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

这个研究项目涉及一个跨学科的主题，是普遍感兴趣的计算数学家和其他领域的科学家。该项目将导致晶体的光学性质的定量理解和成像地球岩石结构清晰，简单，快速。更重要的是，这些新方法将在计算数学方面取得重要进展，从而在材料科学和地震学领域带来实际利益。这项工作的动机是最近的工作冻结高斯近似（FGA）和其独特的解释高频波，建议许多新的数值技术嵌入。国家的最先进的数值方法将被开发和用于各种应用。这项工作将对物理科学和工业领域的一系列重大问题产生影响。具体而言，该项目调查了以下主题：1。从理论上对FGA方法进行更深入的研究。这包括分析FGA的渐近精度，并将其与定制有限点/单元法（TFPM/TFCM）相结合。 2.通过计算具有晶格势的薛定谔方程，研究晶体中电子的动力学。 3.基于FGA及其与TFPM/TFCM的结合，计算地震波在非均质或非横向介质中的传播。 4.调查已开发的数值方法的实际性能，例如，海啸和地震模型。

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.