Monte Carlo and Quasi-Monte Carlo Methods for Statistics

蒙特卡罗和准蒙特卡罗统计方法

• 批准号: 0604939
• 负责人: Art Owen
• 金额: --
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604939&HistoricalAwards=false
• 关键词: Monte; Carlo; Quasi; Methods; Statistics

This project extends and improves Monte Carlo sampling techniques. The main idea is to incorporate ideas that originated in quasi-Monte Carlo sampling into Monte Carlo sampling. In particular this project embeds quasi-Monte Carlo sampling into Markov chain Monte Carlo simulations, a combination that until recently was not known to be possible. The combination can be very effective, bringing variance reductions of over 100 fold in some problems. The project personnel are identifying when quasi-Monte Carlo brings a large improvement in Markov chain Monte Carlo, as well as finding new ways to combine the methods. Another area in which this project is improving Monte Carlo sampling is in integration of unbounded functions. There are versions of randomized quasi-Monte Carlo sampling that attain a better convergence rate than the original quasi-Monte Carlo sampling, at least for well behaved integrands. Unfortunately, problems with unbounded integrands don't see much improvement. This project extends the benefit of randomized quasi-Monte Carlo sampling to unbounded integrands by applying a change of variable formula to bound the integrand while endeavoring to prevent the integrand from becoming too spiky. This project is also investigating tempering methods for speeding up the mixing of Markov chain Monte Carlo as well as applications of Monte Carlo and quasi-Monte Carlo ideas to problems in bioinformatics.Monte Carlo sampling is used in just about every branch of science and engineering. At its simplest it involves simulating a system using random number generators, and recording what happens. In practice Monte Carlo methods are used to solve by brute force computation some problems that are too hard to do mathematically. Real world complications that can easily be introduced into a simulation often make a problem too hard for exact mathematical treatment. Quasi-Monte Carlo methods can be used to drive a simulation byt replacing the random number sequence by very carefully chosen numbers that are much more balanced than random numbers are. The result is often a tremendous speedup for a given level of accuracy, or a tremendous increase in accuracy for a given computation time. This project pushes quasi-Monte Carlo methods into simulations that had hitherto been thought incapable of benefiting from them. Those simulation techniques, known as Markov chain Monte Carlo, are used in many areas including materials science, analysis of educational testing data sets, biomedical research, robotics, computer graphics, and marketing. This project is also looking at other ways to improve Monte Carlo methods with similarly broad potential for benefit.

该项目扩展和改进了蒙特卡罗抽样技术。 其主要思想是将起源于准蒙特卡罗抽样的思想融入蒙特卡罗抽样。特别是这个项目嵌入准蒙特卡罗抽样到马尔可夫链蒙特卡罗模拟，一个组合，直到最近还不知道是可能的。该组合可以非常有效，在某些问题中带来超过100倍的方差减少。 项目人员正在确定准蒙特卡罗何时会给马尔可夫链蒙特卡罗带来很大的改进，以及寻找新的方法来联合收割机的方法。 该项目正在改进蒙特卡洛采样的另一个领域是无界函数的集成。有一些版本的随机化拟蒙特卡罗抽样比原始的拟蒙特卡罗抽样获得更好的收敛速度，至少对于表现良好的被积函数。不幸的是，无界被积函数的问题并没有太大的改善。本计画将随机化拟蒙地卡罗抽样的优点延伸至无界被积函数，借由应用变量的变化公式来限制被积函数，同时努力避免被积函数变得太尖。该项目还研究了用于加速马尔可夫链蒙特卡罗混合的回火方法，以及蒙特卡罗和准蒙特卡罗思想在生物信息学问题中的应用。蒙特卡罗抽样几乎用于科学和工程的每个分支。最简单的是，它涉及使用随机数生成器模拟一个系统，并记录发生的事情。 在实践中，蒙特卡罗方法被用来解决一些很难用数学方法解决的问题。 真实的世界的复杂性很容易被引入到模拟中，这往往会使问题难以进行精确的数学处理。准蒙特卡罗方法可以用来驱动模拟拜特通过用非常仔细选择的比随机数更平衡的数字来替换随机数序列。 其结果通常是在给定精度水平下的巨大加速，或者在给定计算时间下的精度的巨大增加。该项目将准蒙特卡罗方法推进到迄今为止被认为无法从中受益的模拟中。这些模拟技术被称为马尔可夫链蒙特卡罗，用于许多领域，包括材料科学，教育测试数据集分析，生物医学研究，机器人技术，计算机图形学和营销。该项目还在寻找其他方法来改进蒙特卡洛方法，这些方法具有同样广泛的受益潜力。