arXiv subjects

Mario Ullrich

Publications and source records attributed to Mario Ullrich.

At least 37 records · Page 2Linked to original sources

arXiv2020

On the worst-case error of least squares algorithms for $L_2$-approximation with high probability

It was recently shown in [4] that, for $L_2$-approximation of functions from a Hilbert space, function values are almost as powerful as arbitrary linear information, if the approximation numbers are square-summable. That is, we showed that \[ e_n \,\lesssim\, \sqrt{\frac{1}{k_n} \sum_{j\geq k_n} a_j^2} \qquad \text{ with }\quad k_n \asymp \frac{n}{\ln(n)}, \] where $e_n$ are the sampling numbers and $a_k$ are the approximation numbers. In particular, if $(a_k)\in\ell_2$, then $e_n$ and $a_n$ are of the same polynomial order. For this, we presented an explicit (weighted least squares) algorithm based on i.i.d. random points and proved that this works with positive probability. This implies the existence of a good deterministic sampling algorithm. Here, we present a modification of the proof in [4] that shows that the same algorithm works with probability at least $1-{n^{-c}}$ for all $c>0$.

math.NA↗

arXiv2019

On the fixed volume discrepancy of the Fibonacci sets in the integral norms

This paper is devoted to the study of a discrepancy-type characteristic -- the fixed volume discrepancy -- of the Fibonacci point set in the unit square. It was observed recently that this new characteristic allows us to obtain optimal rate of dispersion from numerical integration results. This observation motivates us to thoroughly study this new version of discrepancy, which seems to be interesting by itself. The new ingredient of this paper is the use of the average over the shifts of hat functions instead of taking the supremum over the shifts. We show that this change in the setting results in an improvement of the upper bound for the smooth fixed volume discrepancy, similarly to the well-known results for the usual $L_p$-discrepancy. Interestingly, this shows that ``bad boxes'' for the usual discrepancy cannot be ``too small''. The known results on smooth discrepancy show that the obtained bounds cannot be improved in a certain sense.

math.NA↗

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↗

arXiv2019

Deterministic constructions of high-dimensional sets with small dispersion

The dispersion of a point set $P\subset[0,1]^d$ is the volume of the largest box with sides parallel to the coordinate axes, which does not intersect $P$. Here, we show a construction of low-dispersion point sets, which can be deduced from solutions of certain $k$-restriction problems, which are well-known in coding theory. It was observed only recently that, for any $\varepsilon>0$, certain randomized constructions provide point sets with dispersion smaller than $\varepsilon$ and number of elements growing only logarithmically in $d$. Based on deep results from coding theory, we present explicit, deterministic algorithms to construct such point sets in time that is only polynomial in $d$. Note that, however, the running-time will be super-exponential in $\varepsilon^{-1}$.

cs.CC↗

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

Comparison of hit-and-run, slice sampling and random walk Metropolis

Different Markov chains can be used for approximate sampling of a distribution given by an unnormalized density function with respect to the Lebesgue measure. The hit-and-run, (hybrid) slice sampler and random walk Metropolis algorithm are popular tools to simulate such Markov chains. We develop a general approach to compare the efficiency of these sampling procedures by the use of a partial ordering of their Markov operators, the covariance ordering. In particular, we show that the hit-and-run and the simple slice sampler are more efficient than a hybrid slice sampler based on hit-and-run which, itself, is more efficient than a (lazy) random walk Metropolis algorithm.

math.PR↗

arXiv2018

Lattice rules with random $n$ achieve nearly the optimal $\mathcal{O}(n^{-α-1/2})$ error independently of the dimension

We analyze a new random algorithm for numerical integration of $d$-variate functions over $[0,1]^d$ from a weighted Sobolev space with dominating mixed smoothness $α\ge 0$ and product weights $1\geγ_1\geγ_2\ge\cdots>0$, where the functions are continuous and periodic when $α>1/2$. The algorithm is based on rank-$1$ lattice rules with a random number of points~$n$. For the case $α>1/2$, we prove that the algorithm achieves almost the optimal order of convergence of $\mathcal{O}(n^{-α-1/2})$, where the implied constant is independent of the dimension~$d$ if the weights satisfy $\sum_{j=1}^\infty γ_j^{1/α}<\infty$. The same rate of convergence holds for the more general case $α>0$ by adding a random shift to the lattice rule with random $n$. This shows, in particular, that the exponent of strong tractability in the randomized setting equals $1/(α+1/2)$, if the weights decay fast enough. We obtain a lower bound to indicate that our results are essentially optimal. This paper is a significant advancement over previous related works with respect to the potential for implementation and the independence of error bounds on the problem dimension. Other known algorithms which achieve the optimal error bounds, such as those based on Frolov's method, are very difficult to implement especially in high dimensions. Here we adapt a lesser-known randomization technique introduced by Bakhvalov in 1961. This algorithm is based on rank-$1$ lattice rules which are very easy to implement given the integer generating vectors. A simple probabilistic approach can be used to obtain suitable generating vectors.

