ε-superposition and truncation dimensions in average and probabilistic settings for ∞-variate linear problems

The paper deals with linear problems defined on γ-weighted Hilbert spaces of functions with infinitely many variables. The spaces are endowed with zero-mean Gaussian measures which allows to define and study ε-truncation and ε-superposition dimensions in the average case and probabilistic settings. Roughly speaking, these ε-dimensions quantify the smallest number k=k(ε) of variables that allow to approximate the ∞-variate functions by special ones that depend on at most k-variables with the average error bounded by ε. In the probabilistic setting, given δ∈(0,1), we want the error ≤ε with probability ≥1−δ. We show that the ε-dimensions are surprisingly small which, for anchored spaces, leads to very efficient algorithms, including the Multivariate Decomposition Methods.