Structural Preserving Numerical Methods for Eigenvalue Problems

大型和稀疏矩阵的计算问题通常通过某些子空间投影方法来解决-最常见的是Krylov子空间类型投影。其基本思想是将高维的原始问题（矩阵）投影到特定的子空间中，从而得到更小的、可管理的问题，然后这些更小的简化问题可以通过一种密集矩阵算法来解决，比如LAPACK中的那些。现有的投影技术通常不能保留各种工程应用中特征问题所具有的结构特性，因此简化后的问题不一定能以任何有意义的方式反映其实际背景。可以想象的是，通常情况下，用一个近似的问题会做得更好。的确，在某些情况下，结构保存方法远优于那些无视固有结构的方法。本课题旨在从应用背景出发，探索矩阵的深度结构特性，为特征值及相关问题提供准确、高效的结构保持数值方法。这里将探讨一些有趣的想法，包括执行结构保持子空间投影的一般框架，对所有Krylov子空间类型投影的统一收敛分析，该投影将矩匹配特性与降阶建模和特征值和特征向量收敛理论联系起来，以及将作为设计有效投影基础的亚正交化过程。特征问题在整个应用科学和工程中无处不在，它们的解决方案经常被寻求，并且在各种科学计算任务中以某种方式至关重要。例子包括结构动力学、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计算问题。这项研究将通过使所涉及的矩阵计算更便宜，更准确，最重要的是产生更好地反映底层物理的科学模拟，从而显著推进基础工程应用。在数值特征值计算方面具有新兴专业知识的研究生将参与其中。

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.