Collaborative Research: Optimal Monte Carlo Estimation via Randomized Multilevel Methods

本研究项目将研究一套全面的工具，使有效和无偏蒙特卡罗方法在广泛的设置，如：稳态计算和随机微分方程（SDEs）。pi扩展了最近引入的一种称为多层蒙特卡罗（MLMC）的技术的适用性和功能，该技术迅速普及并显示出非常成功，特别是在sde的数值解的背景下。PIs的战略基于两个基本要素。首先，他们对MLMC的主要思想进行了抽象。这种抽象清楚地表明，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。这种随机化步骤不仅允许完全删除偏差，迄今为止，这种偏差存在于多级方法的每个应用中，而且还允许更容易地优化经典多级应用中出现的参数（通常是用户定义的）。我们对MLMC方法抽象的核心在于在某些度量下构造合适的强（几乎确定）近似序列。构建这种近似隐含的自由产生了丰富的研究项目，涉及现代概率的许多要素，包括随机矩阵、马尔可夫随机场、平均场不动点方程和李亚普诺夫稳定性。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。pi方法将大大扩展最近开发的一种称为多层蒙特卡罗（MLMC）的方法，该方法通常仅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，超出了在工程和科学建模的几乎所有领域中出现的SDEs的重要设置。例如，pi将推广MLMC方法，以准确地执行由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。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，SDEs的稳态分析（即将传统研究领域与新的方法应用相结合）。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。项目负责人计划就该提案的主题编写一门新的联合设计的课程，课程材料将在网上提供，以增加项目结果的传播和潜在的适用性。ppi将尝试从代表性不足的群体中招募高质量的人员，并且除了通过会议和期刊出版物等标准工具之外，还将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.