Mathematical Sciences: Stochastic Matrix Analysis

数学科学：随机矩阵分析

• 批准号: 9403224
• 负责人: Carl Meyer
• 金额: $ 7.84万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-15 至 1997-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403224&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Stochastic; Matrix; Analysis

9403224 Meyer This research focuses on the analysis of stochastic matrices and associated Markov chains concepts. Both theoretical and computational issues are addressed. The theoretical objective is to make a significant impact on the traditional perturbation theory for stochastic matrices and Markov chains. One goal is to completely describe the nature and characteristic features of sensitive Markov chains and to build a clear and concise theory of perturbations in the eigensystems of stochastic matrices. This perturbation theory will be used to better understand the convergence and stability properties of a variety of multilevel algorithms for computing stationary probabilities. The computational aspect is concerned with the numerical determination of stationary probabilities associated with large-scale irreducible Markov chains with special emphasis on those which are nearly uncoupled (systems comprised of a collection of loosely coupled subsystems). The focus is on the development, implementation, and analysis of aggregation/disaggregation algorithms. Both iterative and exact aggregation/disaggregation methods are to be considered along with some hybrid A/D techniques. This research focuses on the analysis of stochastic matrices and associated Markov chains concepts. Markov chain techniques constitute a unifying theme and are the basis for an extremely wide variety of mathematical models which are used to describe, predict, and analyze the dynamics of large evolutionary systems. Markov chain models are fundamental mathematical tools in areas as diverse as engineering, economics, physical science, and social science. In particular, analyzing and computing stationary probabilities associated with large scale Markov chains is a primary concern in problems involving queueing models and networks, telecommunications, computer performance evaluation, economic modeling and forecasting, manufacturing systems modeling, and more generally in applications where stochastic models are used to understand the behavior of systems that evolve with time. This project emphasizes both computational and theoretical issues, and special attention is devoted to the analysis of nearly uncoupled problems (systems comprised of a collection of loosely coupled subsystems). The following specific research topics are to be investigated: (1) As they evolve with time, many physical systems eventually settle down into some sort of steady state. If the physical system is modeled by utilizing the theory of Markov chains, then the steady state nature of the system is characterized by a set of probabilities called "stationary probabilities." Consequently, analyzing the behavior of the stationary probabilities is a fundamental issue. The theoretical component of this research involves studying the stability properties of stationary probabilities. The results of this research should clarify the understanding of the mechanisms which contribute to either the stability (or instability) of the underlying physical system being examined. (2) The computational facet of this project is to develop new algorithms for computing stationary probabilities and to develop new methods by means of which such algorithms can be analyzed. In practical applications such as those mentioned above, it is usually the case that the physical system under question involves an extremely large number of components, but these components can often be grouped into clusters for which there is strong interaction within any given cluster but weaker interaction among the clusters themselves (e.g., consider the economy of the United States). This research effort devotes special attention to analyzing and computing the stability and steady state nature of such systems. Mathematically, this involves computing and analyzing the stationary probabilities of such systems. To this end, numerical algorithms known as aggregation/disaggregation techniques will be designed to specifical ly to exploit the special features of these types of nearly uncoupled systems.

小行星9403224 本研究的重点是分析随机矩阵和相关的马尔可夫链的概念。理论和计算问题都得到解决。 理论目标是对随机矩阵和马尔可夫链的传统扰动理论产生重大影响。一个目标是完整地描述敏感马尔可夫链的性质和特征，并建立一个清晰简洁的随机矩阵特征系统的扰动理论。这种扰动理论将被用来更好地理解的收敛性和稳定性的各种多级算法计算平稳概率。 计算方面关注的是与大规模不可约马尔可夫链，特别强调那些几乎解耦（系统组成的松散耦合的子系统的集合）的固定概率的数值测定。 重点是聚合/解聚算法的开发，实现和分析。 迭代和精确的聚合/解聚方法都将与一些混合A/D技术一起被考虑沿着。 本研究的重点是分析随机矩阵和相关的马尔可夫链的概念。马尔可夫链技术构成了一个统一的主题，是用于描述，预测和分析大型进化系统动态的各种数学模型的基础。 马尔可夫链模型是工程、经济、物理科学和社会科学等领域的基本数学工具。特别地，分析和计算与大规模马尔可夫链相关联的平稳概率是涉及建模模型和网络、电信、计算机性能评估、经济建模和预测、制造系统建模的问题中的主要关注点，并且更一般地，在随机模型用于理解随时间演化的系统的行为的应用中。该项目强调计算和理论问题，并特别关注几乎非耦合问题（由松散耦合的子系统组成的系统）的分析。具体研究内容如下：（1）随着时间的推移，许多物理系统最终会进入某种稳定状态。 如果利用马尔可夫链理论对物理系统进行建模，则系统的稳态性质由一组称为"平稳概率"的概率来表征。“因此，分析平稳概率的行为是一个基本问题。本研究的理论部分涉及研究平稳概率的稳定性。这项研究的结果应该澄清的机制，有助于被检查的基础物理系统的稳定性（或不稳定性）的理解。 (2)该项目的计算方面是开发新的算法计算平稳概率，并开发新的方法，通过这种算法可以分析。在诸如上述那些的实际应用中，通常的情况是，所讨论的物理系统涉及极其大量的组件，但是这些组件通常可以被分组到集群中，对于这些集群，在任何给定集群内存在强交互，但是在集群本身之间存在较弱交互（例如，考虑美国的经济）。 这项研究工作致力于特别注意分析和计算的稳定性和稳态性质的系统。 在数学上，这涉及计算和分析此类系统的平稳概率。为此，被称为聚合/解聚技术的数值算法将被设计为专门利用这些类型的近解耦系统的特殊功能。