Iteratively Constructing Preconditioners via the Conjugate Gradient Method

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Mousa Abadian, Mohammad | 2013

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 44496 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Farhadi, Hamid Reza
Abstract:
The main goal of this work is solving system of linear equations Ax = b, where A is a n_n square matrix, b is a n_1 vector and x is the vector of unknowns. When n is large, using direct methods is not economical. Thus, the system is solved by iterative methods. At first, projection method onto subspace K _ Rn with dimension m _ n is described, and then this subspace K is equalized with the krylov subspace. Then,some samples of projection methods onto the krylov subspace, such as FOM, GMRES and CG (Conjugate Gradient), are considered. The preconditioning of the linear system is explained, that is, instead of solving system Ax = b, the system PAx = Pb (P nonsingular), is solved, such that the condition number of P is less than the condition number of A. In 2007 Dungan and Harvey introduced a new algorithm for designing iterative preconditioner based on one iteration of a new conjugate gradient method. Finally, numerical results of an implementation of this algorithm on some test problems are reported. These results confirm that especially for matrices with large condition number, the algorithm is efficient in error reduction. It is remarkable that a good preconditioning is possible if a good preconditioner is available
Keywords:
Conjuagate Gradient Method ; Preconditioning ; Iteration Method