ITR/AP(DMS): Semidefinite Programming for Electronic Structure

该项目旨在推进大规模半定规划和本征值优化的快速迭代算法的最新技术，以应用于许多电子系统的第一性原理计算。 半定优化代码将被重写，以取代密集的线性代数和直接矩阵分解的迭代求解过程适合大型和稀疏的问题。 数值实验将进行不同的数学公式和稀疏优化程序的实现。 指导应用是一个变分制定的电子结构基态问题，其中的未知数是一个多电子系统的单体和两体约化密度矩阵。 这个应用程序产生了一个大的半定程序具有稀疏的解决方案矩阵和一个非常稀疏的约束集。 在此资助下开发的代码将被编写和记录，目的是广泛传播。半定编程是许多科学和工程应用的一个有价值的框架，包括系统控制，结构分析，组合优化，统计估计和VLSI设计，仅举几个除了电子结构理论之外的成熟的框架。 该项目旨在提高我们计算问题实例的解决方案的能力，这些问题实例的大小超出了当前数值方法的范围。 至于在该项目中的具体应用，基于量子力学的多体系统性质的计算对于理解分子和固态结构、化学和基本生物化学过程以及凝聚态物质的机械和电磁性质是至关重要的。 多体量子力学计算的准确性和规模的进步对技术和社会的潜在好处包括，例如，更好地设计半导体、磁存储材料、高温超导体和化学催化剂，以及改进合理药物设计和碳纳米结构设计等新兴领域。

This project aims to advance the state of the art in fast iterative algorithms for large-scale semidefinite programming and eigenvalue optimization, guided by applications to first-principles computation of properties of systems of many electrons. Semidefinite optimization codes will be rewritten to replace dense linear algebra and direct matrix factorizations by iterative solution processes suited to large and sparse problems. Numerical experiments will be carried out with different mathematical formulations and implementations of the sparse optimization procedures. The guiding application is a variational formulation of the electronic structure ground state problem in which the unknowns are the one-body and two-body reduced density matrices of a many electron system. This application gives rise to a large semidefinite program having sparse solution matrices and an extremely sparse constraint set. The codes that are developed under this grant will be written and documented with the intent of wide dissemination.Semidefinite programming is a valuable framework for many scientific and engineering applications, including systems control, structural analysis, combinatorial optimization, statistical estimation and VLSI design, just to name a few well-established ones besides electronic structure theory. This project aims to increase our ability to compute solutions to problem instances whose size places them beyond the reach of current numerical methods. As to the specific application in this project, the computation of properties of many-body systems on the basis of quantum mechanics is fundamental to the understanding of molecular and solid state structure, chemical and elementary biochemical processes, and mechanical and electromagnetic properties of condensed matter. Potential benefits to technology and society from advances in the accuracy and scale of many-body quantum mechanical computations include, for example, better design of semiconductors, magnetic storage materials, high-temperature superconductors and chemical catalysts, as well as improvements in the emerging areas of rational drug design and the design of carbon nanostructures.