Weighted Korobov Spaces Research Articles

On the power of standard information for tractability for L∞ approximation of periodic functions in the worst case setting

We study multivariate approximation of periodic functions in the worst case setting with the error measured in the L∞ norm. We consider algorithms that use standard information Λstd consisting of function values or general linear information Λall consisting of arbitrary continuous linear functionals. We investigate equivalences of various notions of algebraic and exponential tractability for Λstd and Λall under the absolute or normalized error criterion, and show that the power of Λstd is the same as the one of Λall for various notions of algebraic and exponential tractability. Our results can be applied to weighted Korobov spaces and Korobov spaces with exponential weights. This gives a special solution to Open Problem 145 as posed by Novak and Woźniakowski (2012) [40].

Random-prime–fixed-vector randomised lattice-based algorithm for high-dimensional integration

We show that a very simple randomised algorithm for numerical integration can produce a near optimal rate of convergence for integrals of functions in the d-dimensional weighted Korobov space. This algorithm uses a lattice rule with a fixed generating vector and the only random element is the choice of the number of function evaluations. For a given computational budget n of a maximum allowed number of function evaluations, we uniformly pick a prime p in the range n/2<p≤n. We show error bounds for the randomised error, which is defined as the worst case expected error, of the form O(n−α−1/2+δ), with δ>0, for a Korobov space with smoothness α>1/2 and general weights. The implied constant in the bound is dimension-independent given the usual conditions on the weights. We present an algorithm that can construct suitable generating vectors offline ahead of time at cost O(dn4/ln⁡n) when the weight parameters defining the Korobov spaces are so-called product weights. For this case, numerical experiments confirm our theory that the new randomised algorithm achieves the near optimal rate of the randomised error.

Density Estimation in RKHS with Application to Korobov Spaces in High Dimensions

A kernel method for estimating a probability density function from an independent and identically distributed sample drawn from such density is presented. Our estimator is a linear combination of kernel functions, the coefficients of which are determined by a linear equation. An error analysis for the mean integrated squared error is established in a general reproducing kernel Hilbert space setting. The theory developed is then applied to estimate probability density functions belonging to weighted Korobov spaces, for which a dimension-independent convergence rate is established. Under a suitable smoothness assumption, our method attains a rate arbitrarily close to the optimal rate. Numerical results support our theory.

Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate

We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich [J. Approx. Theory 240 (2019), pp. 96–113] shows that rank-1 lattice rules with a randomly chosen number of points and good generating vector achieve almost the optimal order of the randomized error in weighted Korobov spaces, and moreover, that the error is bounded independently of the dimension if the weight parameters,γj\gamma _j, satisfy the summability condition∑j=1∞γj1/α&gt;∞\sum _{j=1}^{\infty }\gamma _j^{1/\alpha }&gt;\infty, whereα\alphais a smoothness parameter. The argument is based on the existence result that at least half of the possible generating vectors yield almost the optimal order of the worst-case error in the same function spaces.In this paper we provide a component-by-component construction algorithm of such randomized rank-1 lattice rules, without any need to check whether the constructed generating vectors satisfy a desired worst-case error bound. Similarly to the above-mentioned work, we prove that our algorithm achieves almost the optimal order of the randomized error and that the error bound is independent of the dimension if the same condition∑j=1∞γj1/α&gt;∞\sum _{j=1}^{\infty }\gamma _j^{1/\alpha }&gt;\inftyholds. We also provide analogous results for tent-transformed lattice rules for weighted half-period cosine spaces and for polynomial lattice rules in weighted Walsh spaces, respectively.

Construction-Free Median Quasi-Monte Carlo Rules for Function Spaces with Unspecified Smoothness and General Weights

We study quasi-Monte Carlo (QMC) integration of smooth functions defined over the multi-dimensional unit cube. Inspired by a recent work of Pan and Owen, we study a new construction-free median QMC rule which can exploit the smoothness and the weights of function spaces adaptively. For weighted Korobov spaces, we draw a sample of $r$ independent generating vectors of rank-1 lattice rules, compute the integral estimate for each, and approximate the true integral by the median of these $r$ estimates. For weighted Sobolev spaces, we use the same approach but with the rank-1 lattice rules replaced by high-order polynomial lattice rules. A major advantage over the existing approaches is that we do not need to construct good generating vectors by a computer search algorithm, while our median QMC rule achieves almost the optimal worst-case error rate for the respective function space with any smoothness and weights, with a probability that converges to 1 exponentially fast as $r$ increases. Numerical experiments illustrate and support our theoretical findings.

