The two-sided arnoldi algorithm for nonsymmetric eigenvalue problems

1983 ◽  
pp. 104-120 ◽  
Author(s):  
Axel Ruhe
Keyword(s):  
Eigenvalue Problems ◽  
Arnoldi Algorithm

scholarly journals A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems

2016 ◽  
Vol 23 (4) ◽  
pp. 607-628 ◽  
Author(s):  
Roel Van Beeumen ◽  
Elias Jarlebring ◽  
Wim Michiels
Keyword(s):  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problems ◽  
Arnoldi Algorithm ◽  
Nonlinear Eigenvalue

A method for solving stochastic eigenvalue problems II

2013 ◽  
Vol 219 (9) ◽  
pp. 4729-4744 ◽  
Author(s):  
M.M.R. Williams
Keyword(s):  
Eigenvalue Problems

scholarly journals A Revisit to CMFD Schemes: Fourier Analysis and Enhancement

Energies ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 424
Author(s):  
Dean Wang ◽  
Zuolong Zhu
Keyword(s):  
Fourier Analysis ◽  
Eigenvalue Problems ◽  
Neutron Transport ◽  
Coarse Mesh ◽  
Successive Overrelaxation ◽  
Computational Cell ◽  
Artificial Diffusion ◽  
Acceleration Method ◽  
The Stability ◽  
Partial Current

The coarse-mesh finite difference (CMFD) scheme is a very effective nonlinear diffusion acceleration method for neutron transport calculations. CMFD can become unstable and fail to converge when the computational cell optical thickness is relatively large in k-eigenvalue problems or diffusive fixed-source problems. Some variants and fixups have been developed to enhance the stability of CMFD, including the partial current-based CMFD (pCMFD), optimally diffusive CMFD (odCMFD), and linear prolongation-based CMFD (lpCMFD). Linearized Fourier analysis has proven to be a very reliable and accurate tool to investigate the convergence rate and stability of such coupled high-order transport/low-order diffusion iterative schemes. It is shown in this paper that the use of different transport solvers in Fourier analysis may have some potential implications on the development of stabilizing techniques, which is exemplified by the odCMFD scheme. A modification to the artificial diffusion coefficients of odCMFD is proposed to improve its stability. In addition, two explicit expressions are presented to calculate local optimal successive overrelaxation (SOR) factors for lpCMFD to further enhance its acceleration performance for fixed-source problems and k-eigenvalue problems, respectively.

scholarly journals Some transpose-free CG-like solvers for nonsymmetric ill-posed problems

2020 ◽  
Vol 28 (1) ◽  
pp. 15-32
Author(s):  
Silvia Gazzola ◽  
Paolo Novati
Keyword(s):  
Krylov Subspace ◽  
Main Idea ◽  
System Matrix ◽  
Numerical Tests ◽  
New Methods ◽  
Arnoldi Algorithm ◽  
Ill Posed ◽  
Efficient Alternative ◽  
Original Class ◽  
Theoretical Insight

AbstractThis paper introduces and analyzes an original class of Krylov subspace methods that provide an efficient alternative to many well-known conjugate-gradient-like (CG-like) Krylov solvers for square nonsymmetric linear systems arising from discretizations of inverse ill-posed problems. The main idea underlying the new methods is to consider some rank-deficient approximations of the transpose of the system matrix, obtained by running the (transpose-free) Arnoldi algorithm, and then apply some Krylov solvers to a formally right-preconditioned system of equations. Theoretical insight is given, and many numerical tests show that the new solvers outperform classical Arnoldi-based or CG-like methods in a variety of situations.

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