Searcharxiv⌕ Search

arXiv subjects

Friedrich Pillichshammer

Publications and source records attributed to Friedrich Pillichshammer.

At least 19 recordsLinked to original sources
arXiv2026

Tractability versus curse of dimensionality for geometric $L_p$-discrepancies

This paper studies tractability versus the curse of dimensionality for several geometric $L_p$-discrepancies through a unified discrepancy--integration duality framework, where worst case integration errors in suitable function spaces equal the corresponding discrepancies. A general lower bound method for non-negative linear rules in spaces under broad tensor-product assumptions establishes exponential information complexity in the dimension $d$, yielding the curse of dimensionality for the respective discrepancy. We complement this overview by new results on discrepancy--integration duality and the curse of dimensionality for the periodic $L_p$-discrepancy. The current state of research on this general problem is summarized in a clearly laid out table, which also highlights the remaining open questions.

math.NA↗
arXiv2026

Infinite sequences with optimal diaphony, periodic $L_2$-discrepancy, and beyond

We investigate the periodic $L_2$-discrepancy of infinite sequences $§_d$ in $[0,1)^d$ and its analytic counterpart, the diaphony. We prove that infinite order-2 digital sequences over $\mathbb{F}_2$ attain the optimal order $L_{2,N}^{\rm per}(§_d) \le C_d (\log N)^{d/2}/N$ for all $N \in \mathbb{N}\setminus \{1\}$, matching known lower bounds for infinitely many $N \in \mathbb{N}$. This confirms the conjectured optimality of order-2 constructions. By this result, we improve upon previously known constructions using order-5 digital sequences, and reduce the underlying dimension for the interlacing construction from $5d$ to $2d$, significantly improving practicality. We establish our bounds within a broader framework of quasi-Monte Carlo integration for periodic Besov spaces $S_{p,q}^rB(\mathbb{T}^d)$ with dominating mixed smoothness $r \in (1/p,2)$, where $p,q\in [1,\infty]$. Rules based on infinite order-2 digital sequences yield worst-case errors of order $(\log N)^{(d-1)(1-1/q)} / N^{ \min(r,1)}$ for $r \not=1$, and $(\log N)^{d(1-1/q)}/N$ for $r=1$, for all $N \in \mathbb{N}\setminus\{1\}$, while preserving extensibility in $N$.

math.NA↗
arXiv2026

Extreme $L_p$ discrepancy, numerical integration and the curse of dimensionality

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.

math.NA↗
arXiv2026

Spherical Cap $L_2$ Discrepancy -- Blessing of Dimensionality and a Balanced Large-Cap Variant

We prove that the information complexity (i.e., the inverse) of the classical spherical cap $L_2$ discrepancy on the $d$-dimensional sphere $\mathbb{S}^d$ decreases with dimension $d$, indicating a ``blessing of dimensionality'' for the associated numerical integration problem. We then introduce a modified spherical cap $L_2$ discrepancy that emphasizes large caps (close to hemispheres). For this variant, the problem does not become easier with increasing $d$. We also establish a Stolarsky invariance principle which connects the modified spherical cap $L_2$ discrepancy to numerical integration in the Sobolev space $H^{(d+1)/2}(\mathbb{S}^d)$, represented by the reproducing kernel $K(\boldsymbol{x}, \boldsymbol{y}) = 1 - \tfrac{1}{\sqrt{2}} \|\boldsymbol{x} - \boldsymbol{y}\|$. Stolarsky's invariance principle then implies that the worst-case integration error in this space grows polynomially with $d$.

math.NA↗
arXiv2026

The star discrepancy of a union of randomly digitally shifted Korobov polynomial lattice point sets depends polynomially on the dimension