math.NA↗

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

Numerical performance of optimized Frolov lattices in tensor product reproducing kernel Sobolev spaces

In this paper, we deal with several aspects of the universal Frolov cubature method, that is known to achieve optimal asymptotic convergence rates in a broad range of function spaces. Even though every admissible lattice has this favorable asymptotic behavior, there are significant differences concerning the precise numerical behavior of the worst-case error. To this end, we propose new generating polynomials that promise a significant reduction of the integration error compared to the classical polynomials. Moreover, we develop a new algorithm to enumerate the Frolov points from non-orthogonal lattices for numerical cubature in the $d$-dimensional unit cube $[0,1]^d$. Finally, we study Sobolev spaces with anisotropic mixed smoothness and compact support in $[0,1]^d$ and derive explicit formulas for their reproducing kernels. This allows for the simulation of exact worst-case errors which numerically validate our theoretical results.

math.NA↗

arXiv2018

Digital net properties of a polynomial analogue of Frolov's construction

Frolov's cubature formula on the unit hypercube has been considered important since it attains an optimal rate of convergence for various function spaces. Its integration nodes are given by shrinking a suitable full rank $\mathbb{Z}$-lattice in $\mathbb{R}^d$ and taking all points inside the unit cube. The main drawback of these nodes is that they are hard to find computationally, especially in high dimensions.In such situations, quasi-Monte Carlo (QMC) rules based on digital nets have proven to be successful. However, there is still no construction known that leads to QMC rules which are optimal in the same generality as Frolov's. In this paper we investigate a polynomial analog of Frolov's cubature formula, which we expect to be important in this respect. This analog is defined in a field of Laurent series with coefficients in a finite field. A similar approach was previously studied in [M.~B.~Levin. Adelic constructions of low discrepancy sequences. Online Journal of Analytic Combinatorics. Issue 5, 2010.]. We show that our construction is a $(t,m,d)$-net, which also implies bounds on its star-discrepancy and the error of the corresponding cubature rule. Moreover, we show that our cubature rule is a QMC rule, whereas Frolov's is not, and provide an algorithm to determine its integration nodes explicitly. To this end we need to extend the notion of $(t,m,d)$-nets to fit the situation that the points can have infinite digit expansion and develop a duality theory. Additionally, we adapt the notion of admissible lattices to our setting and prove its significance.

math.NA↗

arXiv2017

An upper bound on the minimal dispersion

For $\varepsilon\in(0,1/2)$ and a natural number $d\ge 2$, let $N$ be a natural number with \[ N \,\ge\, 2^9\,\log_2(d)\, \left(\frac{\log_2(1/\varepsilon)}{\varepsilon}\right)^2. \] We prove that there is a set of $N$ points in the unit cube $[0,1]^d$, which intersects all axis-parallel boxes with volume $\varepsilon$. That is, the dispersion of this point set is bounded from above by $\varepsilon$.

math.CA↗

arXiv2017

A note on the dispersion of admissible lattices

In this note we show that the volume of axis-parallel boxes in $\mathbb{R}^d$ which do not intersect an admissible lattice $\mathbb{L}\subset\mathbb{R}^d$ is uniformly bounded. In particular, this implies that the dispersion of the dilated lattices $N^{-1/d}\mathbb{L}$ restricted to the unit cube is of the (optimal) order $N^{-1}$ as $N$ goes to infinity. This result was obtained independently by V.N. Temlyakov (arXiv:1709.08158).

cs.CG↗

arXiv2017

Reproducing Kernels of Sobolev Spaces on $\mathbb{R}^d$ and Applications to Embedding Constants and Tractability

