|Title:||The updated subspaces method in optimization and in solving linear systems of equations|
|Department / Program:||Mathematics|
|Degree Granting Institution:||University of Illinois at Urbana-Champaign|
|Abstract:||The Updated Conjugate Subspaces method, an modified quasi-Newton method for solving nonlinear unconstrained minimization problems, has flexibility of choosing different quadratic approximation at each iteration and therefore has potential of improving the rate of convergence given by the quasi-Newton method. In this dissertation, the convergence behavior of the UCS method is investigated and the further development of this method in solving the nonlinear unconstrained minimizations, the partially separable minimizations and the linear systems of equations is presented.
The first part of the dissertation gives the relation between the UCS method and the choice of the quadratic approximation. Both locally and globally superlinear convergence theorems for the UCS method with some choices of the quadratic approximations are presented. It is also presented that the UCS method has a locally Newton-like convergence behavior when it chooses a proper set of conjugate subspaces and a Newton approximation. A preliminary testing is conducted to evaluate the UCS method and its results are reported.
The second part of the dissertation introduces a parallel Newton method and a parallel modified BFGS method extended from the UCS method for solving a class of nonlinear unconstrained minimizations with a partially separable structure. Testing results are discussed also.
The last part of the dissertation contains two pieces. The first piece explores the convergence behavior of the Conjugate Gradient method for solving linear systems of equations and then gives a definition of a spectral distribution of a system to be favorable to the CG algorithm. The second piece introduces a new algorithm based on the strategy of the conjugate subspaces decomposition for solving the linear systems of equations with many right-hand sides. This algorithm is not only suitable for parallel computation but is also suitable for modifying a spectral distribution if it is not favorable to the CG algorithm so that the resulting spectral distribution is much favorable to the CC algorithm.
|Rights Information:||Copyright 1989 Chen, Mei-Qin|
|Date Available in IDEALS:||2011-05-07|
|Identifier in Online Catalog:||AAI9010826|
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