Mathematical aspects of quasi-Monte Carlo integration

In this talk, I will introduce mathematical aspects of quasi-Monte Carlo (QMC) integration. We aim to approximate the integral of a function on the d-dimensional hypercube [0,1]^d. A useful approach is Monte-Carlo (MC) integration, which uses randomly chosen samples. A drawback of MC is the rate of convergence; the standard deviation of the estimator converges as 1/sqrt(n) asymptotically in n. To have a better rate of convergence as O(log^d N/N) or more, QMC uses deterministic, uniformly distributed points.

In the first part, I will give an overview of QMC, such as star-discrepancy, Koksma-Hlawka inequality, and some explicit constructions as lattices and digital nets.

In the second part, I will show that QMC using lattices and digital nets can achieve a higher rate of convergence for smooth integrands.