# Tag Archives: tractability

## Exponential Convergence and Tractability of Multivariate Integration for Korobov Spaces

I discuss the recently resubmitted manuscript [DLPW] titled `Exponential Convergence and Tractability of Multivariate Integration for Korobov Spaces‘ by J.D., G. Larcher, F. Pillichshammer, and H. Wo\’zniakowski.

The initial aim of the paper is to show that lattice rules can achieve an exponential rate of convergence for infinitely times differentiable functions. The technical difficulty therein lies in the fact that an application of Jensen’s inequality (which states that ${(\sum_{n} |a_n|)^\lambda \le \sum_n |a_n|^\lambda}$ for ${0 < \lambda < 1}$) yields only a convergence of ${O(n^{-\alpha})}$, where ${n}$ is the number of quadrature points. Though ${\alpha}$ can be arbitrarily large, in the land of asymptotia this is still worse than a convergence of, say, ${\omega^{-n^{1/s}}}$, for some ${0 < \omega < 1}$. Hence the first challenge is to find ways to prove convergence rates without relying on Jensen’s inequality.