Henryk Wozniakowski

Aicke Hinrichs, Erich Novak, Mario Ullrich et al.

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.

Aicke Hinrichs, Erich Novak, Henryk Wozniakowski

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.

Erich Novak, Henryk Wozniakowski

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.

Aicke Hinrichs, Erich Novak, Mario Ullrich et al.

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.

Peter Kritzer, Friedrich Pillichshammer, Henryk Wozniakowski

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 $Λ$.

Fred J. Hickernell, Peter Kritzer, Henryk Wozniakowski

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.

Peter Kritzer, Henryk Wozniakowski

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.

Christian Irrgeher, Peter Kritzer, Friedrich Pillichshammer et al.

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.

Christian Irrgeher, Peter Kritzer, Friedrich Pillichshammer et al.

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.

Erich Novak, Henryk Wozniakowski

This is an open letter that we sent to S. Ilie, G. Soederlind and R.M. Corless in August 2008.