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

在这个项目中，研究者和他的学生设计新的高效结构矩阵技术的大型线性系统，包括快速直接求解器和鲁棒有效的预条件。这些技术利用了实际应用中线性系统中某些隐藏的秩结构。高效的多层结构和灵活的秩要求。对于某些偏微分方程的离散化所产生的线性系统，该方法具有近似线性的复杂性。它们也可以作为有效的预处理器。对正定问题的预处理的鲁棒性。该方法适用于经典的直接或迭代求解器难以解决的问题，在微分方程、地震成像、气候、电磁场模拟、信号处理、集成电路模拟等许多复杂的数值问题和工程模拟中具有广泛的影响。在这些应用程序中的主要计算工作往往是解决大规模的线性系统，这可以受益于本项目中开发的高效黑盒求解器或预处理器。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，并在各种数学和工程领域接受培训。研究人员的团队计划建立一个免费的开源软件包，用于实际应用和教育。由研究者组织的Minisymopsia，以及会议，讲座，研讨会和期刊文章，被用来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.