Minimization principles and computation for the generalized linear response eigenvalue problem

2014 ◽  
Vol 54 (1) ◽  
pp. 31-54 ◽  
Author(s):  
Zhaojun Bai ◽  
Ren-Cang Li
Keyword(s):  
Eigenvalue Problem ◽  
Linear Response

scholarly journals Rayleigh-Ritz Majorization Error Bounds for the Linear Response Eigenvalue Problem

2019 ◽  
Vol 17 (1) ◽  
pp. 653-667
Author(s):  
Zhongming Teng ◽  
Hong-Xiu Zhong
Keyword(s):  
Eigenvalue Problem ◽  
Quantum Chemistry ◽  
Error Bounds ◽  
Linear Response ◽  
Computational Quantum Chemistry ◽  
Matrix Anal ◽  
Deflating Subspaces ◽  
Important Notion ◽  
Theoretical Foundations ◽  
Computational Quantum

Abstract In the linear response eigenvalue problem arising from computational quantum chemistry and physics, one needs to compute a few of smallest positive eigenvalues together with the corresponding eigenvectors. For such a task, most of efficient algorithms are based on an important notion that is the so-called pair of deflating subspaces. If a pair of deflating subspaces is at hand, the computed approximated eigenvalues are partial eigenvalues of the linear response eigenvalue problem. In the case the pair of deflating subspaces is not available, only approximate one, in a recent paper [SIAM J. Matrix Anal. Appl., 35(2), pp.765-782, 2014], Zhang, Xue and Li obtained the relationships between the accuracy in eigenvalue approximations and the distances from the exact deflating subspaces to their approximate ones. In this paper, we establish majorization type results for these relationships. From our majorization results, various bounds are readily available to estimate how accurate the approximate eigenvalues based on information on the approximate accuracy of a pair of approximate deflating subspaces. These results will provide theoretical foundations for assessing the relative performance of certain iterative methods in the linear response eigenvalue problem.

scholarly journals Weighted Block Golub-Kahan-Lanczos Algorithms for Linear Response Eigenvalue Problem

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 53 ◽  
Author(s):  
Hongxiu Zhong ◽  
Zhongming Teng ◽  
Guoliang Chen
Keyword(s):  
Eigenvalue Problem ◽  
Error Bounds ◽  
Linear Response ◽  
Numerical Instability ◽  
Lanczos Algorithm ◽  
Numerical Examples ◽  
Block Algorithm ◽  
Restart Strategy ◽  
New Algorithms

In order to solve all or some eigenvalues lied in a cluster, we propose a weighted block Golub-Kahan-Lanczos algorithm for the linear response eigenvalue problem. Error bounds of the approximations to an eigenvalue cluster, as well as their corresponding eigenspace, are established and show the advantages. A practical thick-restart strategy is applied to the block algorithm to eliminate the increasing computational and memory costs, and the numerical instability. Numerical examples illustrate the effectiveness of our new algorithms.

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 FEAST Algorithm for the Linear Response Eigenvalue Problem

Algorithms ◽  
2019 ◽  
Vol 12 (9) ◽  
pp. 181 ◽  
Author(s):  
Zhongming Teng ◽  
Linzhang Lu
Keyword(s):  
Eigenvalue Problem ◽  
Quantum Chemistry ◽  
Linear Response ◽  
Contour Integration ◽  
Numerical Examples ◽  
Complex Contour ◽  
Convergence Results

In the linear response eigenvalue problem arising from quantum chemistry and physics, one needs to compute several positive eigenvalues together with the corresponding eigenvectors. For such a task, in this paper, we present a FEAST algorithm based on complex contour integration for the linear response eigenvalue problem. By simply dividing the spectrum into a collection of disjoint regions, the algorithm is able to parallelize the process of solving the linear response eigenvalue problem. The associated convergence results are established to reveal the accuracy of the approximated eigenspace. Numerical examples are presented to demonstrate the effectiveness of our proposed algorithm.

Minimization Principles for the Linear Response Eigenvalue Problem II: Computation

2013 ◽  
Vol 34 (2) ◽  
pp. 392-416 ◽  
Author(s):  
Zhaojun Bai ◽  
Ren-Cang Li
Keyword(s):  
Eigenvalue Problem ◽  
Linear Response

scholarly journals Convergence analysis of Lanczos-type methods for the linear response eigenvalue problem

2013 ◽  
Vol 247 ◽  
pp. 17-33 ◽  
Author(s):  
Zhongming Teng ◽  
Ren-Cang Li
Keyword(s):  
Eigenvalue Problem ◽  
Convergence Analysis ◽  
Linear Response

Backward perturbation analysis and residual-based error bounds for the linear response eigenvalue problem

2014 ◽  
Vol 55 (3) ◽  
pp. 869-896 ◽  
Author(s):  
Lei-Hong Zhang ◽  
Wen-Wei Lin ◽  
Ren-Cang Li
Keyword(s):  
Eigenvalue Problem ◽  
Error Bounds ◽  
Perturbation Analysis ◽  
Linear Response ◽  
Backward Perturbation

Minimization Principles for the Linear Response Eigenvalue Problem I: Theory

2012 ◽  
Vol 33 (4) ◽  
pp. 1075-1100 ◽  
Author(s):  
Zhaojun Bai ◽  
Ren-Cang Li
Keyword(s):  
Eigenvalue Problem ◽  
Linear Response
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