AF: Medium: Collaborative research: Advanced algorithms and high-performance software for large scale eigenvalue problems

AF：中：协作研究：大规模特征值问题的先进算法和高性能软件

• 批准号: 1505970
• 负责人: Yousef Saad
• 金额: $ 36.07万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-15 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1505970&HistoricalAwards=false
• 关键词: AF; Medium; Collaborative; research; Advanced

Scientists and engineers in areas ranging from physics, chemistry, computer science, to economics, and statistics focus considerable attention on computing "eigenvalues" and "eigenvectors" of matrices. They are central to the study of vibrations when building earthquake-resistant structures, to energy computation in solid-state physics, and to ranking web search results. In spite of the enormous progress that has been made in the last few decades in solution methods for large eigenvalue problems, the current state-of-the-art methods remains unsatisfactory when dealing with the new generation of problems that need tens of thousands of eigenvectors for matrices that can have sizes in the tens of millions.In recent years a new class of techniques has emerged that can compute wanted eigenpairs of large matrices by parts. In these methods, 'windows' or 'slices' of the spectrum can be computed independently of one another and orthogonalization between eigenvectors in different slices is no longer necessary. When the number of eigenpairs to be computed is very large this divide-and-conquer approach becomes mandatory because orthogonalizing very large bases is prohibitive. The resulting interior eigenvalue problems arise in a number of other situations and are now considered by the linear algebra community to be among the most challenging numerical problems to solve, and solution methods for handling them are still lagging.The goal of this project is to advance the state of the art in solution methods for interior eigenvalue problems. The main thrust of the project is the development of novel algorithms based on a combination of Krylov or block-Krylov projection techniques and complex rational filters. A starting point in this investigation is the FEAST approach. This project addresses many interesting questions in several areas, starting with methodologies for solving eigenvalue problems, to approximation theory questions for designing rational filter functions, and ending with effective parallel implementations. Methods based on a domain decomposition framework will also be considered to deal with the common situation where the matrix (or pair of matrices in the generalized case) is (are) distributed.The broader impacts of this project highlight the impact on training, the dissemination of new efficient software, and the use of the software by-products in specific applications. All general-purpose codes that are developed under this project will be freely distributed into the public domain. This project will have an impact on the training of graduate and undergraduate students in a field that is vital to the needs of academia, industry, and government laboratories.

从物理，化学，计算机科学，经济学和统计数据范围内的科学家和工程师都将非常关注的关注在计算矩阵的“特征值”和“特征向量”上。 它们是研究抗震结构，固态物理学中的能量计算以及对Web搜索结果进行排名的振动的核心。 尽管过去几十年来解决了大型特征问题的解决方案方法取得的巨大进展，但在处理需要数万个需要成千上万的矩阵特征媒介的新一代问题时，当前的最新方法仍然不令人满意 部分。 在这些方法中，可以独立计算频谱的“ Windows”或“切片”，并且不再需要在不同切片中的特征向量之间进行正交化。当要计算的本征数的数量非常大时，这种划分和互动方法将变得强制性，因为正交化非常大的基础是过于较大的。 由此产生的内部特征值问题在其他许多情况下都会出现，现在，线性代数社区认为要解决的最具挑战性的数值问题之一，而处理它们的解决方案方法仍在落后。该项目的目的是在室内eigenvalue问题中促进ART方法的ART方法。该项目的主要目的是基于Krylov或Block-Krylov投影技术和复杂有理过滤器的组合的新算法的开发。这项调查的起点是盛宴方法。 该项目从几个领域中解决了许多有趣的问题，从解决特征值问题开始，以设计理论滤波器功能，并以有效的并行实现结束。 还将考虑基于域分解框架的方法来处理（或在广义情况下）分布矩阵（或一对矩阵）的共同情况。该项目的更广泛影响突出了对培训的影响，新有效软件的传播以及在特定应用中使用该软件的使用。该项目下开发的所有通用代码都将自由分配到公共​​领域。 该项目将影响对学术界，工业和政府实验室需求至关重要的领域研究生和本科生的培训。