Domain decomposition methods for electronic structure calculations

电子结构计算在理论化学、物理和材料科学中是非常重要的。事实上，在被引用最多的十篇科学文章中，有两篇论文与这个主题有关。尽管这一领域研究的是特征值问题的离散化，这是数值分析中的一个成熟的主题，但应用数学的专业知识很少涉及。在拟议的项目中，目标是为电子结构计算中出现的特征值问题(如Kohn-Sham DFT(密度泛函理论)方程)开发新的区域分解(DD)算法。这个项目中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的区域分解没有根本的不同。尽管如此，本征值问题的DD方法不如源问题的DD方法受欢迎。另外，最近在隐式溶剂化模型的上下文中的工作和对这些方法的分析表明，即使没有所谓的粗略校正，DD方法对于具有越来越多的固定大小的子域的域，例如链状分子或蛋白质，也是可扩展的。Iii)无界区域上的特征值问题和iv)处理包含类库仑奇异点的势。该项目是一个更广泛的长期计划的第一步，该计划基于局部降阶建模来推导有效的局部基函数，作为广泛使用的但经验压缩的高斯基函数的替代。事实上，区域分解策略允许将方程局部化，并为在第二步中使用经过认证的后验误差估计的降阶建模应用打开了大门。

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.