Projection methods for solving large sparse eigenvalue problems

1983 ◽  
pp. 121-144 ◽  
Author(s):  
Youcef Saad
Keyword(s):  
Eigenvalue Problems ◽  
Projection Methods

Analysis And Efficient Solution Of Stationary Schrödinger Equation Governing Electronic States Of Quantum Dots And Rings In Magnetic Field

2012 ◽  
Vol 11 (5) ◽  
pp. 1591-1617 ◽  
Author(s):  
Marta M. Betcke ◽  
Heinrich Voss
Keyword(s):  
Magnetic Field ◽  
Eigenvalue Problems ◽  
Rotational Symmetry ◽  
Electronic States ◽  
Nonlinear Problems ◽  
Projection Methods ◽  
Computational Time ◽  
Arnoldi Method ◽  
Nonlinear Eigenvalue Problems ◽  
The One

AbstractIn this work the one-band effective Hamiltonian governing the electronic states of a quantum dot/ring in a homogenous magnetic field is used to derive a pair/quadruple of nonlinear eigenvalue problems corresponding to different spin orientations and in case of rotational symmetry additionally to quantum number -L•i. We show, that each of those pair/quadruple of nonlinear problems allows for the min-max characterization of its eigenvalues under certain conditions, which are satisfied for our examples and the common InAs/GaAs heterojunction. Exploiting the minmax property we devise efficient iterative projection methods simultaneously handling the pair/quadruple of nonlinear problems and thereby saving up to 40% of the computational time as compared to the nonlinear Arnoldi method applied to each of the problems separately.

scholarly journals A New Algorithm for Solving Large-Scale Generalized Eigenvalue Problem Based on Projection Methods

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
F. Abbasi Nedamani ◽  
A. H. Refahi Sheikhani ◽  
H. Saberi Najafi
Keyword(s):  
Numerical Stability ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Projection Methods ◽  
Generalized Eigenvalue ◽  
Generalized Eigenvalue Problems ◽  
Ritz Procedure ◽  
The Moment ◽  
Block Version ◽  
Methods Numerical

In this paper, we consider four methods for determining certain eigenvalues and corresponding eigenvectors of large-scale generalized eigenvalue problems which are located in a certain region. In these methods, a small pencil that contains only the desired eigenvalue is derived using moments that have obtained via numerical integration. Our purpose is to improve the numerical stability of the moment-based method and compare its stability with three other methods. Numerical examples show that the block version of the moment-based (SS) method with the Rayleigh–Ritz procedure has higher numerical stability than respect to other methods.

RESTARTING PROJECTION METHODS FOR RATIONAL EIGENPROBLEMS ARISING IN FLUID‐SOLID VIBRATIONS

2008 ◽  
Vol 13 (2) ◽  
pp. 171-182 ◽  
Author(s):  
Marta M. Betcke ◽  
Heinrich Voss
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Projection Methods ◽  
Nonlinear Eigenvalue Problems ◽  
Solid Structure ◽  
Minmax Characterization ◽  
Nonlinear Eigenvalue

For nonlinear eigenvalue problems T(λ)x = 0 satisfying a minmax characterization of its eigenvalues iterative projection methods combined with safeguarded iteration are suitable for computing all eigenvalues in a given interval. Such methods hit their limitations if a large number of eigenvalues is required. In this paper we discuss restart procedures which are able to cope with this problem, and we evaluate them for a rational eigenvalue problem governing vibrations of a fluid‐solid structure.

Structural-Acoustic Vibration Problems in the Presence of Strong Coupling

2012 ◽  
Vol 135 (1) ◽  
Author(s):  
Heinrich Voss ◽  
Markus Stammberger
Keyword(s):  
Eigenvalue Problems ◽  
Acoustic Vibration ◽  
Projection Methods ◽  
Free Vibrations ◽  
Coupled Systems ◽  
Approximation Properties ◽  
Strongly Coupled ◽  
Weakly Coupled ◽  
Weakly Coupled Systems

Free vibrations of fluid–solid structures are governed by unsymmetric eigenvalue problems. A common approach which works fine for weakly coupled systems is to project the problem to a space spanned by modes of the uncoupled system. For strongly coupled systems, however, the approximation properties are not satisfactory. This paper reports on a framework for taking advantage of the structure of the unsymmetric eigenvalue problem allowing for a variational characterization of its eigenvalues and structure preserving iterative projection methods. We further cover an adjusted automated multilevel substructuring (AMLS) method for huge fluid–solid structures. The reliability and efficiency of the method are demonstrated by the free vibrations of a structure completely filled with water.

Solving Huge Gyroscopic Eigenproblems With AMLS and Subspace Iteration

2012 ◽  
Author(s):  
Heinrich Voss ◽  
Jiacong Yin ◽  
Pu Chen
Keyword(s):  
Structural Analysis ◽  
Orthogonal Projection ◽  
Eigenvalue Problems ◽  
Projection Methods ◽  
Powerful Technique ◽  
Subspace Iteration

The Automated Multilevel Sub-structuring (AMLS) method is a powerful technique for computing a large number of eigenpairs with moderate accuracy for huge definite eigenvalue problems in structural analysis. It also turned out to be a useful tool to construct a suitable ansatz space for orthogonal projection methods for gyroscopic problems. This paper takes advantage of information gained from AMLS to improve the obtained eigenpairs via a small number of subspace iteration steps.

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