We study the multivariate L2-approximation problem APPd defined over a Banach space in the average case setting. The space is equipped with a zero-mean Gaussian measure with a weighted covariance kernel, which depends on parameter sequences α={αj}j∈N and β={βj}j∈N with 1<α1≤α2≤⋯ and 1≥β1≥β2≥⋯>0. In this paper, two interesting weighted covariance kernels are considered, which model the importance of the covariance kernels. Under the absolute error criterion or the normalized error criterion, we discuss (s,t)-weak tractability of the L2-approximation problem APP={APPd}d∈N from a Banach space whose zero-mean Gaussian measure has the above two weighted covariance kernels for some positive numbers s and t in the average case setting. Here, (s,t)-weak tractability means how the information complexity behaves as a function of dt and ε−s for large dimension d and small threshold ε. In particular, for all s>0 and t∈(0,1), we find the matching sufficient and necessary condition on the parameter sequences α={αj}j∈N and β={βj}j∈N to obtain (s,t)-weak tractability under the absolute error criterion or the normalized error criterion in the average case setting. We describe (s,t)-weak tractability by the matching sufficient and necessary condition, which reflects symmetry.
We study the multivariate L2-approximation problem APPd defined over a Banach space in the average case setting. The space is equipped with a zero-mean Gaussian measure with a weighted covariance kernel, which depends on parameter sequences α={αj}j∈N and β={βj}j∈N with 1<α1≤α2≤⋯ and 1≥β1≥β2≥⋯>0. In this paper, two interesting weighted covariance kernels are considered, which model the importance of the covariance kernels. Under the absolute error criterion or the normalized error criterion, we discuss (s,t)-weak tractability of the L2-approximation problem APP={APPd}d∈N from a Banach space whose zero-mean Gaussian measure has the above two weighted covariance kernels for some positive numbers s and t in the average case setting. Here, (s,t)-weak tractability means how the information complexity behaves as a function of dt and ε−s for large dimension d and small threshold ε. In particular, for all s>0 and t∈(0,1), we find the matching sufficient and necessary condition on the parameter sequences α={αj}j∈N and β={βj}j∈N to obtain (s,t)-weak tractability under the absolute error criterion or the normalized error criterion in the average case setting. We describe (s,t)-weak tractability by the matching sufficient and necessary condition, which reflects symmetry.
A note about EC-(s,t)-weak tractability of multivariate approximation with analytic Korobov kernels
This note is devoted to discussing multivariate approximation of continuous functions on [0,1]d with analytic Korobov kernels in the worst and average case settings. We only consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We study EC-(s,t)-weak tractability under the absolute or normalized error criterion, and obtain necessary and sufficient conditions for 0<min(s,t)<1 and max(s,t)≤1 in the worst case setting and for s,t>0 in the average case setting.
Worst case tractability of L2-approximation for weighted Korobov spaces
We study L2-approximation problems APPd in the worst case setting in the weighted Korobov spaces $H_{d,\a,{\bm \ga}}$ with parameter sequences ${\bm \ga}=\{\ga_j\}$ and $\a=\{\az_j\}$ of positive real numbers $1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$ and $\frac1 2<\az_1\le \az_2\le \cdots$. We consider the minimal worst case error e(n,APPd) of algorithms that use n arbitrary continuous linear functionals with d variables. We study polynomial convergence of the minimal worst case error, which means that e(n,APPd) converges to zero polynomially fast with increasing n. We recall the notions of polynomial, strongly polynomial, weak and (t1,t2)-weak tractability. In particular, polynomial tractability means that we need a polynomial number of arbitrary continuous linear functionals in d and $\va^{-1}$ with the accuracy $\va$ of the approximation. We obtain that the matching necessary and sufficient condition on the sequences ${\bm \ga}$ and $\a$ for strongly polynomial tractability or polynomial tractability is $$\dz:=\liminf_{j\to\infty}\frac{\ln \ga_j^{-1}}{\ln j}>0,$$ and the exponent of strongly polynomial tractability is $$p^{\text{str}}=2\max\big\{\frac 1 \dz, \frac 1 {2\az_1}\big\}.$$
Tractability of L2-approximation in hybrid function spaces
We consider multivariate L2-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 eL2−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 Λall consisting of all linear functionals and the class Λstd consisting only of function evaluations. The focus lies on the dependence of eL2−app,Λ(N,d) on the dimension d. The main results are conditions for weak, polynomial, and strong polynomial tractability.
Tractability of L2-approximation in hybrid function spaces
We consider multivariate L2-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 eL2−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 Λall consisting of all linear functionals and the class Λstd consisting only of function evaluations. The focus lies on the dependence of eL2−app,Λ(N,d) on the dimension d. The main results are conditions for weak, polynomial, and strong polynomial tractability.
On the power of standard information for tractability for L2-approximation in the randomized setting
We study 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 use the weighted least squares regression algorithm to obtain the upper estimates of the minimal randomized error using Λstd. We investigate the equivalences of various notions of algebraic and exponential tractability for Λstd and Λall for the normalized or absolute error criterion. We show that in the randomized setting for the normalized or absolute error criterion, the power of Λstd is the same as that of Λall for all notions of exponential and algebraic tractability without any condition. Specifically, we solve four Open Problems 98, 100-102 as posed by E.Novak and H.Woźniakowski in the book: Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS Tracts in Mathematics, Zürich, 2012.
We study approximating multivariate functions from a reproducing kernel Hilbert space with the error between the function and its approximation measured in a weighted L2-norm. We consider functions with an arbitrarily large number of variables, d, and we focus on the randomized setting with algorithms using standard information consisting of function values at randomly chosen points. We prove that standard information in the randomized setting is as powerful as linear information in the worst case setting. Linear information means that algorithms may use arbitrary continuous linear functionals, and by the power of information we mean the speed of convergence of the nth minimal errors, i.e., of the minimal errors among all algorithms using n function evaluations. Previously, it was only known that standard information in the randomized setting is no more powerful than the linear information in the worst case setting. We also study (strong) tractability of multivariate approximation in the randomized setting. That is, we study when the minimal number of function evaluations needed to reduce the initial error by a factor ε is polynomial in ε−1 (strong tractability), and polynomial in d and ε−1 (tractability). We prove that these notions in the randomized setting for standard information are equivalent to the same notions in the worst case setting for linear information. This result is useful since for a number of important applications only standard information can be used and verifying (strong) tractability for standard information is in general difficult, whereas (strong) tractability in the worst case setting for linear information is known for many spaces and is relatively easy to check. We illustrate the tractability results for weighted Korobov spaces. In particular, we present necessary and sufficient conditions for strong tractability and tractability. For product weights independent of d, we prove that strong tractability is equivalent to tractability. We stress that all proofs are constructive. That is, we provide randomized algorithms that enjoy the maximal speed of convergence. We also exhibit randomized algorithms which achieve strong tractability and tractability error bounds.