The star discrepancy is a quantitative measure of the uniformity of a point set in the unit cube. A central quantity of interest is the inverse of the star discrepancy, $N(\varepsilon, s)$, defined as the minimum number of points required to achieve a star discrepancy of at most~$\varepsilon$ in dimension~$s$. It is known that $N(\varepsilon, s)$ depends only linearly on the dimension~$s$. All known proofs of this result are non-constructive. Finding explicit point set constructions that achieve this optimal linear dependence on the dimension remains a major open problem. In this paper, we make progress on this question by analyzing point sets constructed from a multiset union of digitally shifted Korobov polynomial lattice point sets. Specifically, we show the following two results. A union of randomly generated Korobov polynomial lattice point sets shifted by a random digital shift of depth $m$ can achieve a star discrepancy whose inverse depends only linearly on $s$. The second result shows that a union of all Korobov polynomial lattice point sets, each shifted by a different random digital shift, achieves the same star discrepancy bound. While our proof relies on a concentration result (Bennett's inequality) and is therefore non-constructive, it significantly reduces the search space for such point sets from a continuum of possibilities to a finite set of candidates, marking a step towards a fully explicit construction.

math.NA↗
arXiv2026

On the quasi-uniformity properties of quasi-Monte Carlo point sets and sequences -- Part I: Lattices and Kronecker sequences

The discrepancy of a point set quantifies how well the points are distributed, with low-discrepancy point sets demonstrating exceptional uniform distribution properties. Such sets are integral to quasi-Monte Carlo methods, which approximate integrals over the unit cube for integrands of bounded variation. In contrast, quasi-uniform point sets are characterized by optimal separation and covering radii, making them well-suited for applications such as radial basis function approximation. This paper explores the quasi-uniformity properties of quasi-Monte Carlo point sets constructed from lattices and also Kronecker sequences. Specifically, we analyze rank-1 lattice point sets, Fibonacci lattice point sets, Frolov point sets, and Kronecker sequences (also referred to as $(n \boldsymbolα)$-sequences), providing insights into their potential for use in applications that require both low-discrepancy and quasi-uniform distribution. As an example, we show that the $(n \boldsymbolα)$-sequence with $α_j = 2^{j/(d+1)}$ for $j \in \{1, 2, \ldots, d\}$ is quasi-uniform and has low-discrepancy. The quasi-uniformity properties of quasi-Monte Carlo digital nets and sequences will be studied in a companion paper.

math.NT↗
arXiv2025

Generalized Discrepancy of Random Points

We study the $L_p$-discrepancy of random point sets in high dimensions, with emphasis on small values of $p$. Although the classical $L_p$-discrepancy suffers from the curse of dimensionality for all $p \in (1,\infty)$, the gap between known upper and lower bounds remains substantial, in particular for small $p \ge 1$. To clarify this picture, we review the existing results for i.i.d.\ uniformly distributed points and derive new upper bounds for \emph{generalized} $L_p$-discrepancies, obtained by allowing non-uniform sampling densities and corresponding non-negative quadrature weights. Using the probabilistic method, we show that random points drawn from optimally chosen product densities lead to significantly improved upper bounds. For $p=2$ these bounds are explicit and optimal; for general $p \in [1,\infty)$ we obtain sharp asymptotic estimates. The improvement can be interpreted as a form of importance sampling for the underlying Sobolev space $F_{d,q}$. Our results also reveal that, even with optimal densities, the curse of dimensionality persists for random points when $p\ge 1$, and it becomes most pronounced for small $p$. This suggests that the curse should also hold for the classical $L_1$-discrepancy for deterministic point sets.

math.NA↗
arXiv2025

Second order interlaced polynomial lattice rules for integration over $\mathbb{R}^s$

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.

math.NA↗
arXiv2024

Lebesgue constants for the Walsh system and the discrepancy of the van der Corput sequence

In this short note we report on a coincidence of two mathematical quantities that, at first glance, have little to do with each other. On the one hand, there are the Lebesgue constants of the Walsh function system that play an important role in approximation theory, and on the other hand there is the star discrepancy of the van der Corput sequence that plays a prominent role in uniform distribution theory. Over the decades, these two quantities have been examined in great detail independently of each other and important results have been proven. Work in these areas has been carried out independently, but as we show here, they actually coincide. Interestingly, many theorems have been discovered in both areas independently, but some results have only been known in one area but not in the other.

