Domain decomposition methods for electronic structure calculations

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，在前十名被引用最多的科学文章中，有两篇论文与这个主题有关。尽管这一领域涉及本征值问题的离散化，这是数值分析中一个成熟的主题，但应用数学的专业知识很少涉及。在拟议的项目中，目的是开发新的区域分解（DD）算法，用于电子结构计算中出现的本征值问题，如Kohn-Sham DFT（密度泛函理论）方程。这个项目的方法是基于这样的想法，即特征值问题的区域分解与源问题没有根本的不同。尽管如此，DD-方法的特征值问题是不太受欢迎的源问题。此外，最近在隐式溶剂化模型的背景下的工作和对这些方法的分析表明，DD方法对于越来越多的固定大小的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。该项目中包含的问题是多方面的，包括：i）非线性特征值问题，ii）大量的待确定的特征值，iii）无界域上的本征值问题和iv）处理包含库仑类奇点的势。该项目是更广泛的长期研究的第一步，长期计划，以获得有效的本地基函数的基础上，本地降阶建模作为替代广泛使用的，但经验收缩高斯基函数。事实上，区域分解策略允许本地化的方程，并打开了大门的应用程序的降阶建模与认证的后验误差估计在第二步。

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.