Structural Preserving Numerical Methods for Eigenvalue Problems

求解大型稀疏矩阵计算问题的方法通常是定义子空间投影法--最常见的是Krylov子空间投影法，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上，得到更小的、可管理的问题，然后用LAPACK中的稠密矩阵算法之一来求解。现有的投影技术往往不能保持各种工程应用中的特征问题所具有的结构性质，因此简化后的问题不一定以任何有意义的方式反映其实际背景。可以想象，就像经常发生的情况一样，用自己的问题来逼近一个问题会更好。事实上，有些情况下，结构保存方法远远优于那些无视固有结构的方法。本文的目的是从矩阵的应用背景出发，挖掘矩阵的深层结构性质，为特征值及相关有实际意义的问题发展准确而有效的结构保值数值方法。这里将探讨一些有趣的想法，包括执行结构保留子空间投影的一般框架，所有Krylov子空间类型投影的统一收敛分析，它将降阶建模中的矩匹配性质与特征值和特征向量收敛理论联系起来，以及将作为设计效率投影的基础的次正交化过程。特征问题普遍存在于应用科学和工程中，其解决方案是例行公事，并且以这样或那样的方式对各种科学计算任务至关重要。例子包括来自结构动力学、控制系统、电路模拟、计算电磁学和微电子机械系统、数据挖掘和网络搜索引擎设计等的计算问题。通过使涉及的矩阵计算更便宜、更准确，并且最重要的是产生更好地反映潜在物理的科学模拟，这项研究将显著推进潜在的工程应用。将涉及在数值特征值计算方面具有新兴专业知识的研究生。

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.