Related papers: On Integration Methods Based on Scrambled Nets of …
Quasi-Monte Carlo methods are designed for integrands of bounded variation, and this excludes singular integrands. Several methods are known for integrands that become singular on the boundary of the unit cube $[0,1]^d$ or at isolated…
Quasi-Monte Carlo (QMC) integration over unbounded domains $\mathbb{R}^s$ remains challenging due to the high dimensionality of sampling space and the boundary growth of the integrand. In applications such as uncertainty quantification…
The $\mathcal{L}_2$ discrepancy is one of several well-known quantitative measures for the equidistribution properties of point sets in the high-dimensional unit cube. The concept of weights was introduced by Sloan and Wo\'{z}niakowski to…
Randomized quasi-Monte Carlo, via certain scramblings of digital nets, produces unbiased estimates of $\int_{[0,1]^d}f(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}$ with a variance that is $o(1/n)$ for any $f\in L^2[0,1]^d$. It also satisfies…
We study quasi-Monte Carlo integration for twice differentiable functions defined over a triangle. We provide an explicit construction of infinite sequences of points including one by Basu and Owen (2015) as a special case, which achieves…
Quasi-Monte Carlo (qMC) methods are a powerful alternative to classical Monte-Carlo (MC) integration. Under certain conditions, they can approximate the desired integral at a faster rate than the usual Central Limit Theorem, resulting in…
Driven by several successful applications such as in stochastic gradient descent or in Bayesian computation, control variates have become a major tool for Monte Carlo integration. However, standard methods do not allow the distribution of…
Let $f:[0,1]^d\to\mathbb{R}$ be a completely monotone integrand as defined by Aistleitner and Dick (2015) and let points $\boldsymbol{x}_0,\dots,\boldsymbol{x}_{n-1}\in[0,1]^d$ have a non-negative local discrepancy (NNLD) everywhere in…
In this article we revisit the problem of numerical integration for monotone bounded functions, with a focus on the class of nonsequential Monte Carlo methods. We first provide new a lower bound on the maximal $L^p$ error of nonsequential…
We show that repulsive random variables can yield Monte Carlo methods with faster convergence rates than the typical $N^{-1/2}$, where $N$ is the number of integrand evaluations. More precisely, we propose stochastic numerical quadratures…
We consider the problem of estimating an expectation $ \mathbb{E}\left[ h(W)\right]$ by quasi-Monte Carlo (QMC) methods, where $ h $ is an unbounded smooth function on $ \mathbb{R}^d $ and $ W$ is a standard normal distributed random…
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…
This paper concerns the use of a particular class of determinantal point processes (DPP), a class of repulsive spatial point processes, for Monte Carlo integration. Let $d\ge 1$, $I\subseteq \overline d=\{1,\dots,d\}$ with $\iota=|I|$.…
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…
We consider the problem of numerical approximation of integrals of random fields over a unit hypercube. We use a stratified Monte Carlo quadrature and measure the approximation performance by the mean squared error. The quadrature is…
We study approximate integration of a function $f$ over $[0,1]^s$ based on taking the median of $2r-1$ integral estimates derived from independently randomized $(t,m,s)$-nets in base $2$. The nets are randomized by Matousek's random linear…
Quantiles and expected shortfalls are usually used to measure risks of stochastic systems, which are often estimated by Monte Carlo methods. This paper focuses on the use of quasi-Monte Carlo (QMC) method, whose convergence rate is…
Niederreiter [H.Niederreiter, Error bounds for quasi-Monte Carlo integration with uniform point sets, Journal of computational and applied mathematics 150 (2003), 283-292] established new bounds for quasi-Monte Carlo integration for nodes…
Recent advances in quasi-Monte Carlo integration have shown that for linearly scrambled digital net estimators, the convergence rate can be dramatically improved by taking the median rather than the mean of multiple independent replicates.…
This paper studies the rate of convergence for conditional quasi-Monte Carlo (QMC), which is a counterpart of conditional Monte Carlo. We focus on discontinuous integrands defined on the whole of $R^d$, which can be unbounded. Under…