Striving for efficiency in high-dimensional integration? This paper explores the applicability of various low-discrepancy sequences for quasi Monte Carlo integration with a large number of variates, addressing the challenge of efficient numerical integration in complex problems. The study introduces modifications to improve the performance of these sequences. The Halton, Sobol, and Faure sequences, along with the Braaten-Weller construction, are studied. Modifications to the Halton sequence and a new construction of the generalized Halton sequence are proposed for unrestricted dimensions. These new generators are shown to significantly improve upon the original Halton sequence. The paper identifies problems in estimating the error in quasi Monte Carlo integration and selecting appropriate test functions. The maximum error of integration of nine test functions is computed for up to 400 dimensions. An empirical formula for the error of quasi Monte Carlo integration is suggested. This research offers valuable insights for researchers and practitioners using quasi Monte Carlo methods in various fields.
Published in ACM Transactions on Mathematical Software, this paper aligns with the journal's focus on the development, analysis, and evaluation of mathematical software. By investigating the performance of low-discrepancy sequences for quasi Monte Carlo integration, the study contributes to the journal's existing body of research on numerical algorithms and computational methods. Its emphasis on practical implementation and error estimation is relevant to the journal's readership.