Design and analysis of efficient quasi-Monte Carlo sampling methods

在大量的科学学科中，随机模型被用来表示必须对其特定的兴趣量进行评估的系统。例如，假设供应和需求都受到某种形式的随机性的影响，生产经理可能需要在预期成本方面比较不同的库存策略，或者电信网络设计人员可能希望比较不同的设计，基于它们处理大型随机信息流的能力，而不会招致损失或速度减慢。蒙特卡罗方法可用于回答具有多个随机相互作用组件的复杂系统的性质问题。这种方法使用随机抽样，以便为所研究的系统“模拟”可能的情景，并对每一种情景的兴趣量的相应数值进行评估。通过多次重复这一过程，创建了该数量的可能平均值的样本，然后可以使用该样本平均值进行推断，例如，样本平均值可以用作感兴趣数量的估计器。准蒙特卡罗方法旨在通过用更结构化的抽样形式取代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低差异序列的使用，低差异序列是试图以非常均匀的方式在定义点的空间中放置点的结构。这些方法在过去15到20年里得到了相当大的关注，因为事实证明，它们对于解决金融中的困难和高维度问题很有用，例如，涉及在很长一段时间内模拟几种金融资产。更准确地说，在相同的计算工作量下，它们可以提供比实际应用蒙特卡罗方法得到的估计量更小的误差。在这个研究计划中，我计划通过与本科生、硕士和博士生合作，在以下三个方面为准蒙特卡罗抽样方案的设计和分析做出贡献：1)提高我们有效设计适用于给定问题的准蒙特卡罗抽样方案(即，低偏差序列)的能力；(2)提出了分析低偏差序列的新方法，以便对这些序列的性能有新的认识；(3)扩展了拟蒙特卡罗方法可以处理的模型类别，使具有复杂依赖结构的模型可以用这些方法来处理。

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.*** **