Structural Preserving Numerical Methods for Eigenvalue Problems

University Of Texas At Arlington, Arlington TX

Investigators

Abstract

Large and sparse matrix computational problems are often solved by certain subspace projection methods -- most commonly Krylov subspace type projections. The basic idea is to project the original problems (matrices) of high dimensions onto certain subspaces to arrive at smaller and manageable ones, and the smaller reduced problems can then be solved by one of the dense matrix algorithms such as those in LAPACK. Existing projection techniques often do not preserve structural properties enjoyed by eigenproblems from various engineering applications, and therefore the reduced problems do not necessarily reflect their practical backgrounds 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 there are cases where structural preserving methods are far superior to those that are blind to the inherent structures. The objective of this proposal is to exploit in depth structural properties of matrices from the standpoint of their application backgrounds and to develop accurate and efficient structural preserving numerical methods for eigenvalue and related problems of practical significance. A number of interesting ideas will be pursued here, including a general framework for carrying out structural preserving subspace projections, an unifying convergence analysis for all Krylov subspace type projections that connects moment matching properties in reduced order modeling and eigenvalue and eigenvector convergence theory, and a sub-orthogonalization process that will serve as the basis to devise efficient projections. Eigenproblems appear ubiquitously all across applied science and engineering, and their solutions are routinely sought and are critical in one way or another to various scientific computational tasks. Examples includes computational problems from structural dynamics, control systems, circuit simulations, computational electromagnetics and microelectromechanical systems, data mining, and web search engine design, etc. This investigation shall advance significantly the underlying engineering applications by making the involved matrix computations much less expensive, more accurate, and most importantly result in scientific simulations that reflect better the underlying physics. Graduate students with emerging expertise in numerical eigenvalue computations will be involved.