Higher-Order Quasi-Monte Carlo

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

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We consider weights of the SPOD form

with $\delta(i,j) = 1$ if $i=j$ and 0 else. For the two cases $t=0,1$, we consider the sequences $\beta_{0,j}=\lambda_j$ and $\beta_{1,j}=\lambda_j\pi \max(k_{1,j},k_{2,j})$, where $\lambda_j$ is given by

and where

is an ordering of $\mathbb{N}^2$ such that $k_{1,j}^2+k_{2,j}^2 \le k_{1,j+1}^2+k_{2,j+1}^2$ for all $j\in\mathbb{N}$, and the ordering is arbitrary in the case of equality. Below, we give the generating vectors for $\beta_{t,j}$. These sequences are then $p_t$-summable with $p_0>1/2$, $p_1>2/3$, which, assuming the limiting values, yields the choice $\alpha=2$ for the interlacing factor.

IDstaircase2d_spod
Generated2015-06-25
Weight TypeSPOD_2dstaircase

Publications using these vectors

  1. Dick, J., Gantner, R.N., Le Gia, Q.T., and Schwab, C. 2015. Higher order Quasi-Monte Carlo integration for Bayesian Estimation. (submitted). (link)
  2. Dick, J., Kuo, F.Y., Le Gia, Q.T., and Schwab, C. 2014. Multi-level higher order QMC Galerkin discretization for affine parametric operator equations. Report 2014-14 Seminar for Applied Mathematics, ETH Zürich. (link)

Generating Vectors

$\alpha=2$
$\boldsymbol{\alpha}$$\boldsymbol{C}$$\boldsymbol{t}$$\boldsymbol{q}$Links
20.100.tar.gz, .tar.bz2, .zip
20.101.tar.gz, .tar.bz2, .zip
20.102.tar.gz, .tar.bz2, .zip
20.103.tar.gz, .tar.bz2, .zip
20.110.tar.gz, .tar.bz2, .zip
20.111.tar.gz, .tar.bz2, .zip
20.112.tar.gz, .tar.bz2, .zip
20.113.tar.gz, .tar.bz2, .zip
$\boldsymbol{\alpha}$$\boldsymbol{C}$$\boldsymbol{t}$$\boldsymbol{q}$Links
20.0100.tar.gz, .tar.bz2, .zip
20.0101.tar.gz, .tar.bz2, .zip
20.0102.tar.gz, .tar.bz2, .zip
20.0103.tar.gz, .tar.bz2, .zip
20.0110.tar.gz, .tar.bz2, .zip
20.0111.tar.gz, .tar.bz2, .zip
20.0112.tar.gz, .tar.bz2, .zip
20.0113.tar.gz, .tar.bz2, .zip