Coalescing systems with random initial conditions

具有随机初始条件的聚结系统

• 批准号: 1612674
• 负责人: Robin Pemantle
• 金额: $ 15万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612674&HistoricalAwards=false
• 关键词: Coalescing; systems; random; initial; conditions

This project concerns probability models evolving and coalescing in one dimension. Such models arise in diverse areas of application; the models in this project are taken from materials science, social science, genetics, and distributed computing. Specific examples are: an interface between regions of liquid in a pipe; candidates sequentially dropping out of a primary election; tree-search for approximate evaluation of a Boolean function. Common to all these models is an underlying mathematical structure in which initial conditions are random, after which the evolution is governed by a non-random mechanism. Analyses of evolutions without randomness is often considerably more difficult than analysis of those with randomness. The focus in this project is on creating tools to circumvent this problem. In most cases the PI seeks qualitative answers to questions such as: what does the system look like after a long time, how long does it take to reach this state, and how robust is the description to changes in the initial conditions? The project's broader impacts beyond the potential applications to the aforementioned areas, include the training of graduate students in the mathematical sciences at the University of Pennsylvania who will have opportunities to engage in research on highly non-trivial topics in probability theory. The PI will also devote some effort to improving STEM education at a broader level through the development of innovative techniques for calculus instruction and curriculum development. The PI will investigate the systematic mathematical study of certain common practices which have suffered from a lack of methodological validation, as a further broader impact.The mathematical techniques to be used in this project involve the introduction of time reversals. In a time reversal, randomness of initial conditions becomes randomness of the time-reversed path. Once there is randomness in the evolution, it is possible to use techniques from Markov chains, statistical mechanics, and other applications of probability theory. There are a number of Markov chains that describe time reversals of a given system. Suppose one's concern is to describe the time-zero distribution of a system evolving deterministically from random conditions at time minus infinity. Among possible Markovian time-reversals, the one that describes this is the one with the maximum entropy. The PI then plans to compute this using a variational principle.

这个项目涉及概率模型在一个维度上的演化和合并。这些模型出现在不同的应用领域；这个项目中的模型取自材料科学、社会科学、遗传学和分布式计算。具体的例子有：管道中液体区域之间的界面；候选人相继退出初选；树搜索布尔函数的近似求值。所有这些模型的共同点是一个潜在的数学结构，其中初始条件是随机的，之后的进化是由非随机机制控制的。分析没有随机性的进化往往比分析有随机性的进化要困难得多。这个项目的重点是创建工具来规避这个问题。在大多数情况下，PI寻求以下问题的定性答案：系统在很长一段时间后是什么样子，需要多长时间才能达到这种状态，以及对初始条件变化的描述有多鲁棒？该项目的广泛影响超出了上述领域的潜在应用，包括宾夕法尼亚大学数学科学研究生的培训，他们将有机会从事概率论中高度重要的主题的研究。PI还将致力于通过开发微积分教学和课程开发的创新技术，在更广泛的层面上改善STEM教育。PI将对缺乏方法验证的某些常见做法进行系统的数学研究，以进一步产生更广泛的影响。在这个项目中使用的数学技术涉及时间反转的引入。在时间反转中，初始条件的随机性变成了时间反转路径的随机性。一旦进化中存在随机性，就可以使用马尔可夫链、统计力学和其他概率论应用中的技术。有许多马尔可夫链描述给定系统的时间反转。假设我们关心的是描述一个系统在负无穷时从随机条件确定性演化的时间零分布。在可能的马尔可夫时间反转中，描述这个的是熵最大的一个。然后PI计划使用变分原理来计算它。