Tractability of infinite-dimensional integration in the worst case and randomized settings

• Authors: L. Plaskota;G. W. Wasilkowski
• Venue: Journal of Complexity
• Year: 2011

We consider approximation of weighted integrals of functions with infinitely many variables in the worst case deterministic and randomized settings. We assume that the integrands f belong to a weighted quasi-reproducing kernel Hilbert space, where the weights have product form and satisfy @c"j=O(j^-^@b) for @b1. The cost of computing f(x) depends on the number Act(x) of active coordinates in x and is equal to $(Act(x)), where $ is a given cost function. We prove, in particular, that if the corresponding univariate problem admits algorithms with errors O(n^-^@k^/^2), where n is the number of function evaluations, then the ~-variate problem is polynomially tractable with the tractability exponent bounded from above by max(2/@k,2/(@b-1)) for all cost functions satisfying $(d)=O(e^k^@?^d), for any k=0. This bound is sharp in the worst case setting if @b and @k are chosen as large as possible and $(d) is at least linear in d. The problem is weakly tractable even for a larger class of cost functions including $(d)=O(e^e^^^k^^^@?^^^d). Moreover, our proofs are constructive.