Collaborative Research: Optimal Monte Carlo Estimation via Randomized Multilevel Methods

该研究项目将研究一套全面的工具，以在广泛的环境中实现高效和无偏的蒙特卡罗方法，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多级Monte Carlo（MLMC）的技术的适用性和能力，该技术已迅速普及，并已被证明是非常成功的，特别是在SDEs的数值解的背景下。PI战略基于两个基本要素。首先，他们抽象了MLMC的主要思想。这种抽象清楚地表明，MLMC可以应用于许多问题设置（超出了上下文），例如：估计马尔可夫随机场的稳态期望和求解分布不动点方程。其次，PI引入了一个简单但强大的额外随机化步骤。这个随机化步骤不仅可以完全删除偏差，这是目前为止存在于每一个单一的应用程序的多级方法，但它也将允许更容易地优化参数（通常是用户定义的），出现在经典的多级应用程序。在我们的MLMC方法的抽象的核心在于建设一个合适的序列强（几乎肯定）近似下的一些度量。自由是隐含在构建这样的近似产生了丰富的研究计划，触及许多现代概率的元素，包括随机矩阵，马尔可夫随机场，平均场不动点方程和李雅普诺夫稳定性。 PI将研究一种方法，使高性能计算的背景下，随机系统的模拟。PI的方法将大大扩展最近开发的方法，称为多级蒙特卡罗（MLMC），这通常只适用于计算随机微分方程（SDES）的数值解。更一般地说，这个研究项目解决了位于现代科学计算中心的广泛问题，超越了在工程和科学建模的几乎所有领域中出现的重要的SDES设置。例如，PI将推广MLMC方法，以准确地执行树索引的马尔可夫链的所谓稳态模拟。这些计算问题经常出现在统计推断应用中，从成像到分类问题。PI研究还通过优化其设计并允许研究（例如）稳态分析的SDES（即将传统研究领域与新的方法应用相结合）来改进经典MLMC技术。PI将特别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。PI计划就本提案的主题开发一门新的联合设计的课程，课程材料将在网上提供，以增加项目调查结果的传播和潜在适用性。PI将尝试从代表性不足的群体中招募高素质的人员，并通过开放获取网站传播研究的科学成果，除了会议和期刊出版物等标准工具之外。

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.