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ParaSLRF: A High Performance Rational Filter Method for Solving Large Scale Eigenvalue Problems

Biyi Wang, Karl Meerbergen, Raf Vandebril, Hengbin An, Zeyao Mo

TL;DR

This work tackles computing all eigenpairs in $\lambda\in(0,\gamma]$ for large-scale generalized eigenvalue problems arising in mechanical vibrations. It introduces ParaSLRF, a two-level parallel rational-filter method that uses shifted Laplace poles to enable efficient iterative solves of the linear systems $$(A-\sigma_j B)Y=BV$$ and a subsequent Rayleigh-Ritz projection on $U^T A U$ and $U^T B U$. Numerical experiments on FE vibration models show ParaSLRF delivering superior CPU time and load balancing compared to contour-based quadrature filters, largely by reducing MPI waiting and balancing inner solves across groups. The authors further enhance ParaSLRF with locking of converged eigenpairs and a scaled Ritz initial guess, plus a fixed inner-iteration cap, achieving 4.7x–9.1x speedups and strong two-level scalability, with recommendations to maximize Level-1 resources for best performance. Overall, the approach provides a robust, scalable framework for large SPD GEPs in engineering applications, outperforming existing rational-filter strategies.

Abstract

In \emph{Wang et al., A Shifted Laplace Rational Filter for Large-Scale Eigenvalue Problems}, the SLRF method was proposed to compute all eigenvalues of a symmetric definite generalized eigenvalue problem lying in an interval on the real positive axis. The current paper discusses a parallel implementation of this method, abbreviated as ParaSLRF. The parallelization consists of two levels: (1) on the highest level, the application of the rational filter to the various vectors is partitioned among groups of processors; (2) within each group, every linear system is solved in parallel. In ParaSLRF, the linear systems are solved by iterative methods instead of direct ones, in contrast to other rational filter methods, such as, PFEAST. Because of the specific selection of poles in ParaSLRF, the computational cost of solving the associated linear systems for each pole, is almost the same. This intrinsically leads to a better load balance between each group of resources, and reduces waiting times of processes. We show numerical experiments from finite element models of mechanical vibrations, and show a detailed parallel performance analysis. ParaSLRF shows the best parallel efficiency, compared to other rational filter methods based on quadrature rules for contour integration. To further improve performance, the converged eigenpairs are locked, and a good initial guess of iterative linear solver is proposed. These enhancements of ParaSLRF show good two-level strong scalability and excellent load balance in our experiments.

ParaSLRF: A High Performance Rational Filter Method for Solving Large Scale Eigenvalue Problems

TL;DR

This work tackles computing all eigenpairs in for large-scale generalized eigenvalue problems arising in mechanical vibrations. It introduces ParaSLRF, a two-level parallel rational-filter method that uses shifted Laplace poles to enable efficient iterative solves of the linear systems and a subsequent Rayleigh-Ritz projection on and . Numerical experiments on FE vibration models show ParaSLRF delivering superior CPU time and load balancing compared to contour-based quadrature filters, largely by reducing MPI waiting and balancing inner solves across groups. The authors further enhance ParaSLRF with locking of converged eigenpairs and a scaled Ritz initial guess, plus a fixed inner-iteration cap, achieving 4.7x–9.1x speedups and strong two-level scalability, with recommendations to maximize Level-1 resources for best performance. Overall, the approach provides a robust, scalable framework for large SPD GEPs in engineering applications, outperforming existing rational-filter strategies.

Abstract

In \emph{Wang et al., A Shifted Laplace Rational Filter for Large-Scale Eigenvalue Problems}, the SLRF method was proposed to compute all eigenvalues of a symmetric definite generalized eigenvalue problem lying in an interval on the real positive axis. The current paper discusses a parallel implementation of this method, abbreviated as ParaSLRF. The parallelization consists of two levels: (1) on the highest level, the application of the rational filter to the various vectors is partitioned among groups of processors; (2) within each group, every linear system is solved in parallel. In ParaSLRF, the linear systems are solved by iterative methods instead of direct ones, in contrast to other rational filter methods, such as, PFEAST. Because of the specific selection of poles in ParaSLRF, the computational cost of solving the associated linear systems for each pole, is almost the same. This intrinsically leads to a better load balance between each group of resources, and reduces waiting times of processes. We show numerical experiments from finite element models of mechanical vibrations, and show a detailed parallel performance analysis. ParaSLRF shows the best parallel efficiency, compared to other rational filter methods based on quadrature rules for contour integration. To further improve performance, the converged eigenpairs are locked, and a good initial guess of iterative linear solver is proposed. These enhancements of ParaSLRF show good two-level strong scalability and excellent load balance in our experiments.
Paper Structure (15 sections, 1 theorem, 11 equations, 8 figures, 4 tables, 2 algorithms)

This paper contains 15 sections, 1 theorem, 11 equations, 8 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Parallel rational filter method
  3. Data distribution and computation flow
  4. The choice of poles
  5. Numerical experiments
  6. Finite element model of vibration
  7. Pyramid model
  8. Pyramid model
  9. Hollow platform model and fish-like model
  10. Setting
  11. Results and efficiency analysis
  12. Enhanced ParaSLRF
  13. Strong scalability
  14. Fixed computing resources
  15. Conclusion

Key Result

Proposition 4.1

Given the pair $(\theta,\mathbf{v})\in\mathbb{R}\times(\mathbb{R}^n\setminus\{0\})$. Let $\mathcal{K}^{T^C}_m(\mathbf{A} - \sigma \mathbf{B},\mathbf{res}_C^{(0)})$ be the (preconditioned) Krylov subspace formed for solving the "Cayley transformation" linear system and let $\mathcal{K}^{T^{SI}}_m(\mathbf{A} - \sigma \mathbf{B},\mathbf{res}_{SI}^{(0)})$ be the Krylov space of dimension $m$ for solv

Figures (8)

  • Figure 1: The 2D representation of $np$ processes.
  • Figure 2: Computational flow of the parallel rational filter method at iteration step $k$, for $np=8, N=n_{part}=4$.
  • Figure 3: The shape of pyramid model.
  • Figure 4: The outer iterations (left) and CPU time (right) of the enhanced ParaSLRF with different $\rm It_{\text{max-linear}}$ for the pyramid S1 model ($N=36339$).
  • Figure 5: The outer iterations (left) and CPU time (right) of the enhanced ParaSLRF with different $\rm It_{\text{max-linear}}$ for hollow platform model ($N=37161$).
  • ...and 3 more figures

Theorems & Definitions (1)

  • Proposition 4.1