Polynomial-Time Algorithms for Multivariate Linear Problems with Finite-Order Weights: Average Case Setting

We study multivariate linear problems in the average case setting with respect to a zero-mean Gaussian measure whose covariance kernel has a finite-order weights structure. This means that the measure is concentrated on a Banach space of d-variate functions that are sums of functions of at most q* variables and the influence of each such term depends on a given weight. Here q* is fixed whereas d varies and can be arbitrarily large. For arbitrary finite-order weights, based on Smolyak’s algorithm, we construct polynomial-time algorithms that use standard information. That is, algorithms that solve the d-variate problem to within ε using of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon^{-p}d^{q^{*}}$\end{document} function values modulo a power of ln ε−1. Here p is the exponent which measures the difficulty of the univariate (d=1) problem, and the power of ln ε−1 is independent of d. We also present a necessary and sufficient condition on finite-order weights for which we obtain strongly polynomial-time algorithms, i.e., when the number of function values is independent of d and polynomial in ε−1. The exponent of ε−1 may be, however, larger than p. We illustrate the results by two multivariate problems: integration and function approximation. For the univariate case we assume the r-folded Wiener measure. Then p=1/(r+1) for integration and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p=1/(r+\frac{1}{2})$\end{document} for approximation.