Quasi-Monte Carlo (QMC) sampling has been developed for integration over $[0,1]^s$ where it has superior accuracy to Monte Carlo (MC) for integrands of bounded variation. Scrambled net quadrature gives allows replication based error estimation for QMC with at least the same accuracy and for smooth enough integrands even better accuracy than plain QMC. Integration over triangles, spheres, disks and Cartesian products of such spaces is more difficult for QMC because the induced integrand on a unit cube may fail to have the desired regularity. In this paper, we present a construction of point sets for numerical integration over Cartesian products of $s$ spaces of dimension $d$, with triangles ($d=2$) being of special interest. The point sets are transformations of randomized $(t,m,s)$-nets using recursive geometric partitions. The resulting integral estimates are unbiased and their variance is $o(1/n)$ for any integrand in $L^2$ of the product space. Under smoothness assumptions on the integrand, our randomized QMC algorithm has variance $O(n^{-1 - 2/d} (\log n)^{s-1})$, for integration over $s$-fold Cartesian products of $d$-dimensional domains, compared to $O(n^{-1})$ for ordinary Monte Carlo.
We introduce a new quality measure to assess randomized low-discrepancy point sets of finite size $n$. This new quality measure, which we call "pairwise sampling dependence index", is based on the concept of negative dependence. A negative value for this index implies that the corresponding point set integrates the indicator function of any unanchored box with smaller variance than the Monte Carlo method. We show that scrambled $(0,m,s)-$nets have a negative pairwise sampling dependence index. We also illustrate through an example that randomizing via a digital shift instead of scrambling may …
When a plain Monte Carlo estimate on $n$ samples has variance $σ^2/n$, then scrambled digital nets attain a variance that is $o(1/n)$ as $n\to\infty$. For finite $n$ and an adversarially selected integrand, the variance of a scrambled $(t,m,s)$-net can be at most $Γσ^2/n$ for a maximal gain coefficient $Γ<\infty$. The most widely used digital nets and sequences are those of Sobol'. It was previously known that $Γ\leqslant 2^t3^s$ for Sobol' points as well as Niederreiter-Xing points. In this paper we study nets in base $2$. We show that $Γ\leqslant2^{t+s-1}$ for nets. This bound is a simple, b…
This article provides a strong law of large numbers for integration on digital nets randomized by a nested uniform scramble. The motivating problem is optimization over some variables of an integral over others, arising in Bayesian optimization. This strong law requires that the integrand have a finite moment of order $p$ for some $p>1$. Previously known results implied a strong law only for Riemann integrable functions. Previous general weak laws of large numbers for scrambled nets require a square integrable integrand. We generalize from $L^2$ to $L^p$ for $p>1$ via the Riesz-Thorin interpol…
Dimension-independent convergence rates of randomized nets using median-of-means
Johann Radon Institute for Computational and Applied Mathematics, ÖAW（约翰·拉登计算与应用数学研究所，奥地利科学院）；Institute of Fundamental and Transdisciplinary Research, Zhejiang University（浙江大学基础与跨学科研究院）
We investigate base $b$ Walsh functions for which the variance of the integral estimator based on a scrambled $(0,m,s)$-net in base $b$ is less than or equal to that of the Monte-Carlo estimator based on the same number of points. First we compute the Walsh decomposition for the joint probability density function of two distinct points randomly chosen from a scrambled $(t,m,s)$-net in base $b$ in terms of certain counting numbers and simplify it in the special case $t$ is zero. Using this, we obtain an expression for the covariance of the integral estimator in terms of the Walsh coefficients o…
Pre-integration is an extension of conditional Monte Carlo to quasi-Monte Carlo and randomized quasi-Monte Carlo. It can reduce but not increase the variance in Monte Carlo. For quasi-Monte Carlo it can bring about improved regularity of the integrand with potentially greatly improved accuracy. Pre-integration is ordinarily done by integrating out one of $d$ input variables to a function. In the common case of a Gaussian integral one can also pre-integrate over any linear combination of variables. We propose to do that and we choose the first eigenvector in an active subspace decomposition to …
Quasi-Monte Carlo methods are a way of improving the efficiency of Monte Carlo methods. Digital nets and sequences are one of the low discrepancy point sets used in quasi-Monte Carlo methods. This thesis presents the three new results pertaining to digital nets and sequences: implementing randomized digital nets, finding the distribution of the discrepancy of scrambled digital nets, and obtaining better quality of digital nets through evolutionary computation. Finally, applications of scrambled and non-scrambled digital nets are provided.
Quasi-Monte Carlo confidence intervals using quantiles of randomized nets
Johann Radon Institute for Computational and Applied Mathematics, ÖAW（约翰·拉多纳计算与应用数学研究所，奥地利科学院）；Institute of Fundamental and Transdisciplinary Research, Zhejiang University（浙江大学基础与交叉研究学院）
Quasi-Monte Carlo point sets with small $t$-values and WAFOM
$(t, m, s)$ -网的$t$ -值是拟蒙特卡罗方法积分点集的重要判据，许多点集都是根据$t$ -值构造的，因为这导致小的积分误差界。最近，Matsumoto，Saito和Matoba提出了Walsh品质因数（WAFOM）作为一个快速计算的点集标准，确保高阶收敛的函数类非常高的光滑度。在本文中，我们考虑的点集的搜索算法，其$t$ -值和WAFOM都很小，以便有效地为更广泛的函数类。为此，我们用小的$t$ -值来固定数字$(t, m, s)$ -网（例如，Sobol或Niederreiter-Xing网），应用随机线性加扰，并根据WAFOM选择加扰的数字$(t, m, s)$网。实验结果表明，所得到的点集提高了光滑函数的收敛速度，并对非光滑函数具有鲁棒性。
Uncertainty quantification using importance-sampled quasi-Monte Carlo with dimension-independent convergence rates
Institute of Fundamental and Transdisciplinary Research, Zhejiang University（浙江大学基础与交叉研究院）；Department of Mathematical Sciences, Tsinghua University（清华大学数学科学系）；School of Mathematics, South China University of Technology（华南理工大学数学学院）
Quasi-Monte Carlo (QMC) methods are equal weight quadrature rules to approximate integrals over the unit cube with respect to the uniform measure. In this paper we discuss QMC integration with respect to general product measures defined on an arbitrary cube. We only require that the cumulative distribution function is invertible. We develop a worst-case error bound and study the dependence of the error on the number of points and the dimension for digital nets and sequences as well as polynomial lattice point sets, which are mapped to the domain using the inverse cumulative distribution functi…
