for all $p$ in $(1,\infty)$ and only the case $p=1$ is still open. This result follows from a more general result that we show for the worst-case error of positive quadrature formulas for an anchored Sobolev space of once differentiable functions in each variable whose first mixed derivative has finite $L_q$-norm, where $q$ is the Hölder conjugate of $p$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_07961 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $L_p$-discrepancy for finite $p>1$ suffers from the curse of dimensionality Novak, Erich Pillichshammer, Friedrich Numerical Analysis 11K38, 65C05, 65Y20 The $L_p$-discrepancy is a classical quantitative measure for the irregularity of distribution of an $N$-element point set in the $d$-dimensional unit cube. Its inverse for dimension $d$ and error threshold $\varepsilon \in (0,1)$ is the number of points in $[0,1)^d$ that is required such that the minimal normalized $L_p$-discrepancy is less or equal $\varepsilon$. It is well known, that the inverse of $L_2$-discrepancy grows exponentially fast with the dimension $d$, i.e., we have the curse of dimensionality, whereas the inverse of $L_{\infty}$-discrepancy depends exactly linearly on $d$. The behavior of inverse of $L_p$-discrepancy for general $p \not\in \{2,\infty\}$ was an open problem since many years. Recently, the curse of dimensionality for the $L_p$-discrepancy was shown for an infinite sequence of values $p$ in $(1,2]$, but the general result seemed to be out of reach. In the present paper we show that the $L_p$-discrepancy suffers from the curse of dimensionality for all $p$ in $(1,\infty)$ and only the case $p=1$ is still open. This result follows from a more general result that we show for the worst-case error of positive quadrature formulas for an anchored Sobolev space of once differentiable functions in each variable whose first mixed derivative has finite $L_q$-norm, where $q$ is the Hölder conjugate of $p$. |
| title | The $L_p$-discrepancy for finite $p>1$ suffers from the curse of dimensionality |
| topic | Numerical Analysis 11K38, 65C05, 65Y20 |
| url | https://arxiv.org/abs/2403.07961 |