Collaborative Research: Optimal Monte Carlo Estimation via Randomized Multilevel Methods

这项研究项目将研究一套全面的工具，以使高效和无偏的蒙特卡罗方法在广泛的环境中，如：稳态计算和随机微分方程(SDE)。PI扩展了最近引入的称为多水平蒙特卡罗(MLMC)的技术的适用性和能力，该技术迅速流行，并已被证明是非常成功的，特别是在SDE的数值解的背景下。绩效指标战略依赖于两个基本要素。首先，他们抽象出MLMC的主要思想。这一抽象清楚地表明，MLMC可以应用于许多问题设置(超出SDE上下文)，例如：估计马尔可夫随机场的稳态期望，以及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。该随机化步骤不仅允许完全消除迄今为止在多水平方法的每个单独应用中存在的偏差，而且还将允许更容易地优化在经典的多水平应用中出现的参数(通常是用户定义的)。我们对MLMC方法的抽象的核心是在一定的度量下构造一个合适的强(几乎肯定的)逼近序列。构造这种近似所隐含的自由度产生了一个丰富的研究程序，它涉及到现代概率的许多元素，包括随机矩阵、马尔可夫随机场、平均场不动点方程和Lyapunov稳定性。PI将研究一种能够在随机系统模拟的背景下实现高性能计算的方法。PIS方法将大大扩展最近发展的一种称为多水平蒙特卡罗(MLMC)的方法，该方法通常只应用于计算随机微分方程(SDE)的数值解。更广泛地说，这项研究项目解决了位于现代科学计算中心的广泛问题，而不仅仅是在工程和科学的几乎所有建模领域中出现的SDE的重要设置。例如，PI将推广MLMC方法，以准确地执行以树为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理应用中，从成像到分类问题。PIS研究还改进了经典的MLMC技术，优化了其设计，并允许研究例如SDE的稳态分析(即将传统研究领域与新的方法应用相结合)。PIS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。私人投资机构计划就这一建议的主题开发一个新的联合设计的课程，课程材料将在网上提供，以增加项目结果的传播和潜在的适用性。私人投资机构将试图从代表性不足的群体中招聘高素质的人员，并将除了会议和期刊出版物等标准工具外，通过开放获取网站传播研究的科学成果。

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.