New scheme for large-scale eigenvalue problems of the RPA-type matrix equation

1992 ◽  
Vol 70 (2) ◽  
pp. 296-300 ◽  
Author(s):  
Susumu Narita ◽  
Tai-ichi Shibuya
Keyword(s):  
Eigenvalue Problem ◽  
Convergence Rate ◽  
Matrix Equation ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Type Equation ◽  
Fast Convergence ◽  
New Method ◽  
Eigenvalues And Eigenvectors ◽  
Numerical Tests

A new method is proposed for obtaining a few eigenvalues and eigenvectors of a large-scale RPA-type equation. Some numerical tests are carried out to study the convergence behaviors of this method. It is found that the convergence rate is very fast and quite satisfactory. It depends strongly on the way of estimating the deviation vectors. Our proposed scheme gives a better estimation for the deviation vectors than Davidson's scheme. This scheme is applicable to the eigenvalue problems of nondiagonally dominant matrices as well. Keywords: large-scale eigenvalue problem, RPA-type equation, fast convergence.

A New Method Tracing Load-Deflection Equilibrium Path of a Doubly Nonlinear Truss

2012 ◽  
Vol 166-169 ◽  
pp. 68-72
Author(s):  
Shu Tang Liu ◽  
Qi Liang Long
Keyword(s):  
Matrix Equation ◽  
Stiffness Matrix ◽  
Large Scale ◽  
New Method ◽  
Truss Structure ◽  
Equilibrium Path ◽  
Nodal Coordinates ◽  
Doubly Nonlinear ◽  
Global Stiffness Matrix

A new method tracing the load-deflection equilibrium path of a truss with doubly nonlinearity is proposed. The total global stiffness matrix equation has been formulated in terms of nodal coordinates, iteration formulations has been written through adopting a single control coordinate, so that an new method tracing the load-deflection equilibrium path has been proposed. Analysis results of Star dome truss and Schwedeler dome truss have shown that the proposed method is stable numerically, quick in convergence, high in degree of accuracy and easy in use. The proposed method can be used for large-scale truss structure.

scholarly journals Trace minimization method via penalty for linear response eigenvalue problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yadan Chen ◽  
Yuan Shen ◽  
Shanshan Liu
Keyword(s):  
Eigenvalue Problem ◽  
Linear Response ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Minimum Principle ◽  
Gradient Descent Method ◽  
Step Size ◽  
Minimization Method ◽  
Energy Excitation ◽  
Orthogonality Constraint

<p style='text-indent:20px;'>In various applications, such as the computation of energy excitation states of electrons and molecules, and the analysis of interstellar clouds, the linear response eigenvalue problem, which is a special type of the Hamiltonian eigenvalue problem, is frequently encountered. However, traditional eigensolvers may not be applicable to this problem owing to its inherently large scale. In fact, we are usually more interested in computing some of the smallest positive eigenvalues. To this end, a trace minimum principle optimization model with orthogonality constraint has been proposed. On this basis, we propose an unconstrained surrogate model called trace minimization via penalty, and we establish its equivalence with the original constrained model, provided that the penalty parameter is larger than a certain threshold. By avoiding the orthogonality constraint, we can use a gradient-type method to solve this model. Specifically, we use the gradient descent method with Barzilai–Borwein step size. Moreover, we develop a restarting strategy for the proposed algorithm whereby higher accuracy and faster convergence can be achieved. This is verified by preliminary experimental results.</p>

scholarly journals A Globally Convergent Matrix-Free Method for Constrained Equations and Its Linear Convergence Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Min Sun ◽  
Jing Liu
Keyword(s):  
Convergence Rate ◽  
Large Scale ◽  
Gradient Methods ◽  
Linear Convergence ◽  
New Method ◽  
Linear Convergence Rate ◽  
Relax Factor ◽  
Constrained Equations ◽  
Accelerate Convergence ◽  
Matrix Free

A matrix-free method for constrained equations is proposed, which is a combination of the well-known PRP (Polak-Ribière-Polyak) conjugate gradient method and the famous hyperplane projection method. The new method is not only derivative-free, but also completely matrix-free, and consequently, it can be applied to solve large-scale constrained equations. We obtain global convergence of the new method without any differentiability requirement on the constrained equations. Compared with the existing gradient methods for solving such problem, the new method possesses linear convergence rate under standard conditions, and a relax factorγis attached in the update step to accelerate convergence. Preliminary numerical results show that it is promising in practice.

