Efficient Sructured Direct Solvers and Robust Structured Preconditioners for Large Linear Systems and Their Applications

在这个项目中，研究者和他的学生为大型线性系统设计了新的高效结构化矩阵技术，包括快速直接求解和鲁棒有效预调节器。这些技术利用了在实际应用中产生的线性系统中的某些隐秩结构。考虑了高效的多层结构和灵活的等级要求。对于由某些偏微分方程离散化引起的线性系统，该方法具有近似线性的复杂度。它们也可以作为有效的调理剂。证明了正定问题预处理的鲁棒性。这些方法对经典直接或迭代求解方法难以解决的问题是有用的。该项目在微分方程、地震成像、气候、电磁场仿真、信号处理、集成电路仿真等许多复杂数值问题和工程仿真中具有广泛的影响。这些应用中的主要计算工作通常是求解大型线性系统，这可以从本项目开发的高效黑盒求解器或预调节器中受益。这些方法有助于打破一些经典的低复杂度界限。学生参与项目的各个方面，并接受各种数学和工程领域的培训。研究者的团队计划为实际应用和教育构建一个免费的开源包。研究者组织的小型专题讨论会，以及会议、讲座、研讨会和期刊文章，用于交流思想和传播结果。

In this project, the investigator and his students design new efficient structured matrix techniques for large linear systems, including fast direct solvers and robust effective preconditioners. These techniques take advantage of certain hidden rank structures in linear systems arising from practical applications. Efficient multi-layer structures and flexible rank requirements are considered. The methods have nearly linear complexity for linear systems arising from the discretization of certain partial differential equations. They can also work as effective preconditioners. Robustness of preconditioning for positive definite problems is shown. The methods are useful for problems which have been considered difficult for classical direct or iterative solvers.This project has broader impacts in many complex numerical problems and engineering simulations, such as differential equations, seismic imaging, climate, electromagnetic field simulation, signal processing, and integrated circuit simulation. The major computational work in these applications is often to solve large-scale linear systems, which can benefit from the efficient black-box solvers or preconditioners developed in this project. These methods help break some classical lower complexity bounds. Students are involved in all aspects of the project, and are trained in various mathematical and engineering areas. The investigator's team plan to build a freely available open source package for both practical applications and education. Minisymposia organized by the investigator, as well as conferences talks, seminars, and journal articles, are used to exchange ideas and to disseminate the results.