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Henryk Wozniakowski

Publications and source records attributed to Henryk Wozniakowski.

14 recordsLinked to original sources
arXiv2018

Exponential tractability of linear tensor product problems

In this article we consider the approximation of compact linear operators defined over tensor product Hilbert spaces. Necessary and sufficient conditions on the singular values of the problem under which we can or cannot achieve different notions of exponential tractability are given in a paper by Papageorgiou, Petras, and Wozniakowski. In this paper, we use the new equivalency conditions shown in a recent paper by the second and third authors of this paper to obtain these results in an alternative way. As opposed to the algebraic setting, quasi-polynomial tractability is not possible for non-trivial cases in the exponential setting.

math.NA↗
arXiv2018

Notes on tractability conditions for linear multivariate problems

We study approximations of compact linear multivariate operators defined over Hilbert spaces. We provide necessary and sufficient conditions on various notions of tractability. These conditions are mainly given in terms of sums of certain functions depending on the singular values of the multivariate problem. They do not require the ordering of these singular values which in many cases is difficult to achieve.

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↗
arXiv2016

$\boldsymbol{L}_{\infty}$-approximation in Korobov spaces with Exponential Weights

We study multivariate $\boldsymbol{L}_{\infty}$-approximation for a weighted Korobov space of periodic functions for which the Fourier coefficients decay exponentially fast. The weights are defined, in particular, in terms of two sequences $\boldsymbol{a}=\{a_j\}$ and $\boldsymbol{b}=\{b_j\}$ of positive real numbers bounded away from zero. We study the minimal worst-case error $e^{\boldsymbol{L}_{\infty}\mathrm{-app},Λ}(n,s)$ of all algorithms that use $n$ information evaluations from a class $Λ$ in the $s$-variate case. We consider two classes $Λ$ in this paper: the class $Λ^{\rm all}$ of all linear functionals and the class $Λ^{\rm std}$ of only function evaluations. We study exponential convergence of the minimal worst-case error, which means that $e^{\boldsymbol{L}_{\infty}\mathrm{-app},Λ}(n,s)$ converges to zero exponentially fast with increasing $n$. Furthermore, we consider how the error depends on the dimension $s$. To this end, we define the notions of $κ$-EC-weak, EC-polynomial and EC-strong polynomial tractability, where EC stands for "exponential convergence". In particular, EC-polynomial tractability means that we need a polynomial number of information evaluations in $s$ and $1+\log\,\varepsilon^{-1}$ to compute an $\varepsilon$-approximation. We derive necessary and sufficient conditions on the sequences $\boldsymbol{a}$ and $\boldsymbol{b}$ for obtaining exponential error convergence, and also for obtaining the various notions of tractability. The results are the same for both classes $Λ$.

math.NA↗
arXiv2015

Tractability of Multivariate Problems for Standard and Linear Information in the Worst Case Setting: Part I

We present a lower error bound for approximating linear multivariate operators defined over Hilbert spaces in terms of the error bounds for appropriately constructed linear functionals as long as algorithms use function values. Furthermore, some of these linear functionals have the same norm as the linear operators. We then apply this error bound for linear (unweighted) tensor products. In this way we use negative tractability results known for linear functionals to conclude the same negative results for linear operators. In particular, we prove that $L_2$-multivariate approximation defined for standard Sobolev space suffers the curse of dimensionality if function values are used although the curse is not present if linear functionals are allowed.

math.NA↗
arXiv2015

Approximation in Hermite spaces of smooth functions

We consider $\mathbb{L}_2$-approximation of elements of a Hermite space of analytic functions over $\mathbb{R}^s$. The Hermite space is a weighted reproducing kernel Hilbert space of real valued functions for which the Hermite coefficients decay exponentially fast. The weights are defined in terms of two sequences $\boldsymbol{a} = \{a_j\}$ and $\boldsymbol{b} = \{b_j\}$ of positive real numbers. We study the $n$th minimal worst-case error $e(n,{\rm APP}_s;Λ^{\rm std})$ of all algorithms that use $n$ information evaluations from the class $Λ^{\rm std}$ which only allows function evaluations to be used. We study (uniform) exponential convergence of the $n$th minimal worst-case error, which means that $e(n,{\rm APP}_s; Λ^{\rm std})$ converges to zero exponentially fast with increasing $n$. Furthermore, we consider how the error depends on the dimension $s$. To this end, we study the minimal number of information evaluations needed to compute an $\varepsilon$-approximation by considering several notions of tractability which are defined with respect to $s$ and $\log \varepsilon^{-1}$. We derive necessary and sufficient conditions on the sequences $\boldsymbol{a}$ and $\boldsymbol{b}$ for obtaining exponential error convergence, and also for obtaining the various notions of tractability. It turns out that the conditions on the weight sequences are almost the same as for the information class $Λ^{\rm all}$ which uses all linear functionals. The results are also constructive as the considered algorithms are based on tensor products of Gauss-Hermite rules for multivariate integration. The obtained results are compared with the analogous results for integration in the same Hermite space. This allows us to give a new sufficient condition for EC-weak tractability for integration.

math.NA↗
arXiv2015

Tractability of Multivariate Approximation Defined over Hilbert Spaces with Exponential Weights

