On the use of dimension reduction techniques in quasi-Monte Carlo methods

It is known that dimension reduction techniques may improve the efficiency of quasi-Monte Carlo (QMC) methods. Different dimension reduction techniques may lead to different efficiencies, even if the nominal dimensions are equal and the same quasi-random sequences are used. To explain this, the degree of additivity and the truncation variance ratio are studied in this paper. Three dimension reduction techniques are compared: Brownian bridge (BB), optimal Brownian bridge (OBB) and principal component analysis (PCA), in which OBB provides a new generating order so that it improves BB. Numerical experiments are performed to compare the efficiency of QMC and Latin hypercube sampling (LHS) combined with different dimension reduction techniques. The importance of the dimension reduction techniques in QMC and the usefulness of the degree of additivity and the truncation variance ratio in characterizing the performance of QMC are confirmed.