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

职业：结构化矩阵计算：基础、方法和应用

基本信息

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

项目摘要

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.

拟议的研究涉及创新和系统的结构化矩阵计算。大规模矩阵在数学计算和工程模拟中经常出现。通过利用固有的结构，人们往往可以开发新的快速和可靠的矩阵方法。这项工作是特别感兴趣的秩结构和数据稀疏矩阵，以及创新的想法，使用它们在解决实际的数值问题（特别是高维离散化问题）。PI建议增强结构化方法的灵活性和适用性。系统分析的结构和方法将包括。研究内容包括：（1）秩性质和新的嵌套层次结构的理论基础;（2）探索矩阵结构的新视角，如无矩阵结构稀疏分解、分解更新、嵌套局部化和稀疏化;（3）有效的秩结构矩阵算法，它们的分析，以及它们在大型线性系统，最小二乘问题，矩阵计算是大多数科学计算任务的核心。该研究将把传统的稠密和稀疏矩阵计算系统地扩展到数据稀疏矩阵计算，并将新的结构矩阵理论和技术引入科学和工程计算。该项目将产生实用的方法来揭示和使用结构，这将进一步产生快速和可靠的算法，如稳定的直接三维PDE求解器，具有接近线性的复杂性。了解结构也将提供新的视角，经典的挑战，在数值解，如大填充，病态，缺乏显式矩阵，重复的解决方案与高度变化的参数。所提出的算法可以用于（说，作为核心求解器）在许多复杂的数值问题，如偏微分方程的解决方案，地震成像，信号处理，纳米结构建模，和VLSI电路模拟。这项工作将为不同领域的研究人员提供参与的多学科机会。该项目将产生免费提供的开放源码软件包，用于实际应用，以及教育推广方案的课程、辅导和测试材料，可以激发来自不同背景的学生的兴趣和成就。