Collaborative Research: Optimal Monte Carlo Estimation via Randomized Multilevel Methods

协作研究：通过随机多级方法进行最优蒙特卡罗估计

• 批准号: 1320550
• 负责人: Jose Blanchet
• 金额: $ 21万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1320550&HistoricalAwards=false
• 关键词: Collaborative; Research; Optimal; Monte; Carlo; Collaborative; Research; Optimal; Monte; Carlo

This research project will investigate a comprehensive set of tools to enable efficient and unbiased Monte Carlo methods in a wide range of settings such as: steady-state computations and stochastic differential equations (SDEs). The PIs extend the applicability and power of a recently introduced technique called multilevel Monte Carlo (MLMC), which has rapidly grown in popularity and has shown to be highly successful, particularly in the context of numerical solutions to SDEs. The PIs strategy rests on two basic ingredients. First, they abstract the main ideas of MLMC. This abstraction makes it clear that MLMC can be applied to many problem settings (beyond the SDE context), for example in problems such as: estimating steady-state expectations of Markov random fields, and solving distributional fixed point equations. Second, the PIs introduce a simple, yet powerful, extra randomization step. This randomization step will permit to not only completely delete the bias, which so far is present in every single application of the multilevel method, but it will also permit to more easily optimize parameters (often user-defined) that arise in classical multilevel applications. At the core of our abstraction of the MLMC method lies the construction of a suitable sequence of strong (almost sure) approximations under some metric. The freedom that is implicit in constructing such approximations yields a rich research program that touches upon many of the elements of modern probability, including random matrices, Markov random fields, mean field fixed point equations and Lyapunov stability. The PIs will investigate a methodology that enables high-performance computing in the context of simulation of stochastic systems. The PIs methodology will substantially extend a recently developed approach, called Multilevel Monte Carlo (MLMC), which has typically been applied only to compute numerical solutions of stochastic differential equations (SDEs). More generally, this research project addresses a wide range of problems that lie at the center of modern scientific computing, beyond the important setting of SDEs which arise in virtually all areas of modeling in engineering and science. For example, the PIs will generalize the MLMC approach to accurately perform so-called steady-state simulation for Markov chains indexed by trees. These computational problems arise very often in statistical inference applications, ranging from imaging to classification problems. The PIs research also improves upon the classical MLMC technique by optimizing its design and allowing the study of, for example, steady-state analysis of SDEs (i.e. combining traditional areas of study with new methodological applications). The PIs will in particular apply these optimized computational techniques to solve problems in service and manufacturing engineering. The PIs plan to develop a new jointly designed course, on the topic of this proposal, and the course material will be made available online to increase the dissemination and the potential applicability of the project's findings. The PIs will attempt to recruit high-quality personnel from under-represented groups and will disseminate the scientific output of the research via open access sites, in addition to the standard vehicles such as conferences and journal publications.

该研究项目将研究一组全面的工具，以在各种环境中实现高效且无偏的蒙特卡洛方法，例如：稳态计算和随机微分方程（SDES）。 PI扩展了最近引入的称为Multevel Monte Carlo（MLMC）的技术的适用性和功能，该技术已迅速增长，并且已证明非常成功，尤其是在SDE的数值解决方案的背景下。 PIS策略基于两种基本成分。首先，他们抽象了MLMC的主要思想。该抽象清楚地表明，MLMC可以应用于许多问题设置（超出SDE上下文之外），例如在诸如：估计马尔可夫随机字段的稳态期望以及求解分布固定点方程的问题中。其次，PI引入了一个简单但功能强大，额外的随机化步骤。此随机分步不仅允许完全删除偏差，到目前为止，在多级方法的每个应用中都存在，而且还可以更轻松地优化经典多级应用程序中出现的参数（通常是用户定义）。我们抽象的MLMC方法的核心是在某些度量标准下构造了合适的强（几乎确定）近似序列。构建这种近似值的隐含自由产生了丰富的研究计划，该计划涉及现代概率的许多要素，包括随机矩阵，马尔可夫随机场，平均场固定点方程和Lyapunov稳定性。 PI将研究一种在随机系统模拟的背景下实现高性能计算的方法。 PIS方法将基本上扩展一种最近开发的方法，称为多级蒙特卡洛（MLMC），该方法通常仅应用于计算随机微分方程（SDE）的数值解。更一般而言，该研究项目解决了现代科学计算中心的广泛问题，除了在工程和科学中几乎所有建模领域中出现的SDE的重要环境之外。例如，PI将概括MLMC方法，以准确执行由树木索引的马尔可夫链进行所谓的稳态模拟。这些计算问题经常出现在统计推理应用中，从成像到分类问题。 PIS研究还通过优化其设计并允许研究SDES的稳态分析（即将传统研究领域与新的方法论应用结合）来改善经典MLMC技术。 PI特别将应用这些优化的计算技术来解决服务和制造工程中的问题。 PIS计划开发一门新的联合设计课程，该课程，该课程，并在线提供该课程材料，以增加项目调查结果的传播和潜在的适用性。 PI将尝试从代表性不足的群体中招募高质量的人员，并将通过开放访问站点（例如会议和期刊出版物）通过开放访问站点来传播研究的科学输出。