scholarly journals Nested Krylov Methods for Shifted Linear Systems

2015 ◽  
Vol 37 (5) ◽  
pp. S90-S112 ◽  
Author(s):  
Manuel Baumann ◽  
Martin B. van Gijzen
Keyword(s):  
Linear Systems ◽  
Krylov Methods ◽  
Shifted Linear Systems

scholarly journals Nested Krylov methods for shifted linear systems

2015 ◽  
Author(s):  
Manuel Baumann ◽  
Martin B. van Gijzen
Keyword(s):  
Linear Systems ◽  
Invariance Property ◽  
Krylov Methods ◽  
Simultaneous Solution ◽  
Krylov Method ◽  
Shifted Linear Systems ◽  
Shift Invariance ◽  
Iterative Framework

Most algorithms for the simultaneous solution of shifted linear systems make use of the shift-invariance property of the underlying Krylov spaces. This particular comes into play when preconditioning is taken into account. We propose a new iterative framework for the solution of shifted systems that uses an inner multi-shift Krylov method as a preconditioner within a flexible outer Krylov method. Shift-invariance is preserved if the inner method yields collinear residuals.

scholarly journals A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

Algorithms ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 100 ◽  
Author(s):  
Luca Bergamaschi
Keyword(s):  
Linear Systems ◽  
Eigenvalue Problems ◽  
Real Life ◽  
Low Rank ◽  
Preconditioned Conjugate Gradient ◽  
Krylov Methods ◽  
Rank Matrix ◽  
Recent Developments ◽  
Speed Up ◽  
Low Rank Matrix

The aim of this survey is to review some recent developments in devising efficient preconditioners for sequences of symmetric positive definite (SPD) linear systems A k x k = b k , k = 1 , … arising in many scientific applications, such as discretization of transient Partial Differential Equations (PDEs), solution of eigenvalue problems, (Inexact) Newton methods applied to nonlinear systems, rational Krylov methods for computing a function of a matrix. In this paper, we will analyze a number of techniques of updating a given initial preconditioner by a low-rank matrix with the aim of improving the clustering of eigenvalues around 1, in order to speed-up the convergence of the Preconditioned Conjugate Gradient (PCG) method. We will also review some techniques to efficiently approximate the linearly independent vectors which constitute the low-rank corrections and whose choice is crucial for the effectiveness of the approach. Numerical results on real-life applications show that the performance of a given iterative solver can be very much enhanced by the use of low-rank updates.

Accelerated preconditioner updates for solving shifted linear systems

2016 ◽  
Vol 94 (4) ◽  
pp. 747-756
Author(s):  
Yu-Qin Bai ◽  
Ting-Zhu Huang ◽  
Wei-Hua Luo
Keyword(s):  
Linear Systems ◽  
Shifted Linear Systems

Solution of generalized shifted linear systems with complex symmetric matrices

2012 ◽  
Vol 231 (17) ◽  
pp. 5669-5684 ◽  
Author(s):  
Tomohiro Sogabe ◽  
Takeo Hoshi ◽  
Shao-Liang Zhang ◽  
Takeo Fujiwara
Keyword(s):  
Linear Systems ◽  
Symmetric Matrices ◽  
Shifted Linear Systems ◽  
Complex Symmetric ◽  
Complex Symmetric Matrices
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