Model Reduction Methods for Extended Quantum Systems: Analysis and Applications

材料科学和化学的最新发展涉及底层系统的量子特性。在研究量子特性时，通常认为所研究的系统是孤立的，是独立发展的，但实际上，量子系统总是与其周围环境相耦合的。直接模拟这样一个系统会导致巨大的计算成本。对于简化模型和高效仿真算法的需求非常强烈，这样人们就可以在研究量子系统（如半导体和纳米电子设备）的同时保持现实的复杂性。本项目通过开发新的模型和数据同化方法，解决了模拟具有量子自由度的扩展系统的基本挑战。该项目将为本科生提供研究培训机会。该项目将侧重于基于伽辽金投影的模型缩减框架。该项目旨在将量子系统与其周围环境分离开来，并以更少的变量描述动力学。该项目将提供模型简化方法的严格误差分析，并考虑一种非参数数据驱动的方法来从数据观测中估计系统属性。作为一个跨学科项目，该项目将整合不同学科的技术，包括数学、物理、化学和量子计算，并产生更准确、更可靠的模型，可以实现，以加速不同学科的计算繁重问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Much recent development in materials science and chemistry involves quantum properties of the underlying systems. Quantum properties are often studied as if the system of interest is isolated and evolves by itself, but in practice, a quantum system is always coupled with its surrounding environment. Direct simulations of such a system can result in overwhelming computational costs. There is a strong demand for simplified models and efficient simulation algorithms so that people can study quantum systems - such as semiconductors and nanoelectronic devices - while retaining realistic complexity. This project tackles the fundamental challenge of simulating an extended system with quantum degrees of freedom by developing new models and data-assimilation methods. The project will provide research training opportunities for undergraduates.This project will focus on a Galerkin-projection-based model-reduction framework. This project will aim to separate the quantum system from its surrounding environment and describe the dynamics with fewer variables. This project will provide rigorous error analysis of the model-reduction method and consider a nonparametric data-driven approach to estimate the system properties from data observations. As an interdisciplinary project, this project will bridge techniques from different disciplines, including mathematics, physics, chemistry, and quantum computing, and give rise to more accurate and reliable models that can be implemented to accelerate computationally heavy problems of different disciplines.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.