Higher-Order Quasi-Monte Carlo

A collection of resources for HOQMC, in particular for interlaced polynomial lattice rules.

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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:

  • Randomly shifted lattice rules: F. Kuo
  • lattice rules and polynomial lattice rules: D. Nuyens

Bibliography

  1. 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)
  2. 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)
  3. 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)
  4. Dick, J. 2009. The decay of the Walsh coefficients of smooth functions. Bull. Aust. Math. Soc. 80, 3, 430–453. (link)
  5. Dick, J., Kuo, F.Y., and Sloan, I.H. 2013. High-dimensional integration: the quasi-Monte Carlo way. Acta Numer. 22, 133–288.
  6. 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)
  7. Dick, J. and Pillichshammer, F. 2010. Digital nets and sequences. Cambridge University Press, Cambridge.
  8. Goda, T. 2015. Good interlaced polynomial lattice rules for numerical integration in weighted Walsh spaces. J. Comput. Appl. Math. 285, 279–294. (link)
  9. 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)
  10. 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)
  11. 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)
  12. Sloan, I.H. and Reztsov, A.V. 2002. Component-by-component construction of good lattice rules. Math. Comp. 71, 237, 263–273. (link)
  13. 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)