Integration and Approximation of Multivariate Functions: Average Case Complexity with Isotropic Wiener Measure

Integration and Approximation of Multivariate Functions: Average Case Complexity with Isotropic Wiener Measure

We study the average case complexity of multivariate integration and L2 function approximation for the class F = C([0, 1]) of continuous functions of d variables. The class F is endowed with the isotropic Wiener measure (Brownian motion in Levy’s sense). Furthermore, for both problems, only function values are used as data.

Integration and L2-Approximation: Average Case Setting with Isotropic Wiener Measure for Smooth Functions

We propose isotropic probability measures deened on classes of smooth multi-variate functions. These provide a natural extension of the classical isotropic Wiener measure to multivariate functions from C 2r. We show that in the corresponding average case setting the minimal errors of algorithms that use n function values are ? n ?(d+4r+1)=2d and ? n ?(4r+1)=2d for the integration and L 2-approx...

Average Case Complexity of Weighted Approximation and Integration over R+

Average Case Complexity of Linear Multivariate

We study the average case complexity of a linear multivariate problem (LMP) defined on functions of d variables. We consider two classes of information. The first Λ std consists of function values and the second Λ all of all continuous linear functionals. Tractability of LMP means that the average case complexity is O((1/ε) p) with p independent of d. We prove that tractability of an LMP in Λ s...

Derandomization, hard functions, and average-case complexity

In this thesis we study questions related to the relative power of randomness in computation, and the connections between the existence of (explicit) hard functions on the average and hard functions in the worst-case. It is well known that these questions are closely related, and in fact, the following three statements are equivalent [19, 65]: 1. There exist a function (computable in exponentia...