Mathematical and Computational Studies of Interfaces and Defects

本项目将专注于数值方法的开发和分析，以探测与各种界面和缺陷问题相关的复杂能量景观，例如发生相变的材料中的微观结构，几何阻挫配置中的量子化涡旋等，以及相关的随机动力学和流体动力学以及它们的数值近似也将被研究。2他也将研究鲁棒性和自适应算法，这些算法在基础结构和重要特征的统计分析中是有用的。这些研究问题，一方面，通过与其他科学家的合作，由实际应用驱动，另一方面，也激发了从几何和拓扑到数值和随机分析的数学学科的新研究。将开展的研究是跨学科性质的，包括计算数学、物理学、信息、材料和生物科学等学科。该研究涉及许多数学和数值挑战，例如理解非线性偏微分方程和随机动力学解的族和路径的集体行为，以及探索模拟结果中隐藏的结构和统计，算法开发和数值模拟的研究有助于提高科学计算的能力和预测能力，它具有潜在的重大科学，社会和经济影响，是国际上最优先的研究项目之一。同时，界面和缺陷也普遍存在于自然界中，它们在物理和生物系统的许多方面起着基础性的作用。更好地理解界面和缺陷的数学和计算，特别是在随机环境中，将丰富科学知识库，这反过来又有助于物理学家，材料科学家和工程师发现具有理想特性的新材料，并开发新的科学设备和商业仪器。该项目还将为未来一代的劳动力和研究人员提供宝贵的跨学科研究机会。强调TEAMS（实验，分析，建模和仿真培训）精神，年轻学生可以更好地准备在职业生涯中进行跨学科研究。

This project will focus on the development and analysis of numerical methods to probe the complex energy landscapes related to various interface and defect problems, such as microstructures in materials undergoing phase transitions, quantized vortices in geometrically frustrated configurations, and so on. The associated stochastic dynamics and hydrodynamics, as well as their numerical approximations will also be investigated.He will also study robust and adaptive algorithms that are useful in the statistical analysis of the underlying structures and important features. These research issues, on one hand, are driven by practical applications through collaborations with other scientists, and on the other hand, also motivate new studies of mathematical subjects ranging from geometry and topology to numerical and stochastic analysis. The research to be carried out is of interdisciplinary nature, encompassing subjects like computational mathematics, physics, information, materials and biological sciences. There are many mathematical and numerical challenges involved in the research such as the understanding of the collective behaviors of families and paths of solutions of nonlinear PDEs and stochastic dynamics, and the exploration of the hidden structures and statistics in the simulated results.The research on the algorithmic development and numerical simulations can help enhancing the capability as well as the predictive power of scientific computations, which has potentially significant scientific, social and economic impact and is one of the top research priorities internationally. Meanwhile, interfaces and defects are also ubiquitous in nature which play fundamental roles in many aspects of physical and biological systems. A better mathematical and computational understanding of interfaces and defects, especially in a stochastic setting, will enrich the scientific knowledge base, which in turn may aid the efforts by physicists, materials scientists and engineers in discovering new materials with desirable properties and in developing new scientific devices and commercial instruments. This project will also provide valuable interdisciplinary research opportunities for the future generation of workforce and researchers. With an emphasis on the TEAMS (Training in Experiments, Analysis, Modeling and Simulations) spirit, young students can be better prepared to conduct interdisciplinary research in their career.