EC-(t1,t2)-tractability of approximation in weighted Korobov spaces in the worst case setting

We investigate multivariate approximation problems in the weighted Korobov spaces in the worst case setting. We consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We discuss the exponential convergence-(t1,t2)-weak tractability (EC-(t1,t2)-WT) under the absolute or normalized error criterion. Especially, we give the sufficient and necessary conditions for EC-WT and EC-(t1,1)-WT with t1<1.

A note on EC-tractability of multivariate approximation in weighted Korobov spaces for the standard information class

In this paper we study exponential tractability of multivariate approximation problem for weighted Korobov spaces in the worst case setting. The considered algorithms are constructive and use the class Λstd of only function evaluations. We give matching necessary and sufficient conditions for notions of EC-quasi-polynomial tractability and EC-uniform weak tractability in terms of two weight parameters of the problem.

Tractability of approximation in the weighted Korobov space in the worst-case setting — a complete picture

In this paper, we study tractability of L2-approximation of one-periodic functions from weighted Korobov spaces in the worst-case setting. The considered weights are of product form. For the algorithms we allow information from the class Λall consisting of all continuous linear functionals and from the class Λstd, which only consists of function evaluations. We provide necessary and sufficient conditions on the weights of the function space for quasi-polynomial tractability, uniform weak tractability, weak tractability and (σ,τ)-weak tractability. Together with the already known results for strong polynomial and polynomial tractability, our findings provide a complete picture of the weight conditions for all current standard notions of tractability.

A note on Korobov lattice rules for integration of analytic functions

We study numerical integration for a weighted Korobov space of analytic periodic functions for which the Fourier coefficients decay exponentially fast. In particular, we are interested in how the error depends on the dimension d. Many recent papers deal with this problem or similar problems and provide matching necessary and sufficient conditions for various notions of tractability. In most cases even simple algorithms are known which allow to achieve these notions of tractability. However, there is a gap in the literature: while for the notion of exponential-weak tractability one knows matching necessary and sufficient conditions, so far no explicit algorithm has been known which yields the desired result.In this paper we close this gap and prove that Korobov lattice rules are suitable algorithms in order to achieve exponential-weak tractability for integration in weighted Korobov spaces of analytic periodic functions.

Stability of lattice rules and polynomial lattice rules constructed by the component-by-component algorithm

We study quasi-Monte Carlo (QMC) methods for numerical integration of multivariate functions defined over the high-dimensional unit cube. Lattice rules and polynomial lattice rules, which are special classes of QMC methods, have been intensively studied and the so-called component-by-component (CBC) algorithm has been well-established to construct rules which achieve the almost optimal rate of convergence with good tractability properties for given smoothness and set of weights. Since the CBC algorithm constructs rules for given smoothness and weights, not much is known when such rules are used for function classes with different smoothness and/or weights.In this paper we prove that a lattice rule constructed by the CBC algorithm for the weighted Korobov space with given smoothness and weights achieves the almost optimal rate of convergence with good tractability properties for general classes of smoothness and weights which satisfy some summability conditions. Such a stability result also can be shown for polynomial lattice rules in weighted Walsh spaces. We further give bounds on the weighted star discrepancy and discuss the tractability properties for these QMC rules. The results are comparable to those obtained for Halton, Sobol and Niederreiter sequences.

Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration

Smolyak’s method, also known as sparse grid method, is a powerful tool to tackle multivariate tensor product problems solely with the help of efficient algorithms for the corresponding univariate problem.In this paper we study the randomized setting, i.e., we randomize Smolyak’s method. We provide upper and lower error bounds for randomized Smolyak algorithms with explicitly given dependence on the number of variables and the number of information evaluations used. The error criteria we consider are the worst case root mean square error (the typical error criterion for randomized algorithms, sometimes referred to as “randomized error”, ) and the root mean square worst case error (sometimes referred to as “worst-case error”).Randomized Smolyak algorithms can be used as building blocks for efficient methods such as multilevel algorithms, multivariate decomposition methods or dimension-wise quadrature methods to tackle successfully high-dimensional or even infinite-dimensional problems. As an example, we provide a very general and sharp result on the convergence rate of Nth minimal errors of infinite-dimensional integration on weighted reproducing kernel Hilbert spaces. Moreover, we are able to characterize the spaces for which randomized algorithms for infinite-dimensional integration are superior to deterministic ones. We illustrate our findings for the special instance of weighted Korobov spaces. We indicate how these results can be extended, e.g., to spaces of functions whose smooth dependence on successive variables increases (“spaces of increasing smoothness”) and to the problem of L2-approximation (function recovery).

EC-tractability of [formula omitted]-approximation in Korobov spaces with exponential weights

This study aims at investigating Lp-approximation with 1≤p≤∞ defined over weighted Korobov spaces. The considered algorithms use finitely many evaluations of arbitrary linear functionals. In the worst case setting, we give matching necessary and sufficient conditions for some notions of EC-tractability in terms of two weight parameters of the problem. Our results give an affirmative answer to the open problem raised recently by Kritzer et al. (2017).

Tractability of $\mathbb{L}_2$-approximation in hybrid function spaces

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 $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ of all algorithms that use $N$ information evaluations from the class $\Lambda$ in the $d$-dimensional case. The two classes $\Lambda$ considered in this paper are the class $\Lambda^{{\rm all}}$ consisting of all linear functionals and the class $\Lambda^{{\rm std}}$ consisting only of function evaluations. The focus lies on the dependence of $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ on the dimension $d$. The main results are conditions for weak, polynomial, and strong polynomial tractability.

Formula omitted]-Approximation in Korobov spaces with exponential weights

We study multivariate L∞-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 a={aj} and b={bj} of positive real numbers bounded away from zero. We study the minimal worst-case error eL∞−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 Λall of all linear functionals and the class Λstd of only function evaluations.We study exponential convergence of the minimal worst-case error, which means that eL∞−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+logε−1 to compute an ε-approximation. We derive necessary and sufficient conditions on the sequences a and b for obtaining exponential error convergence, and also for obtaining the various notions of tractability. The results are the same for both classes Λ.L2-approximation for functions from the same function space has been considered in Dick et al. (2014). It is surprising that most results for L∞-approximation coincide with their counterparts for L2-approximation. This allows us to deduce also results for Lp-approximation for p∈[2,∞].

Tent-transformed lattice rules for integration and approximation of multivariate non-periodic functions

We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use specially constructed tent-transformed rank-1 lattice points as cubature nodes for integration and as sampling points for approximation. For both integration and approximation, we study the connection between the worst-case errors of our algorithms in the cosine space and the worst-case errors of some related algorithms in the well-known weighted Korobov space of smooth periodic functions. By exploiting this connection, we are able to obtain constructive worst-case error bounds with good convergence rates for the cosine space.

A note on tractability of multivariate analytic problems

In this paper we study d-variate approximation for weighted Korobov spaces in the worst-case setting. The considered algorithms use finitely many evaluations of arbitrary linear functionals. We give matching necessary and sufficient conditions for some notions of tractability in terms of two weight parameters. Our result is an affirmative answer to a problem which is left open in a recent paper of Kritzer, Pillichshammer and Woźniakowski.

