Weak Tractability Research Articles

Average Case (s, t)-Weak Tractability of L2-Approximation with Weighted Covariance Kernels

Average Case (s, t)-Weak Tractability of L2-Approximation with Weighted Covariance Kernels

Conditions for tractability of the weighted $$L_p$$-discrepancy and integration in non-homogeneous tensor product spaces

AbstractWe study tractability properties of the weighted $$L_p$$ L p -discrepancy, where the weights model the influence of different coordinates. A discrepancy is said to be tractable if the minimal number N of points, such that the discrepancy is less than the initial discrepancy times an error threshold $$\varepsilon $$ ε , does not grow exponentially fast with the dimension and $$\varepsilon ^{-1}$$ ε - 1 . In this case there are various notions of tractabilities used in order to classify the exact rate. In the present paper we prove matching sufficient conditions (upper bounds) and neccessary conditions (lower bounds) for polynomial and weak tractability for all $$p \in (1, \infty )$$ p ∈ ( 1 , ∞ ) and we prove a necessary condition for strong polynomial tractability. The proofs of the lower bounds are based on a powerful general result for the information complexity of integration with positive quadrature rules for tensor product spaces. As a second application of this general result we consider the integration of polynomials of degree at most 2.

Exponential Convergence-(t,s)-Weak Tractability of Approximation in Weighted Hilbert Spaces

We study L2-approximation problems in the weighted Hilbert spaces in the worst case setting. Three interesting weighted Hilbert spaces appear in this paper, whose weights are equipped with two positive parameters γj and αj for j∈N. We consider algorithms using the class of arbitrary linear functionals. We discuss the exponential convergence-(t,s)-weak tractability of these L2-approximation problems. In particular, we obtain the sufficient and necessary conditions on the weights for exponential convergence-weak tractability and exponential convergence-(t,1)-weak tractability with t<1.

Tractability of Approximation of Functions Defined over Weighted Hilbert Spaces

We investigate L2-approximation problems in the worst case setting in the weighted Hilbert spaces H(KRd,α,γ) with weights Rd,α,γ under parameters 1≥γ1≥γ2≥⋯≥0 and 1<α1≤α2≤⋯. Several interesting weighted Hilbert spaces H(KRd,α,γ) appear in this paper. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss tractability of L2-approximation problems for the involved Hilbert spaces, which describes how the information complexity depends on d and ε−1. As a consequence we study the strongly polynomial tractability (SPT), polynomial tractability (PT), weak tractability (WT), and (t1,t2)-weak tractability ((t1,t2)-WT) for all t1>1 and t2>0 in terms of the introduced weights under the absolute error criterion or the normalized error criterion.

EC-tractability of multivariate approximation in Hermite spaces for the standard information class

In this paper, we study exponential tractability of multivariate approximation problem for Hermite spaces of analytic functions over [Formula: see text] in the worst case setting. The functions are characterized by the property that their Hermite coefficients decay exponentially fast. The considered algorithms based on tensor products of Gauss–Hermite rules for multivariate integration are constructive and use the class [Formula: see text] 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.

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 [formula omitted]-weak tractability of the weighted star discrepancy of regular grids

In this paper we study (s,t)-weak tractability of the weighted star discrepancy with general coefficients of centered regular grids with different mesh-sizes. We give matching necessary and sufficient conditions on (s,t)-weak tractability with s>0 and t∈(0,1] in terms of the weight sequences. We also prove that the weighted star discrepancy has (s,t)-weak tractability for all s>0 and t>1.

Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals

We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an ε-approximation for the d-variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on d and logarithmically on ε−1. The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on d and logarithmic dependence on ε−1, we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential (s,t)-weak tractability with max(s,t)≥1. For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential (s,t)-weak tractability with max(s,t)<1 is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).

Tractability properties of the discrepancy in Orlicz norms

We show that the minimal discrepancy of a point set in the d-dimensional unit cube with respect to Orlicz norms can exhibit both polynomial and weak tractability. In particular, we show that the ψα-norms of exponential Orlicz spaces are polynomially tractable.

Quasi-Monte Carlo tractability of integration problem in function spaces defined over products of balls

Recently, quasi-Monte Carlo (QMC) rules for multivariate integration in some weighted Sobolev spaces of functions defined over unit cubes [Formula: see text], products of [Formula: see text] copies of the simplex [Formula: see text] and products of [Formula: see text] copies of the unit sphere [Formula: see text] have been well-studied in the literature. In this paper, we consider QMC tractability of integrals of functions defined over product of [Formula: see text] copies of the ball [Formula: see text]. The space is a tensor product of [Formula: see text] reproducing kernel Hilbert spaces defined by positive and uniformly bounded weights [Formula: see text] for [Formula: see text]. We obtain matching necessary and sufficient conditions in terms of this weights for various notions of QMC tractability, including strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniformly weak tractability and [Formula: see text]-weak tractability.

