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.
In this dissertation we study the tractability of the information-based complexity n(ε,d) for d-variate function approximation problems. In the deterministic setting for many unweighted problems the curse of dimensionality holds, that means, for some fixed error tolerance ε>0 the complexity n(ε,d) grows exponentially in d. For integration problems one can usually break the curse with the standard Monte Carlo method. For function approximation problems, however, similar effects of randomization have been unknown so far. The thesis contains results on three more or less stand-alone topics. For an extended five page abstract, see the section "Introduction and Results". Chapter 2 is concerned with lower bounds for the Monte Carlo error for general linear problems via Bernstein numbers. This technique is applied to the L∞-approximation of certain classes of C∞-functions, where it turns out that randomization does not affect the tractability classification of the problem. Chapter 3 studies the L∞-approximation of functions from Hilbert spaces with methods that may use arbitrary linear functionals as information. For certain classes of periodic functions from unweighted periodic tensor product spaces, in particular Korobov spaces, we observe the curse of dimensionality in the deterministic setting, while with randomized methods we achieve polynomial tractability. Chapter 4 deals with the L1-approximation of monotone functions via function values. It is known that this problem suffers from the curse in the deterministic setting. An improved lower bound shows that the problem is still intractable in the randomized setting. However, Monte Carlo breaks the curse, in detail, for any fixed error tolerance ε>0 the complexity n(ε,d) grows exponentially in d only.
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.