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  • A Chebyshev Locally Optimal Block Preconditioned Conjugate Gradient Method for Product and Standard Symmetric Eigenvalue Problems
Title: A Chebyshev Locally Optimal Block Preconditioned Conjugate Gradient Method for Product and Standard Symmetric Eigenvalue Problems
The discretized Bethe-Salpeter eigenvalue (BSE) problem arises in many body physics and quantum chemistry. Discretization leads to an {\color{black}algebraic} eigenvalue problem involving a matrix $$H\in \mathbb{C}^{2n\times 2n}$$ with a Hamiltonian-like structure. With proper transformations, {\color{black}the real BSE eigenproblem of form I and the complex BSE eigenproblem of form II} can be transformed into real product eigenvalue problems of order $$n$$ and $2n$, respectively. We propose a new variant of the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) {\color{black}enhanced with polynomial filters} to improve the robustness and effectiveness of a few well-known algorithms for computing the lowest eigenvalues of the product eigenproblems. {\color{black}Furthermore, our proposed method can be easily employed to solve large sparse standard symmetric eigenvalue problems.} We show that our ideal locally optimal algorithm delivers Rayleigh quotient approximation to the desired lowest eigenvalue that satisfies a global quasi-optimality, which is similar to the global optimality of the preconditioned conjugate gradient method for the iterative solution of a symmetric positive definite linear system. The robustness and efficiency of {\color{black}our proposed method} is illustrated by numerical experiments.  more » « less
Award ID(s):
2111496
PAR ID:
10650157
Author(s) / Creator(s):
 ;  
Publisher / Repository:
SIAM
Date Published:
Journal Name:
SIAM Journal on Matrix Analysis and Applications
Volume:
45
Issue:
4
ISSN:
0895-4798
Page Range / eLocation ID:
2211 to 2242
Subject(s) / Keyword(s):
Bethe–Salpeter eigenvalue (BSE) problem, structure preserving, polynomial filter, LOBPCG, Chebyshev-Davidson, global quasi-optimality
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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