scholarly journals An Approach to Some Non-Classical Eigenvalue Problems of Structural Dynamics

2015 ◽  
Vol 11 (4) ◽  
pp. 21-32
Author(s):  
Horea Sandi ◽  
Ioan Sorin Borcia (†)
Keyword(s):  
Eigenvalue Problem ◽  
Structural Dynamics ◽  
Dynamic Systems ◽  
Eigenvalue Problems ◽  
Constitutive Laws ◽  
Illustrative Case ◽  
Eigenvalues And Eigenvectors ◽  
Multiple Eigenvalues ◽  
Circular Frequency ◽  
Linear Problems

Abstract Two main shortcomings of common formulations, encountered in the literature concerning the linear problems of structural dynamics are revealed: the implicit, not discussed, postulation, of the use of Kelvin – Voigt constitutive laws (which is often infirmed by experience) and the calculation difficulties involved by the attempts to use other constitutive laws. In order to overcome these two categories of shortcomings, the use of the bilateral Laplace – Carson transformation is adopted. Instead of the dependence on time, t, of a certain function f (t), the dependence of its image f# (p) on the complex parameter p = χ + iω (ω: circular frequency) will occur. This leads to the formulation of associated non-classical eigenvalue problems. The basic relations satisfied by the eigenvalues λr#(p) and the eigenvectors vr#(p) of dynamic systems are examined (among other, the property of orthogonality of eigenvectors is replaced by the property of pseudo-orthogonality). The case of points p = p’, where multiple eigenvalues occur and where, as a rule, chains of principal vectors are to be considered, is discussed. An illustrative case, concerning a non-classical eigenvalue problem, is presented. Plots of variation along the ω axis, for the real and imaginary components of eigenvalues and eigenvectors, are presented. A brief final discussion closes the paper.

scholarly journals Local defect-correction method based on multilevel discretization for Steklov eigenvalue problem

2021 ◽  
Author(s):  
Fei Xu ◽  
Liu Chen ◽  
Qiumei Huang
Keyword(s):  
Eigenvalue Problem ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Correction Method ◽  
Small Scale ◽  
Local Defect ◽  
Defect Correction ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Steklov Eigenvalue Problems

In this paper, we propose a local defect-correction method for solving the Steklov eigenvalue problem arising from the scalar second order positive definite partial differential equations based on the multilevel discretization. The objective is to avoid solving large-scale equations especially the large-scale Steklov eigenvalue problem whose computational cost increases exponentially. The proposed algorithm transforms the Steklov eigenvalue problem into a series of linear boundary value problems, which are defined in a multigrid space sequence, and a series of small-scale Steklov eigenvalue problems in a coarse correction space. Furthermore, we use the local defect-correction technique to divide the large-scale boundary value problems into small-scale subproblems. Through our proposed algorithm, we avoid solving large-scale Steklov eigenvalue problems. As a result, our proposed algorithm demonstrates significantly improved the solving efficiency. Additionally, we conduct numerical experiments and a rigorous theoretical analysis to verify the effectiveness of our proposed approach.

scholarly journals A Prediction-Correction Dynamic Method for Large-Scale Generalized Eigenvalue Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Xin-long Luo ◽  
Jia-ru Lin ◽  
Wei-ling Wu
Keyword(s):  
Dynamical System ◽  
Eigenvalue Problem ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Correction Method ◽  
Differential Algebraic Equations ◽  
Euler Method ◽  
Generalized Eigenvalue Problem ◽  
Algebraic Equations ◽  
Generalized Eigenvalue

This paper gives a new prediction-correction method based on the dynamical system of differential-algebraic equations for the smallest generalized eigenvalue problem. First, the smallest generalized eigenvalue problem is converted into an equivalent-constrained optimization problem. Second, according to the Karush-Kuhn-Tucker conditions of this special equality-constrained problem, a special continuous dynamical system of differential-algebraic equations is obtained. Third, based on the implicit Euler method and an analogous trust-region technique, a prediction-correction method is constructed to follow this system of differential-algebraic equations to compute its steady-state solution. Consequently, the smallest generalized eigenvalue of the original problem is obtained. The local superlinear convergence property for this new algorithm is also established. Finally, in comparison with other methods, some promising numerical experiments are presented.

