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，是计算数学家和其他领域的科学家普遍感兴趣的。这个项目将导致对晶体光学特性的定量理解，以及清晰、简单、快速地成像地球岩石结构。更重要的是，这些新方法将为计算数学提供重要的进步，从而在材料科学和地震学领域带来实际效益。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，这表明了许多新的数值技术。最先进的数值方法将被开发并用于各种应用。这项工作将对物理科学和工业中广泛的重大问题产生影响。具体而言，该项目调查了以下主题：1。对FGA方法进行了更深入的理论研究。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. 通过拉格朗日和欧拉两种方法计算晶格势Schrödinger方程来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. 考察已开发的数值方法在海啸和地震模型等方面的实际性能。

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.