Stochastic Systems: Theory and Models

随机系统：理论和模型

• 批准号: RGPIN-2015-05909
• 负责人: Madras, Neal
• 金额: $ 1.46万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=658403
• 关键词: Stochastic; Systems; Theory; Models

My research program is focused on probability theory and its applications.  I will work on three different topics, ranging from the pure to the applied.******(1) How reliable are simulation results?  I shall study the efficiency of a class of Monte Carlo simulation algorithms known as Markov chain Monte Carlo methods.  These methods have been used to tackle computationally challenging problems in statistical analysis (for example, in a digital picture that has been blurred by random noise, what is the likely true image?) and other fields.  The user must decide how long to run the algorithm so that the results are not biased by the initial conditions.  This decision is not always clear, and it is easy to get misled by the output.  The only guarantees come with theoretical analysis. To that end, I aim to prove upper bounds on the time required to make the effect of the initial bias arbitrarily small (formally, I bound the rates of convergence of the Markov chains to equilibrium) in some simplified models, which I hope can serve as guidelines for the more complex situations that arise in practice.*******(2) Next, here is problem in pure combinatorics with a probabilistic viewpoint.  A permutation of size N is an arrangement of the numbers from 1 to N (e.g. 86425713 is a permutation of size 8).  We say that a permutation "contains the pattern 4231" if you can find four numbers in the permutation that occur in the same relative order as 4231 (i.e. the first one is largest, the second one is second smallest, the third one is third smallest, and the fourth one is smallest).   E.g., 86425713 contains the pattern 4231 because the numbers 8573 appear in this order.  Consider the set of permutations of size N that do not contain the pattern 4231.  When N is large, mathematicians have tried (with limited success) to determine approximately how big this set is.  I have observed that randomly chosen elements of this set are likely to have some striking structural properties, and a goal of my research is to understand why this happens.***(3) Finally, I will work on aspects of mathematical modelling of the immune system using models that incorporate randomness, to account for uncertainties in outcomes. One question concerns mutations, which are intrinsically random events. When fighting a pathogen such as a flu virus, our immune system uses T cells, which are designed to hunt down certain identifiable signs of the virus. But the pathogen's offspring can escape a T cell if they have a mutation in the right place. I would like to estimate the probability that the T cells can destroy all of the pathogen before some offspring accumulates enough mutations to be able to evade all of the T cells. A second question asks for the probability that our "innate" immune system can control an infection of West Nile virus from a mosquito bite rapidly enough that the T cells are not needed.**

我的研究计划的重点是概率理论及其应用。我将研究三个不同的主题，从纯度到所应用的。******（1）仿真结果的可靠性如何？我将研究一类称为马尔可夫链蒙特卡洛方法的蒙特卡洛模拟算法的效率。这些方法已被用来解决统计分析中的计算挑战问题（例如，在数字图片中被随机噪声模糊，可能的真实图像是什么？）和其他字段。用户必须决定运行多长时间运行算法，以使结果不会受到初始条件的偏见。这个决定并不总是很清楚，很容易被输出误导。唯一的保证包括理论分析。为此，我的目标是在某些简化的模型中，使最初偏见的效果任意小（正式地将马尔可夫链的收敛速率）绑定到最初的偏见所需的时间上，我希望这可以作为在实践中出现的更为复杂的情况。大小为n的排列是从1到n的数字的排列（例如86425713是尺寸8的置换）。我们说，如果您可以在与4231相同的相对顺序中找到四个数字，则“包含图案4231”（即第一个数字最大，第二个数字是第二小，第三个是第三，第三个最小，第四个是最小的）。例如，86425713包含模式4231，因为数字8573以此顺序出现。考虑不包含模式4231的尺寸N的排列集。当N很大时，数学家已经尝试（成功）以确定此组的大约大约。我已经观察到，该集合的随机元素可能具有某种打击结构性，而我的研究的目标是了解为什么会发生这种情况。一个问题涉及突变，这是本质上随机的事件。当与流感病毒之类的病原体作斗争时，我们的免疫系统使用T细胞，这些T细胞旨在追捕某些可识别的病毒迹象。但是，如果病原体的后代在正确的位置有突变，则可以逃脱T细胞。我想估计T细胞可以在某些后代积累足够的突变以能够逃避所有T细胞之前，T细胞可以破坏所有病原体的概率。第二个问题询问我们的“先天”免疫系统可以迅速从蚊子叮咬中控制西尼罗河病毒的感染的可能性，以至于不需要T细胞。**