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

Abstract

We consider approximation of linear multivariate problems defined over weighted tensor product Hilbert spaces with finite-order weights. This means we consider functions of d variables that can be represented as sums of functions of at most q* variables. Here, q* is fixed (and presumably small) and d may be arbitrarily large. For the univariate problem, d = 1, we assume we know algorithms A1,ε that use O(ε−p) function or linear functional evaluations to achieve an error ε in the worst case setting. Based on these algorithms A1,ε, we provide a construction of polynomial-time algorithms Ad,ε for the general d-variate problem with the number of evaluations bounded roughly by ε−pdq* to achieve an error ε in the worst case setting.