The Use of Parallel Polynomial Preconditioners In the Solution of Systems of Linear Equations Yu Liang Books
Download As PDF : The Use of Parallel Polynomial Preconditioners In the Solution of Systems of Linear Equations Yu Liang Books
This book mainly explores the use of polynomial preconditioners in iterative solvers for large-scale sparse linear systems Ax = b. It is well known that preconditioners can significantly improve the convergence of solvers, particularly when the coefficient matrix is ill-conditioned. Further, polynomial preconditioners have several advantages over other popular preconditioners — they may be implemented easily, they are highly parallel, and they are extremely agile. Due to the intrinsic disadvantages of polynomial methods (e.g., spectrum information is needed, poor stability of the large-degree polynomial preconditioning) and the limitation of computing technologies, the polynomial preconditioning technique was somehow ignored in the past ten years. Fortunately, at present, polynomial preconditioners are attracting more and more attention with the development of computer science.
The Use of Parallel Polynomial Preconditioners In the Solution of Systems of Linear Equations Yu Liang Books
This book mainly explores the use of polynomial preconditioners in iterative solvers for large-scale sparse linear systems Ax = b. Polynomial preconditioners have several advantages over other popular preconditioners -- they may be implemented easily, they are highly parallel, and they are extremely agile. Due to the intrinsic disadvantages of polynomial methods (e.g., spectrum information is needed, poor stability of the large-degree polynomial preconditioning) and the limitation of computing technologies, the polynomial preconditioning technique was somehow ignored in the past ten years. Fortunately, at present, polynomial preconditioners are attracting more and more attention with the development of computer science.The construction of appropriate polynomial preconditioners can be transformed into constrained optimization problem, namely that of finding a m-degree polynomial in matrix A, Pm(A), such that Pm(A) A−1. Three typical polynomial preconditioners arise, Neumann-series, Least-squares and Minimax. In this thesis Symmetric linear equations will be mainly discussed, where the spectrum of the coefficient matrix is expressed by a real compact set. In the case of SID (Symmetric Indefinite)linear systems, only Least-squares and Chebyshev methods are applicable. The implementation and detailed performance analysis of the Generalized Least-squaresand the Generalized Minimax preconditioners are the main effort of this work.
In order to demonstrate the practical significance of the polynomial preconditioning technique in scientific and engineering computation, the work outlined in this thesis is incorporated with non-overlapping finite element based domain decomposition methods (i.e., global method and augmented FETI-DP, etc.) to solve the linear and non-linear static and dynamic structural mechanics problems.
All the polynomial preconditioned iterative linear solvers are implemented using MPI (Message Passing Interface) in a highly parallel machine environment (IBM SP2). Experimental results using classical benchmark systems are presented and compared with those obtained using the recently developed SPAI preconditioned Bi-
CGSTAB or ILU (Incomplete LU factorization) preconditioned iterative methods. The performance of polynomial preconditioned iterative linear solvers is critically assessed.
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The Use of Parallel Polynomial Preconditioners In the Solution of Systems of Linear Equations Yu Liang Books Reviews
This book mainly explores the use of polynomial preconditioners in iterative solvers for large-scale sparse linear systems Ax = b. Polynomial preconditioners have several advantages over other popular preconditioners -- they may be implemented easily, they are highly parallel, and they are extremely agile. Due to the intrinsic disadvantages of polynomial methods (e.g., spectrum information is needed, poor stability of the large-degree polynomial preconditioning) and the limitation of computing technologies, the polynomial preconditioning technique was somehow ignored in the past ten years. Fortunately, at present, polynomial preconditioners are attracting more and more attention with the development of computer science.
The construction of appropriate polynomial preconditioners can be transformed into constrained optimization problem, namely that of finding a m-degree polynomial in matrix A, Pm(A), such that Pm(A) A−1. Three typical polynomial preconditioners arise, Neumann-series, Least-squares and Minimax. In this thesis Symmetric linear equations will be mainly discussed, where the spectrum of the coefficient matrix is expressed by a real compact set. In the case of SID (Symmetric Indefinite)linear systems, only Least-squares and Chebyshev methods are applicable. The implementation and detailed performance analysis of the Generalized Least-squaresand the Generalized Minimax preconditioners are the main effort of this work.
In order to demonstrate the practical significance of the polynomial preconditioning technique in scientific and engineering computation, the work outlined in this thesis is incorporated with non-overlapping finite element based domain decomposition methods (i.e., global method and augmented FETI-DP, etc.) to solve the linear and non-linear static and dynamic structural mechanics problems.
All the polynomial preconditioned iterative linear solvers are implemented using MPI (Message Passing Interface) in a highly parallel machine environment (IBM SP2). Experimental results using classical benchmark systems are presented and compared with those obtained using the recently developed SPAI preconditioned Bi-
CGSTAB or ILU (Incomplete LU factorization) preconditioned iterative methods. The performance of polynomial preconditioned iterative linear solvers is critically assessed.
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