Mathematical Algorithms for Computer Simulation

该项目的重点是为基于相互作用的计算机模拟建立数学基础，这些计算机模拟可以表示为有限状态集上的确定性离散时间动态系统。 在这里，基于交互的仿真被认为是一个变量的集合，每个变量都配备了一个更新函数或一组规则，用于根据所有其他变量的状态计算该变量在下一个时间步的状态。 因此，系统的描述是通过局部相互作用给出的，而全局动力学是通过所有局部更新函数的同时迭代产生的。 元胞自动机和布尔网络就是这种模拟的例子. 在元胞自动机的框架下，一个重要的研究问题是如何从局部更新函数的结构中预测系统的全局动力学特性。 通过假设每个变量的状态集是一个有限域，例如通常用于细胞自动机的两个元素的域，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的符号计算软件包的一部分，在计算机代数系统Macaulay2中实现。 基于交互的仿真在大型生物、流行病学和社会技术网络的分析中变得越来越重要，例如免疫系统、传染病在城市地区的传播或道路交通和无线通信网络。 通常情况下，这样的网络是在个人互动的水平上理解的，但全球信息往往是稀疏的。 由于模拟系统的规模和复杂性，这种系统的软件设计是非常具有挑战性的，模拟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。 它还有助于系统地回答有关生物系统和其他系统的问题，例如治疗传染病的最佳方法，或如何控制它们在大量人群中的传播。

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.