Arbitrary Linear Functionals Research Articles

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.

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].

Average case tractability of non-homogeneous tensor product problems with the absolute error criterion

We study average case tractability of non-homogeneous tensor product problems with the absolute error criterion. We consider algorithms that use finitely many evaluations of arbitrary linear functionals. For general non-homogeneous tensor product problems, we obtain the matching necessary and sufficient conditions for strong polynomial tractability in terms of the one-dimensional eigenvalues. We give some examples to show that strong polynomial tractability is not equivalent to polynomial tractability, and polynomial tractability is not equivalent to quasi-polynomial tractability. But for non-homogeneous tensor product problems with decreasing eigenvalues, we prove that strong polynomial tractability is always equivalent to polynomial tractability, and strong polynomial tractability is even equivalent to quasi-polynomial tractability when the one-dimensional largest eigenvalues are less than one. In particular, we find an example that quasi-polynomial tractability with the absolute error criterion is not equivalent to that with the normalized error criterion even if all the one-dimensional largest eigenvalues are one. Finally we consider a special class of non-homogeneous tensor product problems with improved monotonicity condition of the eigenvalues.

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.

On the Power of Standard Information for Tractability for L&lt;sub&gt;2&lt;/sub&gt;- Approximation in the Randomized Setting

We study the approximation of multivariate functions from a separable Hilbert space in the randomized setting with the error measured in the weighted L2 norm. We consider algorithms that use standard information Λstd consisting of function values or general linear information Λall consisting of arbitrary linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability in the randomized setting for Λstd and Λall for the normalized or absolute error criterion. For the normalized error criterion, we show that the power of Λstd is the same as that of Λall for all notions of exponential tractability and some notions of algebraic tractability without any condition. For the absolute error criterion, we show that the power of Λstd is the same as that of Λall for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 98, 101, 102 and almost solve Open Problem 100 as posed by E.Novak and H.Wo´zniakowski in the book: Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS Tracts in Mathematics, Zürich, 2012.

On the power of standard information for tractability for L2-approximation in the average case setting

We study multivariate approximation in the average case setting with the error measured in the weighted L2 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 the equivalences of various notions of algebraic and exponential tractability for Λstd and Λall for the absolute error criterion, and show that the power of Λstd is the same as that of Λall for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 116-118 and almost solve Open Problem 115 as posed by Novak and Woźniakowski (2012) [25].

A partition-enhanced least-squares collocation approach (PE-LSC)

We present a partition-enhanced least-squares collocation (PE-LSC) which comprises several modifications to the classical LSC method. It is our goal to circumvent various problems of the practical application of LSC. While these investigations are focused on the modeling of the exterior gravity field the elaborated methods can also be used in other applications. One of the main drawbacks and current limitations of LSC is its high computational cost which grows cubically with the number of observation points. A common way to mitigate this problem is to tile the target area into sub-regions and solve each tile individually. This procedure assumes a certain locality of the LSC kernel functions which is generally not given and, therefore, results in fringe effects. To avoid this, it is proposed to localize the LSC kernels such that locality is preserved, and the estimated variances are not notably increased in comparison with the classical LSC method. Using global covariance models involves the calculation of a large number of Legendre polynomials which is usually a time-consuming task. Hence, to accelerate the creation of the covariance matrices, as an intermediate step we pre-calculate the covariance function on a two-dimensional grid of isotropic coordinates. Based on this grid, and under the assumption that the covariances are sufficiently smooth, the final covariance matrices are then obtained by a simple and fast interpolation algorithm. Applying the generalized multi-variate chain rule, also cross-covariance matrices among arbitrary linear spherical harmonic functionals can be obtained by this technique. Together with some further minor alterations these modifications are implemented in the PE-LSC method. The new PE-LSC is tested using selected data sets in Antarctica where altogether more than 800,000 observations are available for processing. In this case, PE-LSC yields a speed-up of computation time by a factor of about 55 (i.e., the computation needs only hours instead of weeks) in comparison with the classical unpartitioned LSC. Likewise, the memory requirement is reduced by a factor of about 360 (i.e., allocating memory in the order of GB instead of TB).

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).

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).

Valuations on convex functions and convex sets and Monge–Ampère operators

Abstract The notion of a valuation on convex bodies is very classical; valuations on a class of functions have been introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translation invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite-dimensional kernel. The proof uses the author’s irreducibility theorem and properties of the real Monge–Ampère operators due to A.D. Alexandrov and Z. Blocki. Furthermore we show how to use complex, quaternionic, and octonionic Monge–Ampère operators to construct more examples of continuous valuations on convex functions in an analogous way.