We study multivariate approximation defined over tensor product Hilbert spaces. The domain space is a weighted tensor product Hilbert space with exponential weights which depend on two sequences $\boldsymbol{a}=\{a_j\}_{j\in\mathbb{N}}$ and $\boldsymbol{b}=\{b_j\}_{j\in\mathbb{N}}$ of positive numbers, and on a bounded sequence of positive integers $\boldsymbol{m}=\{m_j\}_{j\in\mathbb{N}}$. The sequence $\boldsymbol{a}$ is non-decreasing and the sequence $\boldsymbol{b}$ is bounded from below by a positive number. We find necessary and sufficient conditions on $\boldsymbol{a},\boldsymbol{b}$ and $\boldsymbol{m}$ to achieve the standard and new notions of tractability in the worst case setting.

math.NA↗
arXiv2014

Tractability of multivariate analytic problems

In the theory of tractability of multivariate problems one usually studies problems with finite smoothness. Then we want to know which $s$-variate problems can be approximated to within $\varepsilon$ by using, say, polynomially many in $s$ and $\varepsilon^{-1}$ function values or arbitrary linear functionals. There is a recent stream of work for multivariate analytic problems for which we want to answer the usual tractability questions with $\varepsilon^{-1}$ replaced by $1+\log \varepsilon^{-1}$. In this vein of research, multivariate integration and approximation have been studied over Korobov spaces with exponentially fast decaying Fourier coefficients. This is work of J. Dick, G. Larcher, and the authors. There is a natural need to analyze more general analytic problems defined over more general spaces and obtain tractability results in terms of $s$ and $1+\log \varepsilon^{-1}$. The goal of this paper is to survey the existing results, present some new results, and propose further questions for the study of tractability of multivariate analytic questions.

math.NA↗
arXiv2013

The Curse of Dimensionality for Numerical Integration of Smooth Functions II

We prove the curse of dimensionality in the worst case setting for numerical integration for a number of classes of smooth $d$-variate functions. Roughly speaking, we consider different bounds for the derivatives of $f \in C^k(D_d)$ and ask whether the curse of dimensionality holds for the respective classes of functions. We always assume that $D_d \subset \mathbb{R}^d$ has volume one and consider various values of $k$ including the case $k=\infty$ which corresponds to infinitely many differentiable functions. We obtain necessary and sufficient conditions, and in some cases a full characterization for the curse of dimensionality. For infinitely many differentiable functions we prove the curse if the bounds on the successive derivatives are appropriately large. The proof technique is based on a volume estimate of a neighborhood of the convex hull of $n$ points which decays exponentially fast if $n$ is small relative to $d$. For $k=\infty$, we also also study conditions for quasi-polynomial, weak and uniform weak tractability.

math.NA↗
arXiv2013

The Curse of Dimensionality for Numerical Integration of Smooth Functions

We prove the curse of dimensionality for multivariate integration of C^r functions: The number of needed function values to achieve an error ε is larger than c_r (1+γ)^d for ε\le ε_0, where c_r,γ>0 and d is the dimension. The proofs are based on volume estimates for r=1 together with smoothing by convolution. This allows us to obtain smooth fooling functions for r>1.

math.NA↗
arXiv2011

Discontinuous information in the worst case and randomized settings

We believe that discontinuous linear information is never more powerful than continuous linear information for approximating continuous operators. We prove such a result in the worst case setting. In the randomized setting we consider compact linear operators defined between Hilbert spaces. In this case, the use of discontinuous linear information in the randomized setting cannot be much more powerful than continuous linear information in the worst case setting. These results can be applied when function evaluations are used even if function values are defined only almost everywhere.

math.NA↗
arXiv2003

Quantum Boolean Summation with Repetitions in the Worst-Average Setting

We study the quantum summation QS algorithm of Brassard, Hoyer, Mosca and Tapp, which approximates the arithmetic mean of a Boolean function defined on $N$ elements. We present sharp error bounds of the QS algorithm in the worst-average setting with the average performance measured in the $L_q$ norm, $q \in [1,\infty]$. We prove that the QS algorithm with $M$ quantum queries, $M<N$, has the worst-average error bounds of the form $Θ(\ln M/M)$ for $q=1$, $Θ(M^{-1/q})$ for $q\in (1,\infty)$, and is equal to 1 for $q=\infty$. We also discuss the asymptotic constants of these estimates. We improve the error bounds by using the QS algorithm with repetitions. Using the number of repetitions which is independent of $M$ and linearly dependent on $q$, we get the error bound of order $M^{-1}$ for any $q \in [1,\infty)$. Since $Ω(M^{-1})$ is a lower bound on the worst-average error of any quantum algorithm with $M$ queries, the QS algorithm with repetitions is optimal in the worst-average setting.

quant-ph↗
arXiv2003

Sharp Error Bounds on Quantum Boolean Summation in Various Settings

We study the quantum summation (QS) algorithm of Brassard, Hoyer, Mosca and Tapp, that approximates the arithmetic mean of a Boolean function defined on N elements. We improve error bounds presented in [1] in the worst-probabilistic setting, and present new error bounds in the average-probabilistic setting. In particular, in the worst-probabilistic setting, we prove that the error of the QS algorithm using $M - 1$ queries is $3π/(4M)$ with probability $8/π^2$, which improves the error bound $πM^{-1} + π^2 M^{-2}$ of Brassard et al. We also present bounds with probabilities $p\in (1/2, 8/π^2]$ and show they are sharp for large $M$ and $NM^{-1}$. In the average-probabilistic setting, we prove that the QS algorithm has error of order $\min\{M^{-1}, N^{-1/2}\}$ if $M$ is divisible by 4. This bound is optimal, as recently shown in [10]. For M not divisible by 4, the QS algorithm is far from being optimal if $M \ll N^{1/2}$ since its error is proportional to $M^{-1}^$.

quant-ph↗