The blog Why Add Q to MC?, explained the advantages of carefully chosen, low discrepancy sampling sites for approximating multivariate integrals, or equivalently, expectations of functions of multivariate random variables \begin{align} \mu: &= \int_{[0,1]^d} f(\boldsymbol{x}) \, \text{d}{\boldsymbol{x}} = {\mathbb{E}}[f(\boldsymbol{x})], \; \text{where } \boldsymbol{x} \sim \mathcal{U}[0,1]^d, \\ \mu \approx \widehat{\mu}_n &: = \frac 1n \sum_{i=1}^n f(\boldsymbol{x}_i),Continue reading “Bayesian Stopping Criteria”
Fred J. Hickernell
Monte Carlo Methods 2021
Hope to see you virtually at Monte Carlo Methods 2021, in August 2021. This the prime odd year Monte Carlo conference featuring computer scientists, mathematicians, and statisticians. We are organizing a special session on Quasi-Monte Carlo software. More info coming soon.
Quasi-Monte Carlo Software Article
We recently uploaded an article on Quasi-Monte Carlo Software to https://arxiv.org/abs/2102.07833. Abstract: Practitioners wishing to experience the efficiency gains from using low discrepancy sequences need correct, well-written software. This article, based on our MCQMC 2020 tutorial, describes some of the better quasi-Monte Carlo (QMC) software available. We highlight the key software components required to approximateContinue reading “Quasi-Monte Carlo Software Article”
QMCPy Version 1.0
The developers of QMCPy are happy to announce the release of version 1.0 on February 12, 2021, Chinese New Year! We would like to thank all those who have made this development possible. A special thank you to Developers: Sou-Cheng T. Choi, Fred J. Hickernell, Michael McCourt, Jagadeeswaran Rathinavel, and Aleksei Sorokin; Collaborators: Mike Giles,Continue reading “QMCPy Version 1.0”
A Tutorial at MCQMC 2020
Fred Hickernell gave a virtual tutorial at MCQMC2020 on August 10, 2020. MCQMC2020 is the premier international conference for Monte Carlo quasi-Monte Carlo in scientific computing. A recording of the talk and the corresponding Google Colab notebook are available.
What Makes a Sequence “Low Discrepancy”?
The first blog post, “Why add Q to MC?”, introduced the concept of evenly spread points, which are commonly referred to as low discrepancy (LD) points. This is in contrast to independent and identically distributed (IID) points. Consider two sequences, $\boldsymbol{T}_1, \boldsymbol{T}_2, \ldots \overset{\text{IID}}{\sim} \mathcal{U}[0,1]^d$ \[\boldsymbol{X}_1, \boldsymbol{X}_2, \ldots \overset{\text{LD}}{\sim} \mathcal{U}[0,1]^d.\] Both sequences are expected toContinue reading “What Makes a Sequence “Low Discrepancy”?”
Why Add Q to MC?
Quasi-Monte Carlo (QMC) methods can sometimes speed up simple Monte Carlo (MC) calculations by orders of magnitude? What makes them work so well? MC methods use computer generated random numbers to generate various scenarios. When computing financial risk, the scenarios may be possible financial market outcomes. When assessing the resiliency of the power grid, theContinue reading “Why Add Q to MC?”