Mathematical Algorithms for Computer Simulation

该项目侧重于开发基于交互的计算机模拟的数学基础，可以表示为确定性，离散时间动态系统的有限状态集。在这里，基于交互的模拟被认为是变量的集合，每个变量都配备了一个更新功能或一组规则，用于根据所有其他变量的状态在下一个时间步计算该变量的状态。因此，通过局部交互给出系统的描述，并通过所有局部更新函数的同时迭代生成全局动态。细胞自动机和布尔网络就是这样的模拟例子。在元胞自动机的框架中，一个被广泛研究的重要问题是如何从局部更新函数的结构来预测这种系统的全局动态特征。通过假设每个变量的状态集是一个有限域，例如具有两个元素的域通常用于元胞自动机，这个问题可以在计算代数的框架内解决。在这个项目中开发的算法将成为有限动力系统符号计算软件包的一部分，在计算机代数系统Macaulay2中实现。基于交互的模拟在分析大型生物、流行病学和社会技术网络（如免疫系统、城市地区传染病的传播或道路交通和无线通信网络）方面变得越来越重要。通常，这种网络是在个体互动的层面上被理解的，但全球信息往往是稀疏的。由于仿真系统的规模和复杂性，此类系统的软件设计非常具有挑战性，仿真输出的分析也是如此。这种模拟的数学基础将为大规模模拟的设计提供工具。它还将有助于系统地回答有关生物和其他系统的问题，例如治疗传染病的最佳方法，或如何在大量人群中控制其传播。

The project focuses on the development of a mathematical foundationfor interaction-based computer simulations that can be represented as deterministic, discrete-time dynamical systems onfinite state sets. Here, an interaction-based simulation isconsidered to be a collection of variables, each equipped with an update function or a set of rules which is used to compute the stateof that variable at the next time step, based on the states of allthe other variables. Thus, the description of the system is givenvia local interactions, and global dynamics is generated throughthe simultaneous iteration of all local update functions. Cellular automataand Boolean networks are examples such simulations. An importantquestion, that has been studied extensively in the framework ofcellular automata is how one can predict global dynamical features ofsuch systems from the structure of the local update functions. By assuming that the state set for each variable is a finite field,such as the field with two elements used typically for cellularautomata,this problem can be addressed within the framework of computationalalgebra. The algorithms developed in this project will become part ofa symbolic computation software package for finite dynamical systems,implemented in the computer algebra system Macaulay2. Interaction-based simulation is becoming increasingly important in theanalysis of large biological, epidemiological, and socio-technical networks, such as the immune system, the spread of infectious diseasesin urban areas, or road traffic and wireless communications networks. Typically, such networks are understood at the level of individualinteractions, but global information tends to be sparse. The softwaredesign of such systems is very challenging, and so is theanalysis of simulation output, due to the size and complexity of thesimulated systems. A mathematical foundation for such simulationswill provide tools for the design of large-scale simulations. Itwill also help to systematically answer questions aboutbiological and other systems, such as optimal ways to treat infectious diseases,or how to contain their spread in large populations.