Algorithms and Computation for Rare Events in Complex Systems

该项目涉及与实际感兴趣的复杂能源景观的平衡、亚稳态和过渡态以及最小能量路径的模拟和分析有关的数学和计算问题，以及相关的随机动力学。即将开展的研究密切受到联邦战略利益领域的应用的推动：开发有效的算法和代码是高性能计算的关键部分，而将要开发的数值方法和软件工具可能对有效的计算材料和药物设计有潜在的帮助。首席研究员将开展跨学科研究，涵盖计算数学、物理、信息、材料和生物科学等学科。他将专注于新算法的开发和分析，这些算法有可能显着改善罕见事件建模和模拟的通常做法。他将考虑一些具体和重要的应用，包括几何限制和受阻构型或可变形几何中相互作用的粒子和界面的系统，这些系统出现在物理、化学和生物学的许多领域(如纳米团簇的形成、双分子构象、囊泡介导的相互作用，以及固态转变中的临界成核)。他将尝试将一些由实践者开发的算法与现有软件代码中实现的算法联系起来，例如用于第一性原理计算和计算化学的算法。研究项目中涉及的大多数问题要么与无限维空间(如确定性或随机性偏微分方程)有关，要么与高维有限维空间(微分方程或涉及大量粒子的粒子系统的离散化)有关，这导致了许多计算挑战。将研究各种数学和数值问题，从有效的局部鞍点搜索及其稳健的数值实现，到对相关动力学和罕见事件的严格分析和有效的多尺度模拟。预计项目期间取得的进展将对对罕见事件研究感兴趣的社区产生广泛影响。该项目还将有助于教育和培训，因为它将在跨学科环境中为学生提供宝贵的培训场地和研究经验。将致力于促进各级学生的积极参与，并将研究成果纳入教学和培训。

The project is concerned with mathematical and computational issues related to the simulation and analysis of equilibria, metastable and transition states and minimum energy paths for complex energy landscapes of practical interests, and associated stochastic dynamics. The research to be carried out is closely motivated by applications in a number of areas of federal strategic interests: the development of effective algorithms and codes is a crucial part of high-performance computing, and numerical methods and software tools to be developed may be potentially useful for effective computational materials and drug design. The principal investigator will carry out interdisciplinary research that encompassing subjects like computational mathematics, physics, information, materials and biological sciences. He will focus on new algorithmic development and analysis which have the potential to significantly improve the usual practice on the modeling and simulation of rare events. He will consider some specific and important applications, including systems of interacting particles and interfaces in geometrically confined and frustrated configurations or deformable geometry which arise in many areas of physics, chemistry and biology (such as formation of nano-clusters, bimolecular conformation, vesicle mediated interactions, and critical nucleation in solid state transformations). He will attempt to draw strong connections with some of the algorithms developed by practitioners implemented in existing software codes such as those for first principle calculation and computational chemistry. Most of the problems involved in the research project are associated with either infinite dimensional spaces such as deterministic or stochastic partial differential equations or finite dimensional spaces with high dimensions (discretization of differential equations or particle systems involving a large number of particles), which lead to many computational challenges. Various mathematical and numerical issues will be studied, ranging from efficient local saddle point search and its robust numerical implementation to rigorous analysis and effective multiscale simulations of relevant dynamics and rare events. It is expected that the progress made during the project will have a broad impact on the community interested in the study of rare events. The project will also contribute to education and training as it will provide students valuable training ground and research experience in an interdisciplinary environment. Much effort will be devoted to promoting active engagement of student participation at all levels and integrating research findings into teaching and training.