Design and analysis of efficient quasi-Monte Carlo sampling methods

高效准蒙特卡罗采样方法的设计与分析

• 批准号: RGPIN-2015-04813
• 负责人: Lemieux, Christiane
• 金额: $ 1.24万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675822
• 关键词: Design; analysis; efficient; quasi; Monte

In a large number of scientific disciplines, stochastic models are used to represent systems for which certain quantities of interest must be evaluated. For example, a production manager might need to compare different inventory policies in terms of their expected cost, assuming that supply and demand are both subjected to some form of randomness, or a telecommunication network designer might want to compare different designs based on their ability to handle large stochastic flows of information without incurring losses or slow-downs. The Monte Carlo method can be used to answer questions of that nature for complex systems which feature several random interacting components. This method uses random sampling in order to "simulate" possible scenarios for the system under study, and for each, the corresponding value of the quantity of interest is evaluated. By repeating this process several times, a sample of possible values for this quantity is created, which can then be used for inference, e.g., the sample mean can be used as an estimator for the quantity of interest. Quasi-Monte Carlo methods aim at improving upon Monte Carlo by replacing the random sampling inherent to the Monte Carlo method by a more structured form of sampling. This improved sampling is based on the use of low-discrepancy sequences, which are constructions that attempt to place points in a very uniform way in the space over which they are defined. These methods have gained considerable attention over the last 15 to 20 years, as they have proven to be useful for solving difficult high-dimensional problems in finance, e.g., involving the simulation of several financial assets over long periods of time. More precisely, with the same computational effort, they can provide estimators with a smaller error than those obtained by applying the Monte Carlo method. ***In this research program, I plan to contribute to the design and analysis of quasi-Monte Carlo sampling schemes, by working with students at the undergraduate, Master's, and doctoral levels on the three following objectives: 1) to improve our ability to efficiently design quasi-Monte Carlo sampling schemes (i.e., low-discrepancy sequences) that work well for a given problem; 2) to propose novel ways to analyze low-discrepancy sequences so that new insight can be gained into the performance of these sequences, and 3) to expand the class of models that can be tackled by quasi-Monte Carlo methods, so that models featuring complex dependence structures can be handled by these methods.*** **

在许多科学学科中，随机模型用于表示必须评估某些感兴趣数量的系统，例如，假设供应和需求，生产经理可能需要根据其预期成本来比较不同的库存策略。都受到某种形式的随机性的影响，或者电信网络设计人员可能希望根据其处理大量随机信息流而不会造成损失或速度下降的能力来比较不同的设计。蒙特卡罗方法可用于回答以下问题：复杂系统的本质该方法使用随机采样来“模拟”所研究系统的可能场景，并通过多次重复此过程来评估每个感兴趣量的相应值。创建该数量的可能值，然后将其用于推理，例如，样本均值可以用作感兴趣数量的估计量，准蒙特卡罗方法旨在通过替换随机方法来改进蒙特卡罗。采样这种改进的采样是蒙特卡罗方法所固有的，它是基于低差异序列的使用，这些序列试图以非常统一的方式将点放置在定义它们的空间中。这些方法在过去 15 到 20 年中获得了相当多的关注，因为它们已被证明对于解决金融领域的高维难题非常有用，例如长时间内模拟多种金融资产。准确地说，通过同样的努力，它们可以提供比应用蒙特卡罗方法获得的误差更小的估计器***在这个研究项目中，我计划为准蒙特卡罗抽样方案的设计和分析做出贡献，通过与本科生、硕士生和博士生合作实现以下三个目标：1）提高我们有效设计适用于以下情况的准蒙特卡罗抽样方案（即低差异序列）的能力：一个给定的问题；2) 提出分析低差异序列的新方法，以便获得对这些序列性能的新见解，以及 3) 扩展可以通过准蒙特卡罗方法解决的模型类别，以便可以通过这些方法处理具有复杂依赖结构的模型。*** **