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Toward genuine efficiency and cluster robustness of preconditioned CG-like eigensolvers

Ming Zhou, Klaus Neymeyr

TL;DR

This work addresses efficient and robust computation of extreme and clustered eigenvalues for Hermitian problems of the form $Ax=\lambda Mx$. It proposes a cluster-robust framework that replaces the LOPCG three-term recurrence with asymptotically equivalent two-term recurrences and augments the trial subspace with a far-previous iterate, leading to fewer total steps. The authors introduce augmentation strategies (LOPCGa) and two-term PCG variants (TPCG/TPCGa) with residual- or peak-based timing to cope with spectral clustering, supported by convergence discussion and numerical experiments. Compared with methods like EPIC and RAP, the new schemes demonstrate reduced iteration counts and improved total runtime, especially for tightly clustered spectra, and maintain modest subspace dimensions (3–4). The results suggest practical gains for both single-vector and potential block extensions and provide a foundation for further theoretical and blockwise developments.

Abstract

The performance of eigenvalue problem solvers (eigensolvers) depends on various factors such as preconditioning and eigenvalue distribution. Developing stable and rapidly converging vectorwise eigensolvers is a crucial step in improving the overall efficiency of their blockwise implementations. The present paper is concerned with the locally optimal block preconditioned conjugate gradient (LOBPCG) method for Hermitian eigenvalue problems, and motivated by two recently proposed alternatives for its single-vector version LOPCG. A common basis of these eigensolvers is the well-known CG method for linear systems. However, the optimality of CG search directions cannot perfectly be transferred to CG-like eigensolvers. In particular, while computing clustered eigenvalues, LOPCG and its alternatives suffer from frequent delays, leading to a staircase-shaped convergence behavior which cannot be explained by the existing estimates. Keeping this in mind, we construct a class of cluster robust vector iterations where LOPCG is replaced by asymptotically equivalent two-term recurrences and the search directions are timely corrected by selecting a far previous iterate as augmentation. The new approach significantly reduces the number of required steps and the total computational time.

Toward genuine efficiency and cluster robustness of preconditioned CG-like eigensolvers

TL;DR

This work addresses efficient and robust computation of extreme and clustered eigenvalues for Hermitian problems of the form . It proposes a cluster-robust framework that replaces the LOPCG three-term recurrence with asymptotically equivalent two-term recurrences and augments the trial subspace with a far-previous iterate, leading to fewer total steps. The authors introduce augmentation strategies (LOPCGa) and two-term PCG variants (TPCG/TPCGa) with residual- or peak-based timing to cope with spectral clustering, supported by convergence discussion and numerical experiments. Compared with methods like EPIC and RAP, the new schemes demonstrate reduced iteration counts and improved total runtime, especially for tightly clustered spectra, and maintain modest subspace dimensions (3–4). The results suggest practical gains for both single-vector and potential block extensions and provide a foundation for further theoretical and blockwise developments.

Abstract

The performance of eigenvalue problem solvers (eigensolvers) depends on various factors such as preconditioning and eigenvalue distribution. Developing stable and rapidly converging vectorwise eigensolvers is a crucial step in improving the overall efficiency of their blockwise implementations. The present paper is concerned with the locally optimal block preconditioned conjugate gradient (LOBPCG) method for Hermitian eigenvalue problems, and motivated by two recently proposed alternatives for its single-vector version LOPCG. A common basis of these eigensolvers is the well-known CG method for linear systems. However, the optimality of CG search directions cannot perfectly be transferred to CG-like eigensolvers. In particular, while computing clustered eigenvalues, LOPCG and its alternatives suffer from frequent delays, leading to a staircase-shaped convergence behavior which cannot be explained by the existing estimates. Keeping this in mind, we construct a class of cluster robust vector iterations where LOPCG is replaced by asymptotically equivalent two-term recurrences and the search directions are timely corrected by selecting a far previous iterate as augmentation. The new approach significantly reduces the number of required steps and the total computational time.
Paper Structure (8 sections, 4 theorems, 56 equations, 8 figures, 2 tables, 5 algorithms)

This paper contains 8 sections, 4 theorems, 56 equations, 8 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. LOPCG and comparable schemes
  3. Benchmark method
  4. Practical eigensolvers
  5. Numerical comparisons
  6. Convergence theory of LOPCG
  7. Two-term PCG (TPCG) schemes
  8. Typical generalized eigenvalue problems

Key Result

Lemma 2.1

Consider Algorithm apcg with $T$ from prec and define $B=T^{1/2}A_{\lambda_1}T^{1/2}$. Let $V$ be an arbitrary orthonormal basis matrix of the image $\hbox{im}(B)$ of $B$, then the vectors $\tilde{x}^{(i)}=V^*T^{-1/2}x^{(i)}$, $i=0,1,\ldots$ are CG-iterates for solving the linear system $\tilde{B}\t

Figures (8)

  • Figure 2.1: Selecting typical cases by observing the convergence history of the heuristic PCG (Algorithm \ref{['apcg']}) for examples \ref{['sp1']} and preconditioners in Table \ref{['tab']}.
  • Figure 2.2: Numerical comparisons between CG-like eigensolvers for the example (\ref{['sp1']}, a); see Section \ref{['sec2c']} for details. The subplots include three cases where LOPCG implementations are entirely/not/partially inferior to EPIC implementations. The augmented scheme LOPCGa (Algorithm \ref{['alopcga']}) is clearly more efficient.
  • Figure 2.3: Numerical comparisons between CG-like eigensolvers for the examples (\ref{['sp1']}, b, c); see Section \ref{['sec2c']} for details. The subplots particularly indicate the cluster robustness of LOPCGa (Algorithm \ref{['alopcga']}) in the sense that the residual norm reduction is straightforward or oscillates more weakly.
  • Figure 3.1: Convergence phenomena beyond the available estimates for CG-like eigensolvers (Remark \ref{['rmka']}). Upper row: convergence history of $\lambda^{(i)}-\lambda_1$ in comparison to $\lambda_2-\lambda_1$ and $(\lambda_2-\lambda_1)/2$ (horizontal lines) which roughly mark the beginning of the final phase. Lower row: single-step convergence factor $(\lambda^{(i+1)}-\lambda_1)/(\lambda^{(i)}-\lambda_1)$ in comparison to the estimated values $\xi$ and $\psi^2$ (horizontal lines) from \ref{['lmpsde1']} and \ref{['lopcge']}.
  • Figure 3.2: Limitation of asymptotic estimates for LOPCG (Remark \ref{['rmkc']}). The asymptotic terms $\delta_1$, $\delta_2$, $\delta_3$ for \ref{['lmlopcge']}, \ref{['lmlopcge1']}, \ref{['gde']} can substantially deviate from the estimated size $\mathcal{O}(\sqrt{\lambda^{(i)}-\lambda_1}\,)$ for clustered eigenvalues (middle and right subplots).
  • ...and 3 more figures

Theorems & Definitions (15)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 5 more