Domain decomposition methods for electronic structure calculations

电子结构计算的域分解方法

• 批准号: 411724963
• 负责人: Professor Dr. Benjamin Stamm
• 金额: --
• 依托单位: Laboratoire Jacques-Louis Lions
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/411724963?language=en
• 关键词: Domain; decomposition; methods; electronic; structure

Electronic structure calculations are paramount in theoretical chemistry, physics, and material science. Indeed, among the top ten most cited scientific articles, two papers are related to this topic. Despite the fact that this field deals with the discretization of eigenvalue problems, which is a well-established subject in numerical analysis, the expertise of applied mathematics is little involved.Within the proposed project, the aim is to develop novel domain decomposition (DD) algorithms for eigenvalue problems which arise in electronic structure calculation like the Kohn-Sham DFT (Density Functional Theory) equations. The approach within this project is based on the idea that domain decomposition for eigenvalue problems is not fundamentally different than for source problems. Despite this fact, DD-methods for eigenvalue problems are less popular than for source problems. Additionally, recent work in the context of implicit solvation models and the analysis of those methods show that the DD-method is scalable for domains of an increasing number of fixed-size sub-domains, like e.g. for chain-like molecules or proteins, even without a so-called coarse-correction.The problems to be embraced within this project are manifold and contain: i) non-linear eigenvalue problems, ii) a large number of eigenvalues to be determined, iii) eigenvalue problems on unbounded domains and iv) dealing with potentials that contain Coulomb-like singularities.The project is a first step within a broader long-term plan to derive efficient local basis functions based on local reduced order modeling as an alternative of the widely-used but empirical contracted Gaussian basis functions. In fact, the domain decomposition strategy allows to localise the equations and opens the door to the application of reduced order modeling with certified a posteriori error estimates in a second step.

电子结构计算在理论化学、物理学和材料科学中至关重要。事实上，在被引用次数最多的十篇科学文章中，有两篇论文与该主题相关。尽管该领域处理特征值问题的离散化，这是数值分析中一个成熟的学科，但很少涉及应用数学的专业知识。在拟议的项目中，目标是开发新颖的域分解（DD）电子结构计算中出现的特征值问题的算法，例如 Kohn-Sham DFT（密度泛函理论）方程。该项目中的方法基于这样的思想：特征值问题的域分解与源问题没有根本上的不同。尽管如此，用于特征值问题的 DD 方法不如用于源问题的流行。此外，最近在隐式溶剂化模型背景下的工作以及对这些方法的分析表明，DD 方法对于越来越多的固定大小子域的域是可扩展的，例如对于链状分子或蛋白质，即使没有所谓的粗校正。该项目要解决的问题是多方面的，包括：i) 非线性特征值问题，ii) 需要确定的大量特征值， iii) 无界域上的特征值问题和 iv) 处理包含类库仑奇点的势。该项目是更广泛的长期计划的第一步，旨在基于局部降阶建模导出有效的局部基函数：广泛使用但经验收缩的高斯基函数的替代方案。事实上，域分解策略允许对方程进行本地化，并为应用降阶建模打开了大门，并在第二步中进行了经过认证的后验误差估计。