Model Reduction Methods for Extended Quantum Systems: Analysis and Applications

材料科学和化学的许多最新发展涉及基础系统的量子特性。量子性质的研究通常是把感兴趣的系统看作是孤立的，并且是自己演化的，但实际上，量子系统总是与周围环境耦合的。直接模拟这样的系统可能会导致巨大的计算成本。人们对简化模型和高效仿真算法的需求非常强烈，这样人们就可以研究量子系统--如半导体和纳米电子器件--同时保持现实的复杂性。该项目通过开发新的模型和数据同化方法来解决模拟具有量子自由度的扩展系统的根本挑战。本计划将为本科生提供研究训练的机会。本计划将集中于一个基于伽辽金投影的模型简化框架。该项目旨在将量子系统与周围环境分离，并以更少的变量描述动力学。这个项目将提供严格的误差分析的模型简化方法，并考虑一个非参数数据驱动的方法来估计系统的性能，从数据观测。作为一个跨学科的项目，该项目将连接不同学科的技术，包括数学，物理，化学和量子计算，该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响进行评估，被认为值得支持审查标准。

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.