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• W13595655 abstract "OF DISSERTATION Eun-Joo Lee The Graduate S hool University of Kentu ky 2008 A urate and Robust Pre onditioning Te hniques for Solving General Sparse Linear Systems ABSTRACT OF DISSERTATION A dissertation submitted in partial ful llment of the requirements for the degree of Do tor of Philosophy in the College of Engineering at the University of Kentu ky By Eun-Joo Lee Lexington, Kentu ky Dire tor: Dr. Jun Zhang, Professor of Computer S ien e Lexington, Kentu ky 2008 Copyright Eun-Joo Lee 2008 ABSTRACT OF DISSERTATION A urate and Robust Pre onditioning Te hniques for Solving General Sparse Linear Systems Pre onditioned Krylov subspa e methods are generally regarded as one lass of the most promising te hniques for solving very large and sparse linear systems, but these methods may produ e instability and/or ina ura y problems for ertain matri es. Ina ura y problems are usually aused by the diAE ulty in determining parameters or threshold values in onstru ting pre onditioners for some parti ular matri es. In addition, small or zero pivots in inde nite matri es may yield pre onditioners that are unstable and ina urate. The purpose of this study is mainly on erned with improving stability and a ura y of pre onditioners. Firstly, for the purpose of improving the a ura y of in omplete lower-upper (ILU) fa torization, two pre onditioning a ura y enhan ement strategies were proposed. The strategies employ the elements that are dropped during ILU fa torization and utilize them in di erent ways with separate algorithms. The rst strategy (error ompensation) applies the dropped elements to the lower and upper parts of the LU fa torization to ompute a new error ompensated LU fa torization. The other strategy (inner-outer iteration), whi h is a variant of the ILU fa torization, embeds the dropped elements in the pre onditioning iteration pro ess. Se ondly, in order to in rease the a ura y and stability of pre onditioners in solving inde nite matri es, hybrid reordering and two-phase pre onditioning strategies based on in omplete LU (ILU) fa torization and a sparse approximate inverse (SAI) pre onditioner, respe tively are onsidered. These strategies attempt to repla e the small or zero pivots with large values, ensuring that the resulting pre onditioners are better onditioned. Spe i ally, the hybrid reordering strategies eAE iently sear h for the entries of single element and/or the maximum absolute value to pla e the elements on the main diagonal of the original matrix, and the two-phase pre onditioning strategy adopts the idea of a shifting method that adds a value to the diagonals of an inde nite matrix. The two-phase pre onditioning strategy produ es an inverse approximation of the shifted matrix by onstru ting a SAI pre onditioner in ea h phase. The two inverse approximation matri es produ ed from ea h phase are then ombined to be used as a pre onditioner.OF DISSERTATION A urate and Robust Pre onditioning Te hniques for Solving General Sparse Linear Systems Pre onditioned Krylov subspa e methods are generally regarded as one lass of the most promising te hniques for solving very large and sparse linear systems, but these methods may produ e instability and/or ina ura y problems for ertain matri es. Ina ura y problems are usually aused by the diAE ulty in determining parameters or threshold values in onstru ting pre onditioners for some parti ular matri es. In addition, small or zero pivots in inde nite matri es may yield pre onditioners that are unstable and ina urate. The purpose of this study is mainly on erned with improving stability and a ura y of pre onditioners. Firstly, for the purpose of improving the a ura y of in omplete lower-upper (ILU) fa torization, two pre onditioning a ura y enhan ement strategies were proposed. The strategies employ the elements that are dropped during ILU fa torization and utilize them in di erent ways with separate algorithms. The rst strategy (error ompensation) applies the dropped elements to the lower and upper parts of the LU fa torization to ompute a new error ompensated LU fa torization. The other strategy (inner-outer iteration), whi h is a variant of the ILU fa torization, embeds the dropped elements in the pre onditioning iteration pro ess. Se ondly, in order to in rease the a ura y and stability of pre onditioners in solving inde nite matri es, hybrid reordering and two-phase pre onditioning strategies based on in omplete LU (ILU) fa torization and a sparse approximate inverse (SAI) pre onditioner, respe tively are onsidered. These strategies attempt to repla e the small or zero pivots with large values, ensuring that the resulting pre onditioners are better onditioned. Spe i ally, the hybrid reordering strategies eAE iently sear h for the entries of single element and/or the maximum absolute value to pla e the elements on the main diagonal of the original matrix, and the two-phase pre onditioning strategy adopts the idea of a shifting method that adds a value to the diagonals of an inde nite matrix. The two-phase pre onditioning strategy produ es an inverse approximation of the shifted matrix by onstru ting a SAI pre onditioner in ea h phase. The two inverse approximation matri es produ ed from ea h phase are then ombined to be used as a pre onditioner. Thirdly, with the intention of enhan ing the onvergen e performan e of the preonditioned iterative solvers, two sparsity pattern sele tion algorithms for a fa tored sparse approximate inverse pre onditioner are onsidered. The sparsity pattern is adaptively updated in the onstru tion phase of a pre onditioner by using ombined information of the inverse and original triangular fa tors of the original matrix. In order to determine the sparsity pattern, the rst algorithm uses the norm of the inverse fa tors multiplied by the largest absolute value of the original fa tors, and the se ond employs the norm of the inverse fa tors divided by the norm of the original fa tors." @default.
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• W13595655 date "2008-01-01" @default.
• W13595655 modified "2023-09-27" @default.
• W13595655 title "Accurate and Robust Preconditioning Techniques for Solving General Sparse Linear Systems" @default.
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