scholarly journals A new Krylov-subspace method for symmetric indefinite linear systems

1994 ◽  
Author(s):  
R.W. Freund ◽  
N.M. Nachtigal
Keyword(s):  
Linear Systems ◽  
Krylov Subspace ◽  
Krylov Subspace Method ◽  
Subspace Method ◽  
Indefinite Linear Systems

scholarly journals Preconditioning Large Indefinite Linear Systems

2011 ◽  
Vol 17 (1) ◽  
pp. 63
Author(s):  
Giovanni Fasano ◽  
Massimo Roma
Keyword(s):  
Numerical Analysis ◽  
Nonlinear Optimization ◽  
Linear Systems ◽  
Krylov Subspace ◽  
Krylov Subspace Method ◽  
The Other ◽  
Subspace Method ◽  
Numerical Experimentation ◽  
Indefinite Linear Systems

After briefly recalling some relevant approaches for preconditioning large symmetric linear systems, we describe a novel class of preconditioners. Our proposal is tailored for large indefinite linear systems, which arise very frequently in many different contexts of numerical analysis and nonlinear optimization. Our preconditioners are built as a byproduct of the Krylov subspace method used to solve the system. We describe theoretical properties of the proposed class of preconditioners, namely their capability of both shifting some eigenvalues of the system’s matrix to controlled values, and reducing the modulus of the other ones. The results of a numerical experimentation give evidence of the good performance of our proposal.  

scholarly journals Krylov subspace methods for estimating operator-vector multiplications in Hilbert spaces

2021 ◽  
Author(s):  
Yuka Hashimoto ◽  
Takashi Nodera
Keyword(s):  
Krylov Subspace ◽  
Time Series Data ◽  
Linear Equations ◽  
Transfer Operator ◽  
Krylov Subspace Methods ◽  
Krylov Subspace Method ◽  
Linear Operators ◽  
Series Data ◽  
Subspace Method ◽  
Subspace Methods

AbstractThe Krylov subspace method has been investigated and refined for approximating the behaviors of finite or infinite dimensional linear operators. It has been used for approximating eigenvalues, solutions of linear equations, and operator functions acting on vectors. Recently, for time-series data analysis, much attention is being paid to the Krylov subspace method as a viable method for estimating the multiplications of a vector by an unknown linear operator referred to as a transfer operator. In this paper, we investigate a convergence analysis for Krylov subspace methods for estimating operator-vector multiplications.

Fast EM Stress Evolution Analysis Using Krylov Subspace Method

2019 ◽  
pp. 47-66
Author(s):  
Sheldon Tan ◽  
Mehdi Tahoori ◽  
Taeyoung Kim ◽  
Shengcheng Wang ◽  
Zeyu Sun ◽  
...  
Keyword(s):  
Krylov Subspace ◽  
Krylov Subspace Method ◽  
Subspace Method ◽  
Stress Evolution ◽  
Evolution Analysis

Refined Algorithm for Solving High-Order Harmonics of Multi Group Neutron Diffusion Equation

2017 ◽  
Author(s):  
Gaoxin Zhou ◽  
Zhi Gang
Keyword(s):  
Diffusion Equation ◽  
Iteration Method ◽  
Krylov Subspace ◽  
Krylov Subspace Method ◽  
Neutron Diffusion ◽  
Subspace Method ◽  
Arnoldi Method ◽  
Arnoldi Algorithm ◽  
Eigenvalue And Eigenvector ◽  
Neutron Diffusion Equation

In recent years, high order harmonic (or eigenvector) of neutron diffusion equation has been widely used in on-line monitoring system of reactor power. There are two kinds of calculation method to solve the equation: corrected power iteration method and Krylov subspace methods. Fu Li used the corrected power iteration method. When solving for the ith harmonic, it tries to eliminate the influence of the front harmonics using the orthogonality of the harmonic function. But its convergence speed depends on the occupation ratio. When the dominant ratios equal to 1 or close to 1, convergence speed of fixed source iteration method is slow or convergence can’t be achieved. Another method is the Krylov subspace method, the main idea of this method is to project the eigenvalue and eigenvector of large-scale matrix to a small one. Then we can solve the small matrix eigenvalue and eigenvector to get the large ones. In recent years, the restart Arnoldi method emerged as a development of Krylov subspace method. The method uses continuous reboot Arnoldi decomposition, limiting expanding subspace, and the orthogonality of the subspace is guaranteed using orthogonalization method. This paper studied the refined algorithms, a method based on the Krylov subspace method of solving eigenvalue problem for large sparse matrix of neutron diffusion equation. Two improvements have been made for a restarted Arnoldi method. One is that using an ingenious linear combination of the refined Ritz vector forms an initial vector and then generates a new Krylov subspace. Another is that retaining the refined Ritz vector in the new subspace, called, augmented Krylov subspace. This way retains useful information and makes the resulting algorithm converge faster. Several numerical examples are the new algorithm with the implicitly restart Arnoldi algorithm (IRA) and the implicitly restarted refined Arnoldi algorithm (IRRA). Numerical results confirm efficiency of the new algorithm.

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