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.

Approximation complexity of sums of random processes

We study approximation properties of additive random fields Yd(t),t∈[0,1]d, d∈N, which are sums of d uncorrelated zero-mean random processes with continuous covariance functions. The average case approximation complexity nYd(ε) is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate Yd, with relative 2-average error not exceeding a given threshold ε∈(0,1). We investigate the growth of nYd(ε) for arbitrary fixed ε∈(0,1) and d→∞. The results are applied to the sums of the Wiener processes with different variance parameters.

Average case tractability of multivariate approximation with Gaussian kernels

We consider d-variate approximation problems in the average case setting in the weighted L2 space with the standard Gaussian weight. This space is equipped with a zero-mean Gaussian measure whose covariance kernel is a product of univariate Gaussian kernels with non-increasing positive shape parameters γj2 for j=1,2,…,d. We study the average case error of algorithms that use finitely many arbitrary continuous linear functionals. We define the information complexity n(ε,d) to be the minimal number of linear functionals which are needed to find an algorithm whose average case error is at most ε. We investigate different notions of tractability or exponential convergence-tractability (EC-tractability), and get necessary and sufficient conditions in terms of the shape parameters. In particular, for fixed s>0 and t∈(0,1) we obtain that the sufficient and necessary condition on γj2 for which limd+ε−1→∞n(ε,d)ε−s+dt=0holds is limj→∞j1−tγj2ln+γj−2=0,where ln+x=max(1,lnx).

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→∞.

Asymptotic analysis of average case approximation complexity of additive random fields

We study approximation properties of centred additive random fields Yd, d∈N. The average case approximation complexity nYd(ε) is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate Yd, with relative 2-average error not exceeding a given threshold ε∈(0,1). We investigate the growth of nYd(ε) for arbitrary fixed ε∈(0,1) and d→∞. Under natural assumptions we obtain general results concerning asymptotics of nYd(ε). We apply our results to additive random fields with marginal random processes corresponding to the Korobov kernels.

Multivariate approximation for analytic functions with Gaussian kernels

We approximate d-variate analytic functions defined on Rd which belong to a tensor product reproducing kernel Hilbert space. The kernel of this space is Gaussian with non-increasing positive shape parameters γj2 for j=1,2,…,d. The error of approximation is defined in the L2 sense with the standard Gaussian weight. We study the worst case error of algorithms that use at most n arbitrary linear functionals on d-variate functions. We prove that for arbitrary shape parameters there are algorithms enjoying exponential convergence, but the exponent of exponential convergence depends on d and goes to zero as d approaches infinity. We study the absolute and normalized error criteria, in which the information complexity n(ε,d) is defined as the minimal number of linear functionals which are needed to find an algorithm whose worst case error is at most ε or at most ε times the norm of the approximation operator. We study different notions of tractability which describe how the information complexity behaves as a function of d and lnε−1. We find necessary and sufficient conditions on various notions of tractability in terms of shape parameters γj2. Surprisingly enough, these conditions are the same for the absolute and normalized error criteria although the norm of the approximation operator may be exponentially small in d making the normalized error much harder than the absolute one. In particular, for any positive t and κ we find conditions on γj2 for which limd+ε−1→∞lnn(ε,d)dt+[lnε−1]κ=0.This holds •for all positive γj2 if max(t,κ)>1,•for t=1 and κ=1 iff limjγj2=0,•for t<1 and κ=1 iff limjln(j)∕ln(1∕γj2)=0,•for t≤1 and κ<1 iff limj→∞j(1−κ)∕κ∕ln(1∕γj2)=0.

A simplified criterion for quasi-polynomial tractability of approximation of random elements and its applications

We study approximation properties of sequences of centered random elements Xd, d∈N, with values in separable Hilbert spaces. We focus on sequences of tensor product-type random elements, which have covariance operators of corresponding tensor product form. The average case approximation complexity nXd(ε) is defined as the minimal number of evaluations of arbitrary linear functionals that is needed to approximate Xd with relative 2-average error not exceeding a given threshold ε∈(0,1). The growth of nXd(ε) as a function of ε−1 and d determines whether a sequence of corresponding approximation problems for Xd, d∈N, is tractable or not. Different types of tractability were studied in the paper by Lifshits et al. (J. Complexity, 2012), where for each type the necessary and sufficient conditions were found in terms of the eigenvalues of the marginal covariance operators. We revise the criterion of quasi-polynomial tractability and provide a simplified version. We illustrate our result by applying it to random elements corresponding to tensor products of squared exponential kernels. We also extend a recent result of Xu (2014) concerning weighted Korobov kernels.

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.