arXiv subjects

Aicke Hinrichs

Publications and source records attributed to Aicke Hinrichs.

At least 19 recordsLinked to original sources

arXiv2023

Infinite-dimensional integration and $L^2$-approximation on Hermite spaces

We study integration and $L^2$-approximation of functions of infinitely many variables in the following setting: The underlying function space is the countably infinite tensor product of univariate Hermite spaces and the probability measure is the corresponding product of the standard normal distribution. The maximal domain of the functions from this tensor product space is necessarily a proper subset of the sequence space $\mathbb{R}^\mathbb{N}$. We establish upper and lower bounds for the minimal worst case errors under general assumptions; these bounds do match for tensor products of well-studied Hermite spaces of functions with finite or with infinite smoothness. In the proofs we employ embedding results, and the upper bounds are attained constructively with the help of multivariate decomposition methods.

math.NA↗

arXiv2022

Lower bounds for integration and recovery in $L_2$

Function values are, in some sense, "almost as good" as general linear information for $L_2$-approximation (optimal recovery, data assimilation) of functions from a reproducing kernel Hilbert space. This was recently proved by new upper bounds on the sampling numbers under the assumption that the singular values of the embedding of this Hilbert space into $L_2$ are square-summable. Here we mainly prove new lower bounds. In particular we prove that the sampling numbers behave worse than the approximation numbers for Sobolev spaces with small smoothness. Hence there can be a logarithmic gap also in the case where the singular numbers of the embedding are square-summable. We first prove new lower bounds for the integration problem, again for rather classical Sobolev spaces of periodic univariate functions.

math.NA↗

arXiv2021

Random sections of $\ell_p$-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators

We study the circumradius of a random section of an $\ell_p$-ellipsoid, $0<p\le \infty$, and compare it with the minimal circumradius over all sections with subspaces of the same codimension. Our main result is an upper bound for random sections, which we prove using techniques from asymptotic geometric analysis if $1\leq p \leq \infty$ and compressed sensing if $0<p \leq 1$. This can be interpreted as a bound on the quality of random (Gaussian) information for the recovery of vectors from an $\ell_p$-ellipsoid for which the radius of optimal information is given by the Gelfand numbers of a diagonal operator. In the case where the semiaxes decay polynomially and $1\le p\le \infty$, we conjecture that, as the amount of information increases, the radius of random information either decays like the radius of optimal information or is bounded from below by a constant, depending on whether the exponent of decay is larger than the critical value $1-\frac{1}{p}$ or not. If $1\leq p\leq 2$, we prove this conjecture by providing a matching lower bound. This extends the recent work of Hinrichs et al. [Random sections of ellipsoids and the power of random information, Trans. Amer. Math. Soc., 2021+] for the case $p=2$.

math.FA↗

arXiv2020

Extreme and periodic $L_2$ discrepancy of plane point sets

In this paper we study the extreme and the periodic $L_2$ discrepancy of plane point sets. The extreme discrepancy is based on arbitrary rectangles as test sets whereas the periodic discrepancy uses "periodic intervals", which can be seen as intervals on the torus. The periodic $L_2$ discrepancy is, up to a multiplicative factor, also known as diaphony. The main results are exact formulas for these kinds of discrepancies for the Hammersley point set and for rational lattices. In order to value the obtained results we also prove a general lower bound on the extreme $L_2$ discrepancy for arbitrary point sets in dimension $d$, which is of order of magnitude $(\log N)^{(d-1)/2}$, like the standard and periodic $L_2$ discrepancies, respectively. Our results confirm that the extreme and periodic $L_2$ discrepancies of the Hammersley point set are of best possible asymptotic order of magnitude. This is in contrast to the standard $L_2$ discrepancy of the Hammersley point set. Furthermore our exact formulas show that also the $L_2$ discrepancies of the Fibonacci lattice are of the optimal order. We also prove that the extreme $L_2$ discrepancy is always dominated by the standard $L_2$ discrepancy, a result that was already conjectured by Morokoff and Caflisch when they introduced the notion of extreme $L_2$ discrepancy in the year 1994.

math.NT↗

arXiv2020

Lower Bounds for the Error of Quadrature Formulas for Hilbert Spaces

We prove lower bounds for the worst case error of quadrature formulas that use given sample points $\X_n = \{ x_1, \dots , x_n \}$. We are mainly interested in optimal point sets $\X_n$, but also prove lower bounds that hold with high probability for sets of independently and uniformly distributed points. As a tool, we use a recent result (and extensions thereof) of Vybíral on the positive semi-definiteness of certain matrices related to the product theorem of Schur. The new technique also works for spaces of analytic functions where known methods based on decomposable kernels cannot be applied.

