Kinetic Approaches for Multi-Scale Problems in Quantum Chemistry and Seismology

这项研究项目涉及一个计算数学家和其他领域的科学家普遍感兴趣的跨学科课题。该项目将导致对晶体光学性质的定量了解，并清晰、简单和快速地成像地球岩石结构。更重要的是，这些新方法将在计算数学方面取得重要进展，在材料科学和地震学领域产生实际效益。这项工作的动机是最近关于冻结高斯近似(FGA)的工作及其对高频波的独特解释，这表明嵌入了许多新的数值技术。最先进的数值方法将被开发和使用在各种应用中。这项工作将对物理科学和工业中非常感兴趣的广泛问题产生影响。具体而言，本课题主要研究了以下几个方面的内容：1.从理论上对FGA方法进行了更深入的研究。这包括分析FGA的渐近精度，并将其与定制的有限点/网格方法(TFPM/TFCM)相结合。2.用拉格朗日方法和欧拉方法计算晶格势薛定谔方程，研究电子在晶体中的动力学。3.基于FGA及其与TFPM/TFCM相结合的方法，计算地震波在非均匀或非横向介质中的传播。4.调查已开发的数值方法在海啸和地震模型上的实际表现。

This research project concerns an interdisciplinary topic that is of general interest to both computational mathematicians and scientists from other areas. This project will lead to quantitative understanding of the optical properties of crystals and in imaging Earth rock structures clearly, simply, and quickly. More importantly, these new methods will provide important advances in computational mathematics, leading to practical benefit in the fields of materials science and seismology. This work is motivated by recent work on Frozen Gaussian Approximation (FGA) and its unique interpretation of high-frequency waves that suggests many new numerical techniques embedded. State-of-the-art numerical methods will be developed and used in various applications. The work will have impact in a wide range of problems of significant interest in physical sciences and industry. Specifically, the project investigates the following topics: 1. To theoretically study the FGA method in more depth. This includes analyzing the asymptotic accuracy of FGA, and integrating it with the Tailored Finite Point/Cell Method (TFPM/TFCM). 2. To study the dynamics of electrons in crystals by computing the Schrödinger equation with a lattice potential in both the Lagrangian and Eulerian approaches. 3. To compute seismic wave propagation in heterogeneous or non-lateral media based on FGA and its combination with TFPM/TFCM. 4. To survey the practical performance of developed numerical methods on, e.g., tsunami and earthquake models.