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，而科学和工业越来越依赖于有效的计算工具。这些工具帮助科学家和分析师评估正在研究的系统的感兴趣的数量。这些工具中的许多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。随机采样通常用于“模拟”系统的场景。对于每个场景，评估感兴趣的量的相应值。通过多次重复此过程，可以创建此量的可能值的样本，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，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，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。 拟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。更确切地说，新的情景是通过隐含地考虑到迄今为止抽样的情景来抽样的。这是通过使用低差异序列来实现的，低差异序列是试图以非常均匀的方式在定义它们的空间中放置点的构造。然后使用复杂的技术将每个点转换为系统的场景。在过去的20到25年里，这些方法得到了相当大的关注，因为它们被证明对解决金融中的高维问题是有用的，例如，涉及在很长一段时间内模拟几种金融资产。也就是说，在相同的计算工作量下，它们提供的估计量比基于蒙特卡罗的估计量误差更小。本研究计划的主要目标是通过关注其基本采样方案中引起的依赖性来推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建近似的算法的设计和分析方面取得重大进展，即，沿着学习系统的一些特征以进一步将采样引导到重要区域中。最后，当使用低差异序列时，更难以应用将“点”转换为“场景”的技术。这限制了可以通过准蒙特卡罗方法处理的模型的种类。我们的研究将试图解决这些局限性。 该研究计划将涉及至少10名来自各个级别的学生，他们将获得蒙特卡洛和准蒙特卡洛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行业或学术界。