EC-[formula omitted]-weak tractability of multivariate linear problems in the average case setting

We study EC-(s,t)-weak tractability of multivariate linear problems in the average case setting. This paper extends earlier work in the worst case setting. The parameters s≥0 and t≥0 allow us to study the information complexity n(ε,d) of a d-variate problem with respect to different powers of lnε−1, corresponding to the bits of accuracy, and d. We consider the absolute and normalized error criteria. In particular, a multivariate problem is EC-(s,t)-weakly tractable iff limd+ε−1→∞lnn(ε,d)∕[dt+lnsε−1]=0. We deal with general linear problems and linear tensor product problems. We show necessary and sufficient conditions for EC-(s,t)-weak tractability. In the case of general linear problem these conditions are matching. For linear tensor product problems, we also show matching conditions with the exception of some cases where s>1, in general.

A note about EC-[formula omitted]-weak tractability of multivariate approximation with analytic Korobov kernels

This paper is devoted to discussing multivariate approximation problems with analytic Korobov kernels in the worst and average case settings. We consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We investigate exponential convergence-(s,t)-weak tractability (EC-(s,t)-WT) under the absolute or normalized error criterion. We completely solve the problem by filling the remaining gaps left open on EC-(s,t)-WT. That is, we obtain necessary and sufficient conditions for EC-(s,t)-WT for 0<min(s,t)<1 and max(s,t)≤1 in the worst case setting and for s,t>0 except s=t=1 in the average case setting.

(S, T)-Weak Tractability of Multivariate Linear Problems in the Average Case Setting

The purpose of this article is to investigate (s, t)-weak tractability of multivariate linear problems in the average case setting. The considered algorithms use finitely many evaluations of arbitrary linear functionals. Generally, we obtained matching necessary and sufficient conditions for (s, t)-weak tractability in terms of the corresponding non-increasing sequence of eigenvalues. Specifically, we discussed (s, t)-weak tractability of linear tensor product problems and obtained necessary and sufficient conditions in terms of the corresponding one-dimensional problem. As an example of applications, we discussed also (s, t)-weak tractability of a multivariate approximation problem.

Tractability of multivariate approximation over weighted standard Sobolev spaces

We study multivariate approximation for complex-valued functions defined over the d-dimensional torus, these functions belonging to weighted standard Sobolev spaces of smoothness r. Algorithms are allowed to use finitely many arbitrary continuous linear functionals, which may be chosen adaptively. The error of an algorithm is measured by the L2 norm, in the worst case setting. No matter how we choose positive weights, we prove that multivariate approximation cannot be quasi-polynomially tractable, meaning that the minimal number of continuous linear functionals needed to find an ε-approximation in the d-variate case grows faster than any polynomial in ε−lnd. We also study (s,t)-weak tractability for positive s and t, meaning that the minimal number of continuous linear functionals is not exponential in ε−t and ds. We restrict our attention to product weights, defined as products of powers of weightlets γj. We obtain conditions on the weightlets that are necessary and sufficient for our problem to be (s,t)-weakly tractable. In particular, if rs<2 and t≤1, then our problem is (s,t)-weakly tractable iff γj=o(j−(2−rs)∕(rs)) as j→∞.

Average case [formula omitted]-weak tractability of non-homogeneous tensor product problems

We study d-variate problems in the average case setting with respect to a zero-mean Gaussian measure. The covariance kernel of this Gaussian measure is a product of univariate kernels and satisfies some special properties. We study (s,t)-weak tractability of this multivariate problem under the absolute or normalized error criterion, and obtain a necessary and sufficient condition for s>0 and t∈(0,1). Our result can apply to the problems with covariance kernels corresponding to Euler and Wiener integrated processes, Korobov kernels, and analytic Korobov kernels.

Tractability properties of the weighted star discrepancy of regular grids

In this paper we study tractability properties of the weighted star discrepancy with general coefficients of centered regular grids with different mesh-sizes. We give exact characterizations of the weight sequences (γj)j≥1 such that the regular grid with different mesh-sizes achieves weak, uniform weak, quasi polynomial, polynomial or strong polynomial tractability for the γ-weighted star discrepancy. For example, a necessary and sufficient condition such that the regular grid with different mesh-sizes achieves weak tractability for the γ-weighted star discrepancy is limj→∞jγj=0.

Formula omitted]-weak tractability of Euler and Wiener integrated processes

We study (s,t)-weak tractability of function approximation in the average case setting with respect to a zero-mean Gaussian measure. A problem is (s,t)-weakly tractable for positive s and t if its information complexity is not an exponential function of the sth power of the reciprocal of the accuracy ε and the tth power of the number d of variables. We give necessary and sufficient conditions on the (s,t)-weak tractability of the approximation of multivariate Euler and Wiener integrated processes.

([formula omitted])-weak tractability of linear problems

We introduce a new notion of tractability for multivariate problems, namely (s,lnκ)-weak tractability for positive s and κ. This allows us to study the information complexity of a d-variate problem with respect to different powers of d and the bits of accuracy lnε−1. We consider the worst case error for the absolute and normalized error criteria. We provide necessary and sufficient conditions for (s,lnκ)-weak tractability for general linear problems and linear tensor product problems defined over Hilbert spaces. In particular, we show that non-trivial linear tensor product problems cannot be (s,lnκ)-weakly tractable when s∈(0,1] and κ∈(0,1]. On the other hand, they are (s,lnκ)-weakly tractable for κ>1 and s>1 if the univariate eigenvalues of the linear tensor product problem enjoy a polynomial decay. Finally, we study (s,lnκ)-weak tractability for the remaining combinations of the values of s and κ.