Peter Kritzer

Peter Kritzer, Nicolas Nagel, Friedrich Pillichshammer

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 $N$.

Josef Dick, Peter Kritzer, Friedrich Pillichshammer et al.

We study multivariate $L_2$-approximation for a weighted Korobov space of analytic 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 numbers no less than one. Let $e^{L_2-\mathrm{app},Λ}(n,s)$ be the minimal worst-case error of all algorithms that use $n$ information functionals from the class $Λ$ in the $s$-variate case. We consider two classes $Λ$: the class $Λ^{\rm all}$ consists of all linear functionals and the class $Λ^{\rm std}$ consists of only function valuations. We study (EXP) exponential convergence. This means that $$ e^{L_2-\mathrm{app},Λ}(n,s) \le C(s)\,q^{\,(n/C_1(s))^{p(s)}}\quad{for all}\quad n, s \in \mathbb{N} $$ where $q\in(0,1)$, and $C,C_1,p:\mathbb{N} \rightarrow (0,\infty)$. If we can take $p(s)=p>0$ for all $s$ then we speak of (UEXP) uniform exponential convergence. We also study EXP and UEXP with (WT) weak, (PT) polynomial and (SPT) strong polynomial tractability. These concepts are defined as follows. Let $n(\e,s)$ be the minimal $n$ for which $e^{L_2-\mathrm{app},Λ}(n,s)\le \e$. Then WT holds iff $\lim_{s+\log\,\e^{-1}\to\infty}(\log n(\e,s))/(s+\log\,\e^{-1})=0$, PT holds iff there are $c,τ_1,τ_2$ such that $n(\e,s)\le cs^{τ_1}(1+\log\,\e^{-1})^{τ_2}$ for all $s$ and $\e\in(0,1)$, and finally SPT holds iff the last estimate holds for $τ_1=0$. The infimum of $τ_2$ for which SPT holds is called the exponent $τ^*$ of SPT. We prove that the results are the same for both classes $Λ$, and obtain conditions for WT, PT, SPT with and without EXP and UEXP.

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 $Λ$.

Peter Kritzer, Frances Y. Kuo, Dirk Nuyens et al.

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.

Peter Kritzer, Friedrich Pillichshammer, G. W. Wasilkowski

We provide lower bounds for the norms of embeddings between $\boldsymbolγ$-weighted Anchored and ANOVA spaces of $s$-variate functions with mixed partial derivatives of order one bounded in $L_p$ norm ($p\in[1,\infty]$). In particular we show that the norms behave polynomially in $s$ for Finite Order Weights and Finite Diameter Weights if $p>1$, and increase faster than any polynomial in $s$ for Product Order-Dependent Weights and any $p$.

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.

Aicke Hinrichs, Peter Kritzer, Friedrich Pillichshammer et al.

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.

Peter Kritzer, Friedrich Pillichshammer, Leszek Plaskota et al.

In this paper, we study the approximation of $d$-dimensional $ρ$-weighted integrals over unbounded domains $\mathbb{R}_+^d$ or $\mathbb{R}^d$ using a special change of variables, so that quasi-Monte Carlo (QMC) or sparse grid rules can be applied to the transformed integrands over the unit cube. We consider a class of integrands with bounded $L_p$ norm of mixed partial derivatives of first order, where $p\in[1,+\infty].$ The main results give sufficient conditions on the change of variables $ν$ which guarantee that the transformed integrand belongs to the standard Sobolev space of functions over the unit cube with mixed smoothness of order one. These conditions depend on $ρ$ and $p$. The proposed change of variables is in general different than the standard change based on the inverse of the cumulative distribution function. We stress that the standard change of variables leads to integrands over a cube; however, those integrands have singularities which make the application of QMC and sparse grids ineffective. Our conclusions are supported by numerical experiments.

Peter Kritzer, Friedrich Pillichshammer, G. W. Wasilkowski

We consider approximation of functions of $s$ variables, where $s$ is very large or infinite, that belong to weighted anchored spaces. We study when such functions can be approximated by algorithms designed for functions with only very small number ${\rm dim^{trnc}}(\varepsilon)$ of variables. Here $\varepsilon$ is the error demand and we refer to ${\rm dim^{trnc}}(\varepsilon)$ as the $\varepsilon$-truncation dimension. We show that for sufficiently fast decaying product weights and modest error demand (up to about $\varepsilon \approx 10^{-5}$) the truncation dimension is surprisingly very small.

Yuhan Ding, Fred J. Hickernell, Peter Kritzer et al.

