Design and analysis of efficient quasi-Monte Carlo sampling methods

在许多科学学科中，随机模型被用来表示必须对某些感兴趣的量进行评估的系统。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他们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具有几个随机相互作用组件的复杂系统的问题。该方法使用随机抽样，以便“模拟”所研究系统的可能情况，并对每种情况评估感兴趣的数量的相应值。通过重复这个过程几次，这个数量的可能值的样本被创建，然后可以用于推理，例如，样本均值可以用作感兴趣数量的估计量。拟蒙特卡罗方法旨在改进蒙特卡罗方法，用更结构化的抽样形式取代蒙特卡罗方法固有的随机抽样。这种改进的采样是基于低差异序列的使用，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，因为它们已被证明对解决金融中的高维问题很有用，例如，涉及长时间内几种金融资产的模拟。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估计误差更小的估计。***在这个研究项目中，我计划通过与本科生、硕士和博士层次的学生合作，为准蒙特卡罗采样方案的设计和分析做出贡献，实现以下三个目标：1)提高我们有效设计准蒙特卡罗采样方案（即低差异序列）的能力，使其能够很好地解决给定问题；2)提出分析低差异序列的新方法，从而对这些序列的性能有新的认识；3)扩展准蒙特卡罗方法可以处理的模型类别，从而可以用这些方法处理具有复杂依赖结构的模型。* * * * *

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.*** **