Advances in sampling methods with a dependence structure

为了在当今世界保持竞争力，加拿大经济依赖于科学和工业的进步，而科学和工业的进步越来越依赖于高效计算工具的可用性。这些工具帮助科学家和分析人员对所研究的系统进行评估。许多这些工具依赖于某种形式的随机抽样来近似没有明确公式存在的数量。随机抽样通常用于“模拟”系统的场景。对于每个场景，计算感兴趣的数量的相应值。通过多次重复这个过程，可以创建这个数量的可能值的样本，然后可以将其用于推理。这种方法通常被称为“蒙特卡罗方法”。这种方法的一个缺点是，从本质上讲，随机抽样会产生不规则性。事实上，由于场景是彼此独立采样的，我们可能会得到太多相似的场景和/或某种类型的场景不够。拟蒙特卡罗方法旨在通过用更结构化的抽样代替随机抽样来解决这个问题。更准确地说，新的场景是通过隐式地考虑到目前为止采样的场景来采样的。这是通过使用低差异序列来实现的，低差异序列是一种试图以非常一致的方式在定义它们的空间中放置点的结构。然后使用复杂的技术将每个点转换为系统的场景。这些方法在过去的20到25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。也就是说，在相同的计算努力下，它们提供的估计器比基于蒙特卡罗的估计器误差更小。本研究计划的主要目标是通过关注其底层抽样方案中诱导的依赖性来提高我们对拟蒙特卡罗方法的理解。这种新方法有可能提高这些方法的有效性。此外，我们的目标是在设计和分析算法方面取得重大进展，这些算法使用低差异序列自适应地构建近似，即在学习系统的一些特征的过程中进一步直接采样到重要区域。最后，当使用低差异序列时，应用将“点”转换为“场景”的技术更加困难。这限制了可以用准蒙特卡罗方法解决的模型种类。我们的研究将试图解决这些限制。该研究项目将涉及至少10名来自各个层次的学生，他们将获得有关蒙特卡罗和准蒙特卡罗方法的宝贵专业知识。该研究将理论与实践相结合，因此学生将很好地将所学知识转移到行业或学术界。

To remain competitive in today's world, the Canadian economy rests on advances in science and industry, which increasingly depend on the availability of efficient computational tools. These tools help scientists and analysts evaluate quantities of interest for a system under study. Many of these tools rely on some form of random sampling to approximate quantities for which no explicit formula exists. Random sampling is typically used to "simulate" scenarios of the system. For each scenario, 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. This approach is often referred to as the "Monte Carlo method". A drawback of this method is that by nature, random sampling can produce irregularities. Indeed, since scenarios are sampled independently from one another, we may get too many that are similar and/or not enough of a certain type. Quasi-Monte Carlo methods aim at addressing this issue by replacing random sampling by more structured sampling. More precisely, new scenarios are sampled by implicitly taking into account the scenarios sampled so far. This is achieved through 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. Sophisticated techniques are then used to transform each point into a scenario of the system. These methods have gained considerable attention over the last 20 to 25 years, as they have proven to be useful for solving high-dimensional problems in finance, e.g., involving the simulation of several financial assets over long periods of time. That is, with the same computational effort, they provide estimators with a smaller error than Monte Carlo-based ones. The main goal of this research program is to advance our understanding of quasi-Monte Carlo methods by focusing on the dependence being induced in their underlying sampling schemes. This new approach has the potential to improve the effectiveness of these methods. In addition, we aim to make significant progress in the design and analysis of algorithms that use low-discrepancy sequences to construct approximations adaptively, i.e., learning along the way some of the features of the system to further direct sampling into important regions. Finally, when using low-discrepancy sequences it is more difficult to apply the techniques by which "points" are transformed into "scenarios". This has limited the kinds of models that can be tackled by quasi-Monte Carlo methods. Our research will attempt to address these limitations. This research program will involve at least 10 students from all levels, who will gain valuable expertise on Monte Carlo and quasi-Monte Carlo methods. This research blends theoretical and practical work, so students will be well equipped to transfer the acquired knowledge to either industry or academia.