We study adaptive approximation algorithms for general multivariate linear problems where the sets of input functions are non-convex cones. While it is known that adaptive algorithms perform essentially no better than non-adaptive algorithms for convex input sets, the situation may be different for non-convex sets. A typical example considered here is function approximation based on series expansions. Given an error tolerance, we use series coefficients of the input to construct an approximate solution such that the error does not exceed this tolerance. We study the situation where we can bound the norm of the input based on a pilot sample, and the situation where we keep track of the decay rate of the series coefficients of the input. Moreover, we consider situations where it makes sense to infer coordinate and smoothness importance. Besides performing an error analysis, we also study the information cost of our algorithms and the computational complexity of our problems, and we identify conditions under which we can avoid a curse of dimensionality.

Christian Irrgeher, Peter Kritzer, Friedrich Pillichshammer

We study multivariate integration and approximation for functions belonging to a weighted reproducing kernel Hilbert space based on half-period cosine functions in the worst-case setting. The weights in the norm of the function space depend on two sequences of real numbers and decay exponentially. As a consequence the functions are infinitely often differentiable, and therefore it is natural to expect exponential convergence of the worst-case error. We give conditions on the weight sequences under which we have exponential convergence for the integration as well as the approximation problem. Furthermore, we investigate the dependence of the errors on the dimension by considering various notions of tractability. We prove sufficient and necessary conditions to achieve these tractability notions.

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.

Adrian Ebert, Peter Kritzer, Dirk Nuyens

In the analysis of using quasi-Monte Carlo (QMC) methods to approximate expectations of a linear functional of the solution of an elliptic PDE with random diffusion coefficient the sensitivity w.r.t. the parameters is often stated in terms of product-and-order-dependent (POD) weights. The (offline) fast component-by-component (CBC) construction of an $N$-point QMC method making use of these POD weights leads to a cost of $\mathcal{O}(s N \log(N) + s^2 N)$ with $s$ the parameter truncation dimension. When $s$ is large this cost is prohibitive. As an alternative Herrmann and Schwab introduced an analysis resulting in product weights to reduce the construction cost to $\mathcal{O}(s N \log(N))$. We here show how the reduced CBC method can be used for POD weights to reduce the cost to $\mathcal{O}(\sum_{j=1}^{\min\{s,s^*\}} (m-w_j+j) \, b^{m-w_j})$, where $N=b^m$ with prime $b$, $w_1 \le \cdots \le w_s$ are nonnegative integers and $s^*$ can be chosen much smaller than $s$ depending on the regularity of the random field expansion as such making it possible to use the POD weights directly. We show a total error estimate for using randomly shifted lattice rules constructed through the reduced CBC construction.

Peter Kritzer, Helene Laimer, Friedrich Pillichshammer

We consider multivariate $\mathbb{L}_2$-approximation in reproducing kernel Hilbert spaces which are tensor products of weighted Walsh spaces and weighted Korobov spaces. We study the minimal worst-case error $e^{\mathbb{L}_2-\mathrm{app},Λ}(N,d)$ of all algorithms that use $N$ information evaluations from the class $Λ$ in the $d$-dimensional case. The two classes $Λ$ considered in this paper are the class $Λ^{\rm all}$ consisting of all linear functionals and the class $Λ^{\rm std}$ consisting only of function evaluations. The focus lies on the dependence of $e^{\mathbb{L}_2-\mathrm{app},Λ}(N,d)$ on the dimension $d$. The main results are conditions for weak, polynomial, and strong polynomial tractability.

Josef Dick, Peter Kritzer

Higher order nets were introduced by Dick as a generalisation of classical $(t,m,s)$-nets, which are point sets frequently used in quasi-Monte Carlo integration algorithms. Essential tools in finding such point sets of high quality are propagation rules, which make it possible to generate new higher order nets from existing higher order nets and even classical $(t,m,s)$-nets. Such propagation rules for higher order nets were first considered by the authors in [J. Dick, P. Kritzer. Duality theory and propagation rules for generalized digital nets. Math. Comp. 79, 993--1017, 2010] and further developed in [J. Baldeaux, J. Dick, F. Pillichshammer. Duality theory and propagation rules for higher order nets. Discrete Math. 311, 362--386, 2011]. In [E.L. Blokh, V.V. Zyablov. Coding of generalized concatenated codes. Problems of Information Transmission, 10, 218--222, 1974] Blokh and Zyablov established a very general propagation rule for linear codes. This propagation rule has been extended to $(t,m,s)$-nets by Schürer and Schmid in [R. Schürer, W.Ch. Schmid. \textit{MinT---the database of optimal net, code, OA, and OOA parameters}. Available at: \texttt{http://mint.sbg.ac.at}]. In this paper we show that this propagation rule can also be extended to higher order nets. Examples indicate that this propagation rule yields new higher order nets with significantly higher quality.

