On Weak Tractability of the Clenshaw-Curtis Smolyak Algorithm

TL;DR

This paper proves that the Clenshaw-Curtis Smolyak algorithm achieves weak tractability for integrating certain high-dimensional analytic functions, marking a novel positive result for this class of algorithms.

Contribution

It establishes the first positive weak tractability result for the Smolyak algorithm on a normalized, unweighted problem with non-tensor product integrand space.

Findings

Clenshaw-Curtis Smolyak algorithm is weakly tractable for the problem.

The proof uses polynomial exactness and operator norm bounds.

First such positive result for this class of problems.

Abstract

We consider the problem of integration of d-variate analytic functions defined on the unit cube with directional derivatives of all orders bounded by 1. We prove that the Clenshaw Curtis Smolyak algorithm leads to weak tractability of the problem. This seems to be the first positive tractability result for the Smolyak algorithm for a normalized and unweighted problem. The space of integrands is not a tensor product space and therefore we have to develop a different proof technique. We use the polynomial exactness of the algorithm as well as an explicit bound on the operator norm of the algorithm.