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.*** **

在大量的科学学科中，随机模型被用来表示系统，必须对某些感兴趣的量进行评估。例如，生产经理可能需要根据预期成本比较不同的库存策略，假设供应和需求都受到某种形式的随机性的影响，或者电信网络设计师可能希望根据处理大量随机信息流的能力来比较不同的设计，而不会导致损失或减速。蒙特卡罗方法可以用来回答复杂系统的性质问题，其特征是多个随机相互作用的组件。该方法使用随机抽样，以“模拟”所研究系统的可能场景，并对每个场景评估感兴趣的数量的相应值。通过多次重复该过程，创建了该量的可能值的样本，然后可以将其用于推断，例如，样本平均值可以用作感兴趣的量的估计器。准蒙特卡罗方法旨在通过用更结构化的抽样形式代替蒙特卡罗方法固有的随机抽样来改进蒙特卡罗方法。这种改进的采样是基于使用低差异序列，这是一种试图以非常均匀的方式在定义它们的空间中放置点的结构。这些方法在过去的15到20年中获得了相当大的关注，因为它们已经被证明对于解决金融中的困难高维问题是有用的，例如，涉及在很长一段时间内模拟几种金融资产。更确切地说，在相同的计算工作量下，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量。* 在本研究计划中，我计划通过与本科生，硕士和博士生合作，为准蒙特卡罗抽样方案的设计和分析做出贡献。1）提高我们有效设计准蒙特卡罗抽样方案的能力（即，低差异序列），对于给定的问题很好地工作; 2）提出新的方法来分析低差异序列，以便可以对这些序列的性能获得新的见解，以及3）扩展可以通过准蒙特卡罗方法处理的模型类别，以便可以通过这些方法处理具有复杂依赖结构的模型。 **