CAREER: Structured Matrix Computations: Foundations, Methods, and Applications

职业：结构化矩阵计算：基础、方法和应用

• 批准号: 1255416
• 负责人: Jianlin Xia
• 金额: $ 41.39万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1255416&HistoricalAwards=false
• 关键词: CAREER; Structured; Matrix; Computations; Foundations; CAREER; Structured; Matrix; Computations; Foundations

The proposed research is concerned with innovative and systematic structured matrix computations. Large-scale matrices arise frequently in mathematical computations and engineering simulations. By exploiting the inherent structures, one can often develop new fast and reliable matrix methods. This work is especially interested in rank structures and data-sparse matrices, as well as innovative ideas of using them in the solutions of practical numerical problems (especially high dimensional discretized problems). The PI proposes to enhance both the flexibility and the applicability of structured methods. Systematic analysis for the structures and the methods will be included. The following aspects will be studied: (1) theoretical foundations of rank properties and new nested hierarchical structures; (2) new perspectives for exploring matrix structures, such as matrix-free structured sparse factorization, factorization update, and nested localization and sparsification; (3) efficient rank structured matrix algorithms, their analysis, and their use in fast solutions of large linear systems, least squares problems, eigenvalue problems, discretized PDEs, and numerous engineering applications.Matrix computations lie at the heart of most scientific computation tasks. This research will systematically extend classical dense and sparse matrix computations to data-sparse ones, and will introduce new structured matrix theories and techniques into scientific and engineering computations. The project will result in practical ways to reveal and use structures, which will further yield fast and reliable algorithms such as stable direct three dimensional PDE solvers with nearly linear complexity. Understanding the structures will also provide new perspectives to classical challenges in numerical solutions, such as large fill-in, ill conditioning, lack of explicit matrices, and repeated solutions with highly varying parameters. The proposed algorithms can be used (say, as kernel solvers) in many complex numerical problems such as PDE solution, seismic imaging, signal processing, nanostructure modeling, and VLSI circuit simulation. The work will provide a multidisciplinary opportunity for researchers in different areas to participate. The project will result in freely available open source packages for practical applications, as well as courses ad tutorial and test materials for educational outreach programs that can stimulate the interest and achievement of students from diverse backgrounds.

拟议的研究与创新和系统的结构矩阵计算有关。大规模矩阵经常出现在数学计算和工程模拟中。通过利用固有结构，通常可以开发新的快速可靠矩阵方法。这项工作对等级结构和数据质量矩阵以及在实用数值问题解决方案（尤其是高维离散问题）中使用的创新思想特别感兴趣。 PI提议提高结构化方法的灵活性和适用性。将包括对结构和方法的系统分析。将研究以下方面：（1）等级属性和新嵌套层次结构的理论基础； （2）探索矩阵结构的新观点，例如无基质结构化稀疏分解，分解更新以及嵌套的定位和稀疏； （3）有效的等级结构矩阵算法，它们的分析以及它们在大型线性系统的快速解决方案，最小二乘问题，特征值问题，离散的PDE和众多工程应用中的使用。matrix计算在于大多数科学计算任务的核心。这项研究将系统地将经典密集和稀疏的矩阵计算扩展到数据范围的计算，并将将新的结构化矩阵理论和技术引入科学和工程计算中。该项目将以实用的方式揭示和使用结构，这将进一步产生快速可靠的算法，例如稳定的直接三维PDE求解器，具有几乎线性的复杂性。了解结构还将为数值解决方案中的经典挑战提供新的观点，例如较大的填充，不良调节，缺乏明确的矩阵以及具有高度不同参数的重复解决方案。在许多复杂的数值问题（例如PDE溶液，地震成像，信号处理，纳米结构建模和VLSI电路仿真）中，可以使用（例如，作为内核求解器）（例如内核求解器）。这项工作将为不同领域的研究人员参与的多学科机会。该项目将为实用应用程序提供免费的开源包，以及用于教育外展计划的AD教程和测试材料，这些课程可以激发来自不同背景的学生的兴趣和成就。