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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=657182
• 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.*** **

在大量科学学科中，随机模型用于表示必须评估一定量的感兴趣的系统。例如，假设供求既经历了某种形式的随机性，否则生产经理可能需要根据其预期成本进行比较不同的库存策略，或者电信网络设计师可能希望根据其处理大量的随机流量而不会损失或慢下来比较不同的设计。蒙特卡洛方法可用于回答具有多种随机交互成分的复杂系统的性质问题。该方法使用随机抽样来“模拟”正在研究的系统的可能场景，并且对于每种系统，都评估了感兴趣量的相应值。通过重复此过程多次，创建了该数量的可能值的样本，然后可以将其用于推断，例如，样本均值可以用作兴趣量的估计器。准蒙特·卡洛方法旨在通过通过更结构化的采样形式代替蒙特卡洛法的随机采样来改善蒙特卡洛。这种改进的采样基于使用低分配序列的使用，这些序列是试图以非常统一的方式将点放置在其定义的空间中的结构。在过去的15到20年中，这些方法引起了人们的关注，因为事实证明，这些方法可用于解决财务上的困难高维问题，例如，涉及长期内的几个金融资产的模拟。更确切地说，通过相同的计算工作，它们可以为估计器提供比应用蒙特卡洛方法获得的误差。 ***在这项研究计划中，我计划通过与以下三个目标的本科，硕士和博士学位的学生合作，为准蒙特·卡洛取样方案的设计和分析做出贡献：1），以提高我们有效地设计Quasi-Monte Carlo Sampling方案（即低discrepance sequesences sequecess）的能力。 2）提出新的方法来分析低静止序列序列，以便可以在这些序列的性能中获得新的见解，以及3）扩展可以通过准蒙特卡罗方法来解决的模型类别，以便可以通过这些方法来处理具有复杂依赖结构的模型。