Model Reduction Methods for Extended Quantum Systems: Analysis and Applications

材料科学和化学的最新发展涉及到底层系统的量子性质。量子性质的研究通常被认为是感兴趣的系统是孤立的，并自行演化，但在实践中，量子系统总是与其周围环境耦合。对这样一个系统的直接模拟可能会导致巨大的计算成本。人们迫切需要简化的模型和高效的模拟算法，以便人们在研究量子系统--如半导体和纳米电子器件--的同时，保持现实的复杂性。这个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。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Galerkin投影的模型简化框架。这个项目的目标是将量子系统与其周围环境分开，并用更少的变量来描述动力学。该项目将对模型简化方法进行严格的误差分析，并考虑使用非参数数据驱动的方法来从数据观测中估计系统特性。作为一个跨学科的项目，这个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，并产生更准确和可靠的模型，这些模型可以用来加速不同学科的计算繁重的问题。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.