CAREER: Sequential Monte Carlo Methods for High Dimensional Systems

发展能够令人满意地描述和分析高维系统的模型和方法的进展对于许多不同的学科，包括生物学、气象学、经济学、社会科学和工程学，都是非常有价值的。这些系统具有非线性且难以理解的特点。受简单局部规则支配的计算方法具有提供有洞察力的解释的潜力，并为定量和定性描述和理解复杂系统铺平道路。本项目致力于开发此类方法，特别是开发基于顺序蒙特卡罗的高维系统信号处理的协同研究和教育计划。本研究旨在为高维系统的顺序蒙特卡罗方法奠定基础。在文献中，有声称粒子过滤器不能用于复杂系统，因为它们的随机测量退化为单个粒子。虽然这对于这些过滤器的标准实现是正确的，但对于替代方法则不适用。提出了一种基于分而治之原则的新方法。特别是，通过设置一个相互连接的过滤器网络，每个过滤器都在较低维度的空间工作，从而避免了传统粒子过滤的崩溃。研究任务包括发展方法的理论基础，建立从业人员使用该方法的指导方针，分析其准确性、稳定性和可扩展性，以及在广泛的复杂系统上进行验证。提出的方法是原创性的，并为粒子过滤这一最大的公开问题提供了解决方案。

CAREER: Sequential Monte Carlo Methods for High Dimensional SystemsAbstractAdvances in the development of models and methods that can satisfactorily describe and analyze high dimensional systems are extremely valuable for many different disciplines including biology, meteorology, economics,social sciences, and engineering. These systems are characterized by nonlinearities and are difficult to understand.Computational methods governed by simple local rules have the potential of providing insightful interpretationsand of paving the way towards quantitative and qualitative descriptions and understanding of complex systems.This project is focused on development of such methods and in particular on the development of a synergistic research and educational program in sequential Monte Carlo-based signal processing for high dimensional systems.This research aims at laying the foundations of a sequential Monte Carlo methodology for high dimensional systems. In the literature, there are claims stating that particle filters cannot be used for complex systems because their random measures degenerate to single particles. While this is true for standard implementation of these filters, it does not hold true for alternative approaches. A new methodology based on the principle of divide and conquer is developed. In particular, the collapse of traditional particle filltering is avoided by setting an interconnected network of filters, each of them working on lower dimensional spaces. Research tasks include development of the theoretical grounds of the methods, establishment of guidelines for its use by practitioners,analysis of its accuracy, stability and scalability, and validation on a wide range of complex systems. The proposed methods are original and provide solutions to arguably the biggest open problem of particle filtering.