math.NT↗
arXiv2024

Conditions for tractability of the weighted $L_p$-discrepancy and integration in non-homogeneous tensor product spaces

We study tractability properties of the weighted $L_p$-discrepancy. The concept of {\it weighted} discrepancy was introduced by Sloan and Woź\-nia\-kowski in 1998 in order to prove a weighted version of the Koksma-Hlawka inequality for the error of quasi-Monte Carlo integration rules. The weights have the aim to model the influence of different coordinates of integrands on the error. A discrepancy is said to be tractable if the information complexity, i.e., the minimal number $N$ of points such that the discrepancy is less than the initial discrepancy times an error threshold $\varepsilon$, does not grow exponentially fast with the dimension. In this case there are various notions of tractabilities used in order to classify the exact rate. For even integer parameters $p$ there are sufficient conditions on the weights available in literature, which guarantee the one or other notion of tractability. In the present paper we prove matching sufficient conditions (upper bounds) and neccessary conditions (lower bounds) for polynomial and weak tractability for all $p \in (1, \infty)$. The proofs of the lower bounds are based on a general result for the information complexity of integration with positive quadrature formulas for tensor product spaces. In order to demonstrate this lower bound we consider as a second application the integration of tensor products of polynomials of degree at most 2.

math.NA↗
arXiv2024

Intractability results for integration in tensor product spaces

We study lower bounds on the worst-case error of numerical integration in tensor product spaces. As reference we use the $N$-th minimal error of linear rules that use $N$ function values. The information complexity is the minimal number $N$ of function evaluations that is necessary such that the $N$-th minimal error is less than a factor $\varepsilon$ times the initial error. We are interested to which extent the information complexity depends on the number $d$ of variables of the integrands. If the information complexity grows exponentially fast in $d$, then the integration problem is said to suffer from the curse of dimensionality. Under the assumption of the existence of a worst-case function for the uni-variate problem we present two methods for providing good lower bounds on the information complexity. The first method is based on a suitable decomposition of the worst-case function. This method can be seen as a generalization of the method of decomposable reproducing kernels, that is often successfully applied when integration in Hilbert spaces with a reproducing kernel is studied. The second method, although only applicable for positive quadrature rules, has the advantage, that it does not require a suitable decomposition of the worst-case function. Rather, it is based on a spline approximation of the worst-case function and can be used for analytic functions. The methods presented can be applied to problems beyond the Hilbert space setting. For demonstration purposes we apply them to several examples, notably to uniform integration over the unit-cube, weighted integration over the whole space, and integration of infinitely smooth functions over the cube. Some of these results have interesting consequences in discrepancy theory.

math.NA↗
arXiv2024

The $L_p$-discrepancy for finite $p>1$ suffers from the curse of dimensionality

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 $p$.

math.NA↗
arXiv2023

Bernstein approximation and beyond: proofs by means of elementary probability theory

Bernstein polynomials provide a constructive proof for the Weierstrass approximation theorem, which states that every continuous function on a closed bounded interval can be uniformly approximated by polynomials with arbitrary accuracy. Interestingly the proof of this result can be done using elementary probability theory. This way one can even get error bounds for Lipschitz functions. In this note, we present these techniques and show how the method can be extended naturally to other interesting situations. As examples, we obtain in an elementary way results for the Szász-Mirakjan operator and the Baskakov operator.

math.NA↗
arXiv2023

The curse of dimensionality for the $L_p$-discrepancy with finite $p$

The $L_p$-discrepancy is a quantitative measure for the irregularity of distribution of an $N$-element point set in the $d$-dimensional unit cube, which is closely related to the worst-case error of quasi-Monte Carlo algorithms for numerical integration. Its inverse for dimension $d$ and error threshold $\varepsilon \in (0,1)$ is the minimal number of points in $[0,1)^d$ 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 $p \not\in \{2,\infty\}$ has been an open problem for many years. In this paper we show that the $L_p$-discrepancy suffers from the curse of dimensionality for all $p$ in $(1,2]$ which are of the form $p=2 \ell/(2 \ell -1)$ with $\ell \in \mathbb{N}$. This result follows from a more general result that we show for the worst-case error of numerical integration in an anchored Sobolev space with anchor 0 of once differentiable functions in each variable whose first derivative has finite $L_q$-norm, where $q$ is an even positive integer satisfying $1/p+1/q=1$.

