Effective dimension of some weighted pre-Sobolev spaces with dominating mixed partial derivatives

TL;DR

This paper investigates the effective dimension in weighted pre-Sobolev spaces with dominating mixed derivatives, providing bounds and conditions under which quadrature methods are tractable, especially highlighting low effective dimensions in various settings.

Contribution

It introduces new notions of effective dimension in weighted pre-Sobolev spaces and derives bounds using Poincaré inequalities, demonstrating low effective dimension in non-periodic integrand spaces.

Findings

Superposition dimension is logarithmic in 1/ε

Spaces with product weights exhibit strong tractability

Even uniform weight spaces have low superposition dimension

Abstract

This paper considers two notions of effective dimension for quadrature in weighted pre-Sobolev spaces with dominating mixed partial derivatives. We begin by finding a ball in those spaces just barely large enough to contain a function with unit variance. If no function in that ball has more than $\varepsilon$ of its variance from ANOVA components involving interactions of order $s$ or more, then the space has effective dimension at most $s$ in the superposition sense. A similar truncation sense notion replaces the cardinality of the ANOVA component by the largest index it contains. Some Poincar\'e type inequalities are used to bound variance components by multiples of these space's squared norm and those in turn provide bounds on effective dimension. Very low effective dimension in the superposition sense holds for some spaces defined by product weights in which quadrature is strongly tractable. The superposition dimension is $O( \log(1/\varepsilon)/\log(\log(1/\varepsilon)))$ just like the superposition dimension used in the multidimensional decomposition method. Surprisingly, even spaces where all subset weights are equal, regardless of their cardinality or included indices, have low superposition dimension in this sense. This paper does not require periodicity of the integrands.