Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension

TL;DR

This paper demonstrates that Chen-Skriganov point sets achieve optimal discrepancy rates in Besov spaces with dominating mixed smoothness, extending their optimality to a broader functional setting.

Contribution

The paper extends the known optimal discrepancy properties of Chen-Skriganov point sets to Besov spaces with dominating mixed smoothness, using a $b$-adic Haar basis approach.

Findings

Chen-Skriganov point sets achieve optimal discrepancy in Besov spaces

Extension of discrepancy results to other function spaces

Implications for numerical integration accuracy

Abstract

In a celebrated construction, Chen and Skriganov gave explicit examples of point sets achieving the best possible $L_2$-norm of the discrepancy function. We consider the discrepancy function of the Chen-Skriganov point sets in Besov spaces with dominating mixed smoothness and show that they also achieve the best possible rate in this setting. The proof uses a $b$-adic generalization of the Haar system and corresponding characterizations of the Besov space norm. Results for further function spaces and integration errors are concluded.