Michael Gnewuch, Peter Kritzer, Klaus Ritter

We study embeddings between reproducing kernel Hilbert spaces $H(K)$ of functions of $d \in \mathbb{N} \cup \{\infty\}$ variables. The kernels $K$ are superpositions of weighted finite tensor products of a fixed univariate kernel. The basic idea for the embeddings is to compensate a change of the univariate kernel by a suitable transformation of the weights. For the proofs we employ ($d \in \mathbb{N}$) and develop ($d = \infty$) a discrete calculus on the cone of all weights, where completely monotone weights play a particular role. We sketch how to apply the embedding results to computational problems, as, e.g., numerical integration or function recovery.

Peter Kritzer, Friedrich Pillichshammer, G. W. Wasilkowski

The paper considers truncation errors for functions of the form $f(x_1,x_2,\dots)=g(\sum_{j=1}^\infty x_j\,ξ_j)$, i.e., errors of approximating $f$ by $f_k(x_1,\dots,x_k)=g(\sum_{j=1}^k x_j\,ξ_j)$, where the numbers $ξ_j$ converge to zero sufficiently fast and $x_j$'s are i.i.d. random variables. As explained in the introduction, functions $f$ of the form above appear in a number of important applications. To have positive results for possibly large classes of such functions, the paper provides sharp bounds on truncation errors in both the average and worst case settings. In the former case, the functions $g$ are from a Hilbert space $G$ endowed with a zero mean probability measure with a given covariance kernel. In the latter case, the functions $g$ are from a reproducing kernel Hilbert space, or a space of functions satisfying a Hölder condition.

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.

Josef Dick, Peter Kritzer

The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several authors have reported that in numerical experiments the generating vector sometimes has repeated components. We study a variation of the classical component-by-component algorithm for the construction of lattice or polynomial lattice point sets where the components are forced to differ from each other. This avoids the problem of having projections where all quadrature points lie on the main diagonal. Since the previous results on the worst-case error do not apply to this modified algorithm, we prove such an error bound here. We also discuss further restrictions on the choice of components in the component-by-component algorithm.

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.

Peter Hellekalek, Peter Kritzer, Friedrich Pillichshammer

In this paper, we study quasi-Monte Carlo (QMC) integration in weighted Sobolev spaces. In contrast to many previous results the QMC algorithms considered here are of open type, i.e., they are extensible in the number of sample points without having to discard the samples already used. As the underlying integration nodes we consider randomized Halton sequences in prime bases $\boldsymbol{p}=(p_1,...,p_s)$ for which we study the root mean square (RMS) worst-case error. The randomization method is a $\boldsymbol{p}$-adic shift which is based on $\boldsymbol{p}$-adic arithmetic. The obtained error bounds are optimal in the order of magnitude of the number of sample nodes. Furthermore we obtain conditions on the coordinate weights under which the error bounds are independent of the dimension $s$. In terms of the field of Information-Based Complexity this means that the corresponding QMC rule achieves a strong polynomial tractability error bound. Our findings on the RMS worst-case error of randomized Halton sequences can be carried over to the RMS $L_2$-discrepancy. Except for the $\boldsymbol{p}$-adic shift our results are fully constructive and no search algorithms (such as the component-by-component algorithm) are required.

Josef Dick, Peter Kritzer, Gunther Leobacher et al.

QMC rules are equal weight quadrature rules for approximating integrals over $[0,1]^s$. One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic functions on $[0,1]^s$ with square integrable partial mixed derivatives of order $α$. Using Parseval's identity, this smoothness can be defined for all real numbers $α> 1/2$. This condition is necessary as otherwise the Korobov space contains discontinuous functions for which function evaluation is not well defined. This paper is concerned with more precise endpoint estimates of the integration error using QMC rules for Korobov spaces with $α$ arbitrarily close to $1/2$. To obtain such estimates we introduce a $\log$-scale for functions with smoothness close to $1/2$, which we call $\log$-Korobov spaces. We show that lattice rules can be used to obtain an integration error of order $\mathcal{O}(N^{-1/2} (\log N)^{-μ(1-λ)/2})$ for any $1/μ<λ\le 1$, where $μ>1$ is a power in the $\log$-scale. We also consider tractability of numerical integration for weighted Korobov spaces with product weights $(γ_j)_{j \in \mathbb{N}}$. It is known that if $\sum_{j=1}^\infty γ_j^τ< \infty$ for some $1/(2α) < τ\le 1$ one can obtain error bounds which are independent of the dimension. In this paper we give a more refined estimate for the case where $τ$ is close to $1/(2 α)$, namely we show dimension independent error bounds under the condition that $\sum_{j=1}^\infty γ_j \max\{1, \log γ_j^{-1}\}^{μ(1-λ)} < \infty$ for some $1/μ< λ\le 1$. The essential tool in our analysis is a $\log$-scale Jensen's inequality. The results described above also apply to integration in $\log$-cosine spaces using tent-transformed lattice rules.