MULTISCALE STOCHASTIC REACTION-DIFFUSION ALGORITHMS

随机反应扩散过程广泛应用于癌症、免疫系统和发育生物学的建模。这些过程描述了在空间中扩散的化学反应物质的浓度变化。随机反应扩散过程的数值模拟具有挑战性，因为潜在的生化系统通常很大，并且自然涉及化学反应和扩散速度的不同尺度及其在不同位置的数量。多尺度特性会降低数值模拟的速度。此外，随机模拟需要一组重复的模拟来获得化学物质的平均行为。本项目的目标是通过基于尺度的不同数值格式的耦合和应用优化策略来提高准确性和效率，为随机反应扩散过程开发有效的数值算法。模拟大型随机系统的能力将为复杂生化系统的建模和理解提供有效的工具。与此项目相关的研究生课程将被重新设计。本科生和研究生将参与该项目并接受指导。本项目主要研究结合不同数值格式的随机反应扩散过程的多尺度数值算法的开发和分析。马尔可夫链模型被广泛用于模拟具有扩散的化学反应物质，但当系统涉及多尺度现象时，马尔可夫链模型的精确模拟计算成本很高。利用马尔可夫链模型开发和理解随机反应扩散过程的多尺度方法有许多研究，但现有方法的主要缺点是它们不能完全考虑化学物质丰度在时间和空间上的显著变化，从而降低了近似的准确性。在这个项目中，一个感兴趣的空间域将根据化学物质的丰度划分为几个子集，马尔可夫链模型和随机偏微分方程将分别应用于不同的区域。然后，将该方法扩展到包含随时间变化的数值格式之间的移动界面，并使用最优策略找到下一个反应的位置。大型随机反应扩散系统的快速模拟技术将提高实验科学和工程研究的效率。此外，该提案的结果将应用于探索对人类发育或疾病的影响，例如，发现癌症或免疫系统中的关键信号通路，并发现组织和器官如何在胚胎中发育具有不同的功能。

Stochastic reaction-diffusion processes are widely used in the modeling of cancer, immune systems, and developmental biology. These processes describe the concentration changes of the chemically reacting species that diffuse through space. Numerical simulation of stochastic reaction-diffusion processes is challenging since the underlying biochemical systems are large in general and naturally involve various scales in the speed of chemical reaction and diffusion and their quantities in different locations. The multiscale nature can slow down the speed of the numerical simulation. Moreover, stochastic simulation requires a set of repeated performance of the simulations to obtain averaged behavior of chemical species. The goal of this project is to develop efficient numerical algorithms for stochastic reaction-diffusion processes by coupling different numerical schemes based on scales and by applying optimal strategies to increase accuracy and efficiency. The ability to simulate large stochastic systems will provide an efficient tool for modeling and understanding of complex biochemical systems. A graduate-level course in the field related to this project will be redesigned. The undergraduate and graduate students will participate in the project and will receive mentorships. This project focuses on the development and the analysis of multiscale numerical algorithms for stochastic reaction-diffusion processes combining different numerical schemes. Markov chain models are widely used to model chemically reacting species with diffusion, but the exact simulation of Markov chain models for large systems are computationally expensive when the systems involve multiscale phenomena. There are many studies to develop and to understand multiscale methods for stochastic reaction-diffusion processes using Markov chain models, but the major drawback in the existing methodologies is that they do not fully account for significant changes in the abundances of chemical species in time and space, which reduce the accuracy of the approximations. In this project, a spatial domain of interest will be divided into several subsets based on the abundance of chemical species and Markov chain models and stochastic partial differential equations will be respectively applied to the different regions. Then, the method will be extended to incorporate moving interfaces between the numerical schemes, which change in time, and to use optimal strategies to find a location of the next reaction. The fast simulation skills of large stochastic reaction-diffusion systems will increase efficiency in the study of experimental science and engineering. Moreover, the results of the proposal will be applied to explore the impact on human development or disease, for example, to find key signaling pathways in cancer or immune systems and to find how the tissues and organs develop to have different functions in the embryo.