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

大型线性系统的高效结构化直接求解器和鲁棒结构化预处理器及其应用

• 批准号: 1115572
• 负责人: Jianlin Xia
• 金额: $ 6.5万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115572&HistoricalAwards=false
• 关键词: Efficient; Sructured; Direct; Solvers; Robust

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.

在这个项目中，研究人员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，包括快速直接求解器和强大有效的预处理器。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。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的离散化产生的线性系统，该方法具有接近线性的复杂度。它们还可以作为有效的预处理器。显示了针对正定问题的预处理的鲁棒性。这些方法对于经典直接或迭代求解器被认为困难的问题很有用。该项目在许多复杂的数值问题和工程模拟中具有更广泛的影响，例如微分方程、地震成像、气候、电磁场模拟、信号处理和集成电路模拟。这些应用中的主要计算工作通常是求解大规模线性系统，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较低复杂性界限。学生参与该项目的各个方面，并接受各种数学和工程领域的培训。研究人员的团队计划为实际应用和教育构建一个免费的开源包。研究者组织的小型研讨会以及会议演讲、研讨会和期刊文章用于交流想法和传播结果。