Structural Preserving Numerical Methods for Eigenvalue Problems

大型稀疏矩阵的计算问题通常用某些子空间投影方法来解决--最常见的是Krylov子空间类型的投影，其基本思想是将高维的原始问题（矩阵）投影到某些子空间上，得到更小的可管理的子空间，然后可以用LAPACK中的稠密矩阵算法来解决更小的约化问题。现有的投影技术往往不保留结构属性享有的特征问题从各种工程应用，因此，减少的问题不一定反映其实际背景在任何有意义的方式。可以想象，用一个同类问题来近似一个问题会做得更好，这是常有的事。事实上，在某些情况下，结构保护方法远远上级那些对固有结构视而不见的方法。本文的目的是从矩阵的应用背景出发，深入研究矩阵的结构性质，并为特征值及相关问题发展具有实际意义的精确、高效的保结构数值方法。本文提出了一系列有趣的思想，包括：构造保持结构的子空间投影的一般框架，将降阶模型中的矩匹配性质与特征值和特征向量收敛理论联系起来的所有Krylov子空间投影的统一收敛性分析，和，正交化过程，这将作为基础，设计efficientprojects.Eigenproblems出现无处不在所有应用科学和工程，和他们的解决方案是例行寻求和关键的一种或另一种方式，以各种科学计算任务。例子包括计算问题，从结构动力学，控制系统，电路仿真，计算电磁学和微机电系统，数据挖掘，和网络搜索引擎设计等。这项调查将大大推进底层的工程应用，使所涉及的矩阵计算更便宜，更准确，最重要的是在scientificsimulations，更好地反映了底层的物理。研究生与新兴的专业知识，在数值特征值计算将参与。

Large and sparse matrix computational problems are often solved by certainsubspace projection methods -- most commonly Krylov subspace type projections.The basic idea is to project the original problems (matrices) of high dimensionsonto certain subspaces to arrive at smaller and manageable ones, and the smallerreduced problems can then be solved by one of the dense matrix algorithms such asthose in LAPACK. Existing projection techniques often do not preserve structuralproperties enjoyed by eigenproblems from various engineering applications, andtherefore the reduced problems do not necessarily reflect their practicalbackgrounds in any meaningful ways. It is conceivable, as it is often the case,that approximating a problem by one of its own kind would do better. Indeed thereare cases where structural preserving methods are far superior to those that areblind to the inherent structures. The objective of this proposal is to exploitin depth structural properties of matrices from the standpoint of their applicationbackgrounds and to develop accurate and efficient structural preserving numericalmethods for eigenvalue and related problems of practical significance. A number ofinteresting ideas will be pursued here, including a general framework for carryingout structural preserving subspace projections, an unifying convergence analysisfor all Krylov subspace type projections that connects moment matching propertiesin reduced order modeling and eigenvalue and eigenvector convergence theory, and asub-orthogonalization process that will serve as the basis to devise efficientprojections.Eigenproblems appear ubiquitously all across applied science and engineering,and their solutions are routinely sought and are critical in one way or anotherto various scientific computational tasks. Examples includes computationalproblems from structural dynamics, control systems, circuit simulations,computational electromagnetics and microelectromechanical systems, data mining,and web search engine design, etc. This investigation shall advance significantlythe underlying engineering applications by making the involved matrix computationsmuch less expensive, more accurate, and most importantly result in scientificsimulations that reflect better the underlying physics. Graduate students withemerging expertise in numerical eigenvalue computations will be involved.