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arXiv · 1201.2311

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

Lev Markhasin
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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.

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Lev Markhasin. 2012-01-11. Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension. https://doi.org/10.1016/j.jco.2013.04.007

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