EFFECTIVE DIMENSION OF SOME WEIGHTED PRE-SOBOLEV SPACES WITH DOMINATING MIXED PARTIAL DERIVATIVES

摘要

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 epsilon of its variance from analysis of variance (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 an ANOVA component by the largest index it contains. These effective dimension definitions for the integration problem coincide with some of the definitions in information-based complexity for the function approximation problem. Some Poincare-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/epsilon)/ log(log(1/epsilon))) just like the superposition dimension used in the multivariate 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.