Numerical integration in log-Korobov and log-cosine spaces

Quasi-Monte Carlo (QMC) rules are equal weight quadrature rules for approximating integrals over the s-dimensional unit cube [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. In this setting, the condition α > 1/2 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 function spaces 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 \(\mathcal {O}(N^{-1/2} (\log N)^{-\mu (1-\lambda )/2})\) for any 1/μ < λ ≤ 1, where μ > 1 is a certain power in the log-scale. A second result is concerned with tractability of numerical integration for weighted Korobov spaces with product weights \((\gamma _{j})_{j \in \mathbb {N}}\). Previous results have shown that if \({\sum }_{j=1}^{\infty } \gamma _{j}^{\tau } < \infty \) for some 1/(2α) < τ ≤ 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/(2α), namely we show dimension independent error bounds under the condition that \({\sum }_{j=1}^{\infty } \gamma _{j} \max \{1, \log \gamma _{j}^{-1}\}^{\mu (1-\lambda )} < \infty \) for some 1/μ < λ ≤ 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.

Multivariate integration of infinitely many times differentiable functions in weighted Korobov spaces

We study multivariate integration for a weighted Korobov space of periodic infinitely many times differentiable functions for which the Fourier coefficients decay exponentially fast. The weights are defined in terms of two non-decreasing sequences a = { a i } \mathbf {a}=\{a_i\} and b = { b i } \mathbf {b}=\{b_i\} of numbers no less than one and a parameter ω ∈ ( 0 , 1 ) \omega \in (0,1) . Let e ( n , s ) e(n,s) be the minimal worst-case error of all algorithms that use n n function values in the s s -variate case. We would like to check conditions on a \mathbf {a} , b \mathbf {b} and ω \omega such that e ( n , s ) e(n,s) decays exponentially fast, i.e., for some q ∈ ( 0 , 1 ) q\in (0,1) and p &gt; 0 p&gt;0 we have e ( n , s ) = O ( q n p ) e(n,s)=\mathcal {O}(q^{\,n^{\,p}}) as n n goes to infinity. The factor in the O \mathcal {O} notation may depend on s s in an arbitrary way. We prove that exponential convergence holds iff B := ∑ i = 1 ∞ 1 / b i &gt; ∞ B:=\sum _{i=1}^\infty 1/b_i&gt;\infty independently of a \mathbf {a} and ω \omega . Furthermore, the largest p p of exponential convergence is 1 / B 1/B . We also study exponential convergence with weak, polynomial and strong polynomial tractability. This means that e ( n , s ) ≤ C ( s ) q n p e(n,s)\le C(s)\,q^{\,n^{\,p}} for all n n and s s and with log C ( s ) = exp ⁡ ( o ( s ) ) \log \,C(s)=\exp (o(s)) for weak tractability, with a polynomial bound on log C ( s ) \log \,C(s) for polynomial tractability, and with uniformly bounded C ( s ) C(s) for strong polynomial tractability. We prove that the notions of weak, polynomial and strong polynomial tractability are equivalent, and hold iff B &gt; ∞ B&gt;\infty and a i a_i are exponentially growing with i i . We also prove that the largest (or the supremum of) p p for exponential convergence with strong polynomial tractability belongs to [ 1 / ( 2 B ) , 1 / B ] [1/(2B),1/B] .

Approximation of analytic functions in Korobov spaces

We study multivariate L2-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 a={aj} and b={bj} of positive real numbers bounded away from zero. We study the minimal worst-case error eL2−app,Λ(n,s) of all algorithms that use n information evaluations from the class Λ in the s-variate case. We consider two classes Λ in this paper: the class Λall of all linear functionals and the class Λstd of only function evaluations.We study exponential convergence of the minimal worst-case error, which means that eL2−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 weak, polynomial and strong polynomial tractability. In particular, polynomial tractability means that we need a polynomial number of information evaluations in s and 1+logε−1 to compute an ε-approximation. We derive necessary and sufficient conditions on the sequences a and b for obtaining exponential error convergence, and also for obtaining the various notions of tractability. The results are the same for both classes Λ. They are also constructive with the exception of one particular sub-case for which we provide a semi-constructive algorithm.

Worst case error for integro-differential equations by a lattice-Nyström method

In this paper, we make an offer of the lattice approximate method for solving a class of multi-dimensional integro-differential equations with the initial conditions. Also, we analyze the worst case error measured in weighted Korobov spaces for these equations. Finally, numerical examples complete this work.