math.NA↗

arXiv2020

Gelfand numbers of embeddings of Schatten classes

Let $0<p,q\leq \infty$ and denote by $\mathcal{S}_p^N$ and $\mathcal{S}_q^N$ the corresponding Schatten classes of real $N\times N$ matrices. We study the Gelfand numbers of natural identities $\mathcal{S}_p^N\hookrightarrow \mathcal{S}_q^N$ between Schatten classes and prove asymptotically sharp bounds up to constants only depending on $p$ and $q$. This extends classical results for finite-dimensional $\ell_p$ sequence spaces by E. Gluskin to the non-commutative setting and complements bounds previously obtained by B. Carl and A. Defant, A. Hinrichs and C. Michels, and J. Chávez-Domínguez and D. Kutzarova.

math.FA↗

arXiv2020

Random sections of ellipsoids and the power of random information

We study the circumradius of the intersection of an $m$-dimensional ellipsoid $\mathcal E$ with semi-axes $σ_1\geq\dots\geq σ_m$ with random subspaces of codimension $n$. We find that, under certain assumptions on $σ$, this random radius $\mathcal{R}_n=\mathcal{R}_n(σ)$ is of the same order as the minimal such radius $σ_{n+1}$ with high probability. In other situations $\mathcal{R}_n$ is close to the maximum $σ_1$. The random variable $\mathcal{R}_n$ naturally corresponds to the worst-case error of the best algorithm based on random information for $L_2$-approximation of functions from a compactly embedded Hilbert space $H$ with unit ball $\mathcal E$. In particular, $σ_k$ is the $k$th largest singular value of the embedding $H\hookrightarrow L_2$. In this formulation, one can also consider the case $m=\infty$, and we prove that random information behaves very differently depending on whether $σ\in \ell_2$ or not. For $σ\notin \ell_2$ random information is completely useless, i.e., $\mathbb E[\mathcal{R}_n] = σ_1$. For $σ\in \ell_2$ the expected radius of random information tends to zero at least at rate $o(1/\sqrt{n})$ as $n\to\infty$. In the important case $σ_k \asymp k^{-α} \ln^{-β}(k+1)$, where $α> 0$ and $β\in\mathbb R$, we obtain that $$ \mathbb E [\mathcal{R}_n(σ)] \asymp \begin{cases} σ_1 & : α<1/2 \,\text{ or }\, β\leqα=1/2 \\ σ_n \, \sqrt{\ln(n+1)} & : β>α=1/2 \\ σ_{n+1} & : α>1/2. \end{cases} $$ In the proofs we use a comparison result for Gaussian processes à la Gordon, exponential estimates for sums of chi-squared random variables, and estimates for the extreme singular values of (structured) Gaussian random matrices. The upper bound is constructive. It is proven for the worst case error of a least squares estimator.

math.FA↗

arXiv2020

Expected dispersion of uniformly distributed points

The dispersion of a point set in $[0,1]^d$ is the volume of the largest axis parallel box inside the unit cube that does not intersect with the point set. We study the expected dispersion with respect to a random set of $n$ points determined by an i.i.d. sequence of uniformly distributed random variables. Depending on the number of points $n$ and the dimension $d$ we provide an upper and lower bound of the expected dispersion. In particular, we show that the minimal number of points required to achieve an expected dispersion less than $\varepsilon\in(0,1)$ depends linearly on the dimension $d$.

math.PR↗

arXiv2020

Tractability properties of the discrepancy in Orlicz norms

We show that the minimal discrepancy of a point set in the $d$-dimensional unit cube with respect to Orlicz norms can exhibit both polynomial and weak tractability. In particular, we show that the $ψ_α$-norms of exponential Orlicz spaces are polynomially tractable.

math.NA↗

arXiv2020

A note on the periodic $L_2$-discrepancy of Korobov's $p$-sets

We study the periodic $L_2$-discrepancy of point sets in the $d$-dimensional torus. This discrepancy is intimately connected with the root-mean-square $L_2$-discrepancy of shifted point sets, with the notion of diaphony, and with the worst case error of cubature formulas for the integration of periodic functions in Sobolev spaces of mixed smoothness. In discrepancy theory many results are based on averaging arguments. In order to make such results relevant for applications one requires explicit constructions of point sets with ``average'' discrepancy. In our main result we study Korobov's $p$-sets and show that this point sets have periodic $L_2$-discrepancy of average order. This result is related to an open question of Novak and Woźniakowski.

math.NT↗

arXiv2019

Fibonacci lattices have minimal dispersion on the two-dimensional torus

We study the size of the largest rectangle containing no point of a given point set in the two-dimensional torus, the dispersion of the point set. A known lower bound for the dispersion of any point set of cardinality $n\ge 2$ in this setting is $2/n$. We show that if $n$ is a Fibonacci number then the Fibonacci lattice has dispersion exactly $2/n$ meeting the lower bound. Moreover, we completely characterize integration lattices achieving the lower bound and provide insight into the structure of other optimal sets. We also treat related results in the nonperiodic setting.

