多項式固有値問題における櫻井-杉浦法に対するバランシング及びスケーリング手法の研究

筑波大学 (University of Tsukuba)201

Parallel symmetric eigenvalue problem solvers

Sparse symmetric eigenvalue problems arise in many computational science and engineering applications: in structural mechanics, nanoelectronics, and spectral reordering, for example. Often, the large size of these problems requires the development of eigensolvers that scale well on parallel computing platforms. In this dissertation, we describe two such eigensolvers, TraceMin and TraceMin-Davidson. These methods are different from many other eigensolvers in that they do not require accurate linear solves to be performed at each iteration in order to find the smallest eigenvalues and their associated eigenvectors. After introducing these closely related eigensolvers, we discuss alternative methods for solving the saddle point problems arising at each iteration, which can improve the overall running time. Additionally, we present TraceMin-Multisectioning, a new TraceMin implementation geared towards finding large numbers of eigenpairs in any given interval of the spectrum. We conclude with numerical experiments comparing our trace-minimization solvers to other popular eigensolvers (such as Krylov-Schur, LOBPCG, Jacobi-Davidson, and FEAST), establishing the competitiveness of our methods

Projections methods for oversized linear algebra problems

Everybody who has some experience in doing mathematics knows that di mensional reduction and projection are useful tools to confront problems that are too complicated to solve without any simplication Who hasnt occasion ally but notwithstanding timidly suggested that perhaps it would be a good idea to study the simple onedimensional case rst before trying to understand the realworld threedimensional problem Apparently it is a widespread faith that such simplications will not damage the essential mathematical or physical truth that is hidden in the original problem But is this faith founded Re gardless of the answer one should realize that in many applications there is no plausible alternative so it would be unfair to judge too harshly on those who solve reduced problems and with due mathematical care formulate interesting and strong theorems and hypotheses on the full problem Among them are the people from the eld of numerical linear algebr