math.NA↗
arXiv2023

Quasi-Monte Carlo methods for mixture distributions and approximated distributions via piecewise linear interpolation

We study numerical integration over bounded regions in $\mathbb{R}^s, s\ge1$ with respect to some probability measure. We replace random sampling with quasi-Monte Carlo methods, where the underlying point set is derived from deterministic constructions that aim to fill the space more evenly than random points. Such quasi-Monte Carlo point sets are ordinarily designed for the uniform measure, and the theory only works for product measures when a coordinate-wise transformation is applied. Going beyond this setting, we first consider the case where the target density is a mixture distribution where each term in the mixture comes from a product distribution. Next we consider target densities which can be approximated with such mixture distributions. We require the approximation to be a sum of coordinate-wise products and the approximation to be positive everywhere (so that they can be re-scaled to probability density functions). We use tensor product hat function approximations for this purpose here, since a hat function approximation of a positive function is itself positive. We also study more complex algorithms, where we first approximate the target density with a general Gaussian mixture distribution and approximate the mixtures with an adaptive hat function approximation on rotated intervals. The Gaussian mixture approximation allows us to locate the essential parts of the target density, whereas the adaptive hat function approximation allows us to approximate the finer structure of the target density. We prove convergence rates for each of the integration techniques based on quasi-Monte Carlo sampling for integrands with bounded partial mixed derivatives. The employed algorithms are based on digital $(t,s)$-sequences over the finite field $\mathbb{F}_2$ and an inversion method. Numerical examples illustrate the performance of the algorithms for some target densities and integrands.

math.NA↗
arXiv2023

Grid-Based Decimation for Wavelet Transforms with Stably Invertible Implementation

The constant center frequency to bandwidth ratio (Q-factor) of wavelet transforms provides a very natural representation for audio data. However, invertible wavelet transforms have either required non-uniform decimation -- leading to irregular data structures that are cumbersome to work with -- or require excessively high oversampling with unacceptable computational overhead. Here, we present a novel decimation strategy for wavelet transforms that leads to stable representations with oversampling rates close to one and uniform decimation. Specifically, we show that finite implementations of the resulting representation are energy-preserving in the sense of frame theory. The obtained wavelet coefficients can be stored in a timefrequency matrix with a natural interpretation of columns as time frames and rows as frequency channels. This matrix structure immediately grants access to a large number of algorithms that are successfully used in time-frequency audio processing, but could not previously be used jointly with wavelet transforms. We demonstrate the application of our method in processing based on nonnegative matrix factorization, in onset detection, and in phaseless reconstruction.

eess.AS↗
arXiv2022

Optimal periodic $L_2$-discrepancy and diaphony bounds for higher order digital sequences

We present an explicit construction of infinite sequences of points $(\boldsymbol{x}_0,\boldsymbol{x}_1, \boldsymbol{x}_2, \ldots)$ in the $d$-dimensional unit-cube whose periodic $L_2$-discrepancy satisfies $$L_{2,N}^{\rm per}(\{\boldsymbol{x}_0,\boldsymbol{x}_1,\ldots, \boldsymbol{x}_{N-1}\}) \le C_d N^{-1} (\log N)^{d/2} \quad \mbox{for all } N \ge 2,$$ where the factor $C_d > 0$ depends only on the dimension $d$. The construction is based on higher order digital sequences as introduced by J. Dick in the year 2008. The result is best possible in the order of magnitude in $N$ according to a Roth-type lower bound shown first by P.D. Proinov. Since the periodic $L_2$-discrepancy is equivalent to P. Zinterhof's diaphony the result also applies to this alternative quantitative measure for the irregularity of distribution.

math.NT↗