math.MG↗

arXiv2019

On the power of random information

We study approximation and integration problems and compare the quality of optimal information with the quality of random information. For some problems random information is almost optimal and for some other problems random information is much worse than optimal information. We prove new results and give a short survey of known results.

math.NA↗

arXiv2018

On a multi-dimensional Poissonian pair correlation concept and uniform distribution

The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, in the one-dimensional setting, the pair correlation statistics measures the distribution of spacings between sequence elements in the unit interval at distances of order of the mean spacing $1/N$. In the $d$-dimensional case, of course, the order of the mean spacing is $1/N^{\frac{1}{d}}$, and --in our concept-- the distance of sequence elements will be measured by the supremum-norm. Additionally, we show that, in some sense, almost all sequences satisfy this new concept and we examine the link to uniform distribution. The metrical pair correlation theory is investigated and it is proven that a class of typical low-discrepancy sequences in the high-dimensional unit cube do not have Poissonian pair correlations, which fits the existing results in the one-dimensional case.

math.NT↗

arXiv2018

The minimal $k$-dispersion of point sets in high-dimensions

In this manuscript we introduce and study an extended version of the minimal dispersion of point sets, which has recently attracted considerable attention. Given a set $\mathscr P_n=\{x_1,\dots,x_n\}\subset [0,1]^d$ and $k\in\{0,1,\dots,n\}$, we define the $k$-dispersion to be the volume of the largest box amidst a point set containing at most $k$ points. The minimal $k$-dispersion is then given by the infimum over all possible point sets of cardinality $n$. We provide both upper and lower bounds for the minimal $k$-dispersion that coincide with the known bounds for the classical minimal dispersion for a surprisingly large range of $k$'s.

math.NA↗

arXiv2018

The curse of dimensionality for numerical integration on general domains

We prove the curse of dimensionality in the worst case setting for multivariate numerical integration for various classes of smooth functions. We prove the results when the domains are isotropic convex bodies with small diameter satisfying a universal $ψ_2$-estimate. In particular, we obtain the result for the important class of volume-normalized $\ell_p^d$-balls in the complete regime $2\leq p \leq \infty$. This extends a result in a work of A. Hinrichs, E. Novak, M. Ullrich and H. Woźniakowski [J. Complexity, 30(2), 117-143, 2014] to the whole range $2\leq p \leq \infty$, and additionally provides a unified approach. The key ingredient in the proof is a deep result from the theory of Asymptotic Geometric Analysis, the thin-shell volume concentration estimate due to O. Guédon and E. Milman. The connection of Asymptotic Geometric Analysis and Information-based Complexity revealed in this work seems promising and is of independent interest.

math.NA↗

arXiv2018

Tractability properties of the weighted star discrepancy of the Halton sequence

We study the weighted star discrepancy of the Halton sequence. In particular, we show that the Halton sequence achieves strong polynomial tractability for the weighted star discrepancy for product weights $(γ_j)_{j \ge 1}$ under the mildest condition on the weight sequence known so far for explicitly constructive sequences. The condition requires $\sup_{d \ge 1} \max_{\emptyset \not= \mathfrak{u} \subseteq [d]} \prod_{j \in \mathfrak{u}} (j γ_j) < \infty$. The same result holds for Niederreiter sequences and for other types of digital sequences. Our results are true also for the weighted unanchored discrepancy.

math.NA↗

arXiv2017

Truncation Dimension for Linear Problems on Multivariate Function Spaces

The paper considers linear problems on weighted spaces of multivariate functions of many variables. The main questions addressed are: When is it possible to approximate the solution for the original function of very many variables by the solution for the same function; however with all but the first $k$ variables set to zero, so that the corresponding error is small? What is the truncation dimension, i.e., the smallest number $k=k(\varepsilon)$ such that the corresponding error is bounded by a given error demand $\varepsilon$? Surprisingly, $k(\varepsilon)$ could be very small even for weights with a modest speed of convergence to zero.

math.NA↗

arXiv2017

Irregularities of distributions and extremal sets in combinatorial complexity theory

In 2004 the second author of the present paper proved that a point set in $[0,1]^d$ which has star-discrepancy at most $\varepsilon$ must necessarily consist of at least $c_{abs} d \varepsilon^{-1}$ points. Equivalently, every set of $n$ points in $[0,1]^d$ must have star-discrepancy at least $c_{abs} d n^{-1}$. The original proof of this result uses methods from Vapnik--Chervonenkis theory and from metric entropy theory. In the present paper we give an elementary combinatorial proof for the same result, which is based on identifying a sub-box of $[0,1]^d$ which has approximately $d$ elements of the point set on its boundary. Furthermore, we show that a point set for which no such box exists is rather irregular, and must necessarily have a large star-discrepancy.

math.NA↗