New Harvesting and Curing Techniques in the Production of Breeder's Seed Peanuts1

1979 ◽  
Vol 6 (2) ◽  
pp. 70-72
Author(s):  
T. A. Coffelt ◽  
F. S. Wright ◽  
J. L. Steele
Keyword(s):  
Large Scale ◽  
Combine Method ◽  
Frost Damage ◽  
New Method ◽  
Experimental Equipment ◽  
Harvesting Method ◽  
Curing Methods ◽  
Labor Requirements ◽  
High Level ◽  
Varietal Purity

Abstract A new method of harvesting and curing breeder's seed peanuts in Virginia was initiated that would 1) reduce the labor requirements, 2) maintain a high level of germination, 3) maintain varietal purity at 100%, and 4) reduce the risk of frost damage. Three possible harvesting and curing methods were studied. The traditional stack-pole method satisfied the latter 3 objectives, but not the first. The windrow-combine method satisfied the first 2 objectives, but not the last 2. The direct harvesting method satisfied all four objectives. The experimental equipment and curing procedures for direct harvesting had been developed but not tested on a large scale for seed harvesting. This method has been used in Virginia to produce breeder's seed of 3 peanut varieties (Florigiant, VA 72R and VA 61R) during five years. Compared to the stackpole method, labor requirements have been reduced, satisfactory levels of germination and varietal purity have been obtained, and the risk of frost damage has been minimized.

scholarly journals TreeMerge: a new method for improving the scalability of species tree estimation methods

2019 ◽  
Vol 35 (14) ◽  
pp. i417-i426 ◽  
Author(s):  
Erin K Molloy ◽  
Tandy Warnow
Keyword(s):  
Large Scale ◽  
Species Tree ◽  
New Method ◽  
Divide And Conquer ◽  
Supplementary Information ◽  
Estimation Methods ◽  
Running Time ◽  
Tree Estimation ◽  
Computationally Intensive ◽  
A Minor

Abstract Motivation At RECOMB-CG 2018, we presented NJMerge and showed that it could be used within a divide-and-conquer framework to scale computationally intensive methods for species tree estimation to larger datasets. However, NJMerge has two significant limitations: it can fail to return a tree and, when used within the proposed divide-and-conquer framework, has O(n5) running time for datasets with n species. Results Here we present a new method called ‘TreeMerge’ that improves on NJMerge in two ways: it is guaranteed to return a tree and it has dramatically faster running time within the same divide-and-conquer framework—only O(n2) time. We use a simulation study to evaluate TreeMerge in the context of multi-locus species tree estimation with two leading methods, ASTRAL-III and RAxML. We find that the divide-and-conquer framework using TreeMerge has a minor impact on species tree accuracy, dramatically reduces running time, and enables both ASTRAL-III and RAxML to complete on datasets (that they would otherwise fail on), when given 64 GB of memory and 48 h maximum running time. Thus, TreeMerge is a step toward a larger vision of enabling researchers with limited computational resources to perform large-scale species tree estimation, which we call Phylogenomics for All. Availability and implementation TreeMerge is publicly available on Github (http://github.com/ekmolloy/treemerge). Supplementary information Supplementary data are available at Bioinformatics online.

scholarly journals A Generalized Inexact Newton Method for Inverse Eigenvalue Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Weiping Shen
Keyword(s):  
Newton Method ◽  
Quadratic Convergence ◽  
Eigenvalue Problems ◽  
Generalized Newton Method ◽  
Inexact Newton Method ◽  
Numerical Tests ◽  
Inverse Eigenvalue Problems ◽  
Inverse Eigenvalue ◽  
Generalized Newton ◽  
Special Case

We propose a generalized inexact Newton method for solving the inverse eigenvalue problems, which includes the generalized Newton method as a special case. Under the nonsingularity assumption of the Jacobian matrices at the solutionc*, a convergence analysis covering both the distinct and multiple eigenvalue cases is provided and the quadratic convergence property is proved. Moreover, numerical tests are given in the last section and comparisons with the generalized Newton method are made.

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