Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Rational Points on Curves over Finite Fields: Theory and Applications
Rational Points on Curves over Finite Fields: Theory and Applications
Optimal quadrature for Haar wavelet spaces
Mathematics of Computation
On the mean square weighted L2 discrepancy of randomized digital nets in prime base
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration
Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration
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Quasi-Monte Carlo rules are equal weight quadrature rules defined over the domain $[0,1]^s$. Here we introduce quasi-Monte Carlo-type rules for numerical integration of functions defined on $\mathbb{R}^s$. These rules are obtained by way of some transformation of digital nets such that locally one obtains quasi-Monte Carlo rules, but at the same time, globally one also has the required distribution. We prove that these rules are optimal for numerical integration in spaces of bounded fractional variation. The analysis is based on certain tilings of the Walsh phase plane. As a proof of concept, some numerical examples are included for dimensions between 3 and 10. We compare our method with quasi-Monte Carlo rules transformed to $\mathbb{R}^s$ using the inverse cumulative distribution function. In these examples, the new method significantly improves upon both methods for dimensions up to 5; no improvement is seen for dimension 10.