Literature
We refer to quasi-Monte Carlo methods with orders of convergence higher than $\mathcal{O}(N^{-1})$ as “Higher-order Quasi-Monte Carlo (HOQMC)” methods. This page contains a list of relevant literature that contributed to the development of HOQMC methods, in particular interlaced polynomial lattice rules developed in [Dick et al. 2014].
The concept of weighted function spaces come from [Sloan and Woźniakowski 1998], and the setting required for higher-order methods is based on [Kuo et al. 2012]. Originally, component-by-component (CBC) construction was proposed in [Sloan and Reztsov 2002]. Fast CBC construction was introduced in [Nuyens and Cools 2006; Cools et al. 2006].
The fundamental developments required for HOQMC stem from [Dick 2007; Dick 2008; Dick 2009], as well as [Goda and Dick 2015; Goda 2015]. A good introduction to the theory can be found in [Dick and Pillichshammer 2010]. Some newer review review articles about quasi-Monte Carlo integration are [Dick et al. 2013].
Other Generating Vector Websites
The following websites contain other generating vectors:
Bibliography
- Cools, R., Kuo, F.Y., and Nuyens, D. 2006. Constructing embedded lattice rules for multivariable integration. SIAM J. Sci. Comput. 28, 6, 2162–2188 (electronic). (link)
- Dick, J. 2007. Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions. SIAM J. Numer. Anal. 45, 5, 2141–2176 (electronic). (link)
- Dick, J. 2008. Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal. 46, 3, 1519–1553. (link)
- Dick, J. 2009. The decay of the Walsh coefficients of smooth functions. Bull. Aust. Math. Soc. 80, 3, 430–453. (link)
- Dick, J., Kuo, F.Y., and Sloan, I.H. 2013. High-dimensional integration: the quasi-Monte Carlo way. Acta Numer. 22, 133–288.
- Dick, J., Kuo, F.Y., Le Gia, Q.T., Nuyens, D., and Schwab, C. 2014. Higher order QMC Petrov-Galerkin discretization for affine parametric operator equations with random field inputs. SIAM J. Numer. Anal. 52, 6, 2676–2702. (link)
- Dick, J. and Pillichshammer, F. 2010. Digital nets and sequences. Cambridge University Press, Cambridge.
- Goda, T. 2015. Good interlaced polynomial lattice rules for numerical integration in weighted Walsh spaces. J. Comput. Appl. Math. 285, 279–294. (link)
- Goda, T. and Dick, J. 2015. Construction of interlaced scrambled polynomial lattice rules of arbitrary high order. Found. Comput. Math. 15, 5, 1245–1278. (link)
- Kuo, F., Schwab, C., and Sloan, I. 2012. Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond. ANZIAM Journal 53, 0, 1–37. (link)
- Nuyens, D. and Cools, R. 2006. Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comp. 75, 254, 903–920 (electronic). (link)
- Sloan, I.H. and Reztsov, A.V. 2002. Component-by-component construction of good lattice rules. Math. Comp. 71, 237, 263–273. (link)
- Sloan, I.H. and Woźniakowski, H. 1998. When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals? J. Complexity 14, 1, 1–33. (link)