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Variants of thick-restart Lanczos for the Bethe-Salpeter eigenvalue problem

Fernando Alvarruiz, Blanca Mellado-Pinto, Jose E. Roman

TL;DR

Three variants of structure-preserving Lanczos eigensolvers are devised to compute a subset of eigenvalues (those of either smallest or largest magnitude) together with their corresponding right and left eigenvectors together with their corresponding right and left eigenvectors.

Abstract

The non-Hermitian Bethe-Salpeter eigenvalue problem is a structured eigenproblem, with real eigenvalues coming in pairs $\{λ,-λ\}$ where the corresponding pair of eigenvectors are closely related, and furthermore the left eigenvectors can be trivially obtained from the right ones. We exploit these properties to devise three variants of structure-preserving Lanczos eigensolvers to compute a subset of eigenvalues (those of either smallest or largest magnitude) together with their corresponding right and left eigenvectors. For this to be effective in real applications, we need to incorporate a thick-restart technique in a way that the overall computation preserves the problem structure. The new methods are validated in an implementation within the SLEPc library using several test matrices, some of them coming from the Yambo materials science code.

Variants of thick-restart Lanczos for the Bethe-Salpeter eigenvalue problem

TL;DR

Three variants of structure-preserving Lanczos eigensolvers are devised to compute a subset of eigenvalues (those of either smallest or largest magnitude) together with their corresponding right and left eigenvectors together with their corresponding right and left eigenvectors.

Abstract

The non-Hermitian Bethe-Salpeter eigenvalue problem is a structured eigenproblem, with real eigenvalues coming in pairs where the corresponding pair of eigenvectors are closely related, and furthermore the left eigenvectors can be trivially obtained from the right ones. We exploit these properties to devise three variants of structure-preserving Lanczos eigensolvers to compute a subset of eigenvalues (those of either smallest or largest magnitude) together with their corresponding right and left eigenvectors. For this to be effective in real applications, we need to incorporate a thick-restart technique in a way that the overall computation preserves the problem structure. The new methods are validated in an implementation within the SLEPc library using several test matrices, some of them coming from the Yambo materials science code.

Paper Structure

This paper contains 28 sections, 2 theorems, 85 equations, 2 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Problem definition
  3. The Bethe--Salpeter eigenproblem
  4. Possible solution approaches and related work
  5. Thick restart Lanczos
  6. Method based on Shao et al
  7. Lanczos recurrence
  8. Reorthogonalization
  9. Restart
  10. Obtaining approximate eigentriplets
  11. Method based on Grüning et al
  12. Lanczos recurrence
  13. Reorthogonalization
  14. Restart
  15. Obtaining the eigenvectors
  16. ...and 13 more sections

Key Result

Theorem 1

Let H be of the form eq:bse1 satisfying that $\hat{H}$ in eq:factored is positive definite. Then there exist $X_1$, $X_2\in\mathbb{C}^{n\times n}$ and positive numbers $\lambda_1,\lambda_2,\dots,\lambda_n\in\mathbb{R}$ such that where and $\Lambda_+=\operatorname{diag}\{\lambda_1,\dots,\lambda_n\}$.

Figures (2)

  • Figure 1: Illustration of the steps of thick restart in the Grüning method: (1) initial Lanczos factorization of order $k$, (2) solve projected problem, sort and check convergence, (3) truncate to factorization of order $r$, and (4) extend to a factorization of order $k$.
  • Figure 2: Parallel execution time on CPU for the pentadiag large test on the left, and CrI$_3$ large test on the right, for an increasing number of MPI processes.

Theorems & Definitions (3)

  • Theorem 1: Shao et al. Shao:2016:SPP
  • Proposition 1
  • proof