Advances in sampling methods with a dependence structure

具有依赖结构的采样方法的进展

• 批准号: RGPIN-2020-04019
• 负责人: Lemieux, Christiane
• 金额: $ 1.75万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=765768
• 关键词: Advances; sampling; methods; dependence; structure

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.

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师评估所研究系统的兴趣量。其中许多工具依赖某种形式的随机抽样来近似不存在明确公式的数量。随机采样通常用于“模拟”系统的场景。对于每个场景，都会评估感兴趣数量的相应值。通过多次重复此过程，可以创建该数量的可能值样本，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”。 这种方法的缺点是随机抽样本质上会产生不规则性。事实上，由于场景是彼此独立采样的，因此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多。 准蒙特卡罗方法旨在通过用更结构化的采样代替随机采样来解决这个问题。更准确地说，通过隐式考虑到目前为止采样的场景来对新场景进行采样。这是通过使用低差异序列来实现的，这些序列是试图以非常统一的方式将点放置在定义点的空间中的结构。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 20 到 25 年中获得了相当多的关注，因为它们已被证明对于解决金融中的高维问题很有用，例如涉及长时间模拟多种金融资产。也就是说，在相同的计算量下，它们为估计器提供的误差比基于蒙特卡罗的估计器更小。该研究计划的主要目标是通过关注准蒙特卡罗方法在其基础采样方案中引起的依赖性来增进我们对准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，我们的目标是在使用低差异序列自适应地构造近似值的算法的设计和分析方面取得重大进展，即沿途学习系统的一些特征以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”的技术变得更加困难。这限制了准蒙特卡罗方法可以处理的模型种类。我们的研究将尝试解决这些限制。 该研究项目将涉及至少 10 名各个级别的学生，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。这项研究融合了理论和实践工作，因此学生将有能力将所获得的知识转移到工业界或学术界。