The standard Sobolev space $W^s_2(\mathbb{R}^d)$, with arbitrary positive integers $s$ and $d$ for which $s>d/2$, has the reproducing kernel $$ K_{d,s}(x,t)=\int_{\mathbb{R}^d}\frac{\prod_{j=1}^d\cos\left(2π\,(x_j-t_j)u_j\right)} {1+\sum_{0<|α|_1\le s}\prod_{j=1}^d(2π\,u_j)^{2α_j}}\,{\rm d}u $$ for all $x,t\in\mathbb{R}^d$, where $x_j,t_j,u_j,α_j$ are components of $d$-variate $x,t,u,α$, and $|α|_1=\sum_{j=1}^dα_j$ with non-negative integers $α_j$. We obtain a more explicit form for the reproducing kernel $K_{1,s}$ and find a closed form for the kernel $K_{d, \infty}$. Knowing the form of $K_{d,s}$, we present applications on the best embedding constants between the Sobolev space $W^s_2(\mathbb{R}^d)$ and $L_\infty(\mathbb{R}^d)$, and on strong polynomial tractability of integration with an arbitrary probability density. We prove that the best embedding constants are exponentially small in $d$, whereas worst case integration errors of algorithms using $n$ function values are also exponentially small in $d$ and decay at least like $n^{-1/2}$. This yields strong polynomial tractability in the worst case setting for the absolute error criterion.

math.NA↗

arXiv2017

A Monte Carlo method for integration of multivariate smooth functions

We study a Monte Carlo algorithm that is based on a specific (randomly shifted and dilated) lattice point set. The main result of this paper is that the mean squared error for a given compactly supported, square-integrable function is bounded by $n^{-1/2}$ times the $L_2$-norm of the Fourier transform outside a region around the origin, where $n$ is the expected number of function evaluations. As corollaries we obtain the optimal order of convergence for functions from the Sobolev spaces $H^s_p$ with isotropic, anisotropic, or mixed smoothness with given compact support for all values of the parameters. If the region of integration is the unit cube, we obtain the same optimal orders for functions without boundary conditions. This proves, in particular, that the optimal order of convergence in the latter case is $n^{-s-1/2}$ for $p\ge2$, which is, in contrast to the case of deterministic algorithms, independent of the dimension. This shows that Monte Carlo algorithms can improve the order by more than $n^{-1/2}$ for a whole class of natural function spaces.

math.NA↗

arXiv2016

Lattice based integration algorithms: Kronecker sequences and rank-1 lattices

We prove upper bounds on the order of convergence of lattice based algorithms for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov spaces of dominating mixed smoothness $\mathring{\mathbf{B}}^s_{p,θ}$, which also comprise the concept of Sobolev spaces of dominating mixed smoothness $\mathring{\mathbf{H}}^s_{p}$ as special cases. The considered algorithms are quasi-Monte Carlo rules with underlying nodes from $T_N(\mathbb{Z}^d) \cap [0,1)^d$, where $T_N$ is a real invertible generator matrix of size $d$. For such rules the worst-case error can be bounded in terms of the Zaremba index of the lattice $\mathbb{X}_N=T_N(\mathbb{Z}^d)$. We apply this result to Kronecker lattices and to rank-1 lattice point sets, which both lead to optimal error bounds up to $\log N$-factors for arbitrary smoothness $s$. The advantage of Kronecker lattices and classical lattice point sets is that the run-time of algorithms generating these point sets is very short.

math.NA↗

arXiv2016

The role of Frolov's cubature formula for functions with bounded mixed derivative

We prove upper bounds on the order of convergence of Frolov's cubature formula for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov $\mathbf{B}^s_{p,θ}$ and Triebel-Lizorkin spaces $\mathbf{F}^s_{p,θ}$ and our results treat the whole range of admissible parameters $(s\geq 1/p)$. In particular, we obtain upper bounds for the difficult the case of small smoothness which is given for Triebel-Lizorkin spaces $\mathbf{F}^s_{p,θ}$ in case $1<θ 1/θ$. In the latter case the presented upper bounds are optimal, i.e., they can not be improved by any other cubature formula. The optimality for "small" smoothness is open.

math.NA↗

arXiv2016

Product rules are optimal for numerical integration in classical smoothness spaces

We mainly study numerical integration of real valued functions defined on the $d$-dimensional unit cube with all partial derivatives up to some finite order $r\ge1$ bounded by one. It is well known that optimal algorithms that use $n$ function values achieve the error rate $n^{-r/d}$, where the hidden constant depends on $r$ and $d$. Here we prove explicit error bounds without hidden constants and, in particular, show that the optimal order of the error is $\min \bigl\{1, d \, n^{-r/d}\bigr\}$, where now the hidden constant only depends on $r$, not on $d$. For $n=m^d$, this optimal order can be achieved by (tensor) product rules. We also provide lower bounds for integration defined over an arbitrary open domain of volume one. We briefly discuss how lower bounds for integration may be applied for other problems such as multivariate approximation and optimization.

math.NA↗