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Adaptive Polynomial Filtering for Hermitian Interior Eigenproblems: Convergence Analysis

Xiaofei Xu, Yuhui Ni, Shengguo Li, Juan Zhang

Abstract

Interior eigenvalue problems for large-scale sparse Hermitian matrices are fundamental in computational science. We propose an adaptive polynomial filtering strategy based on Chebyshev expansion of a step function, integrated into a filtered subspace iteration framework. We establish pointwise convergence bounds in both undamped and damped settings and incorporate an enhanced spurious eigenvalue detection technique to improve efficiency and robustness. At the implementation level, we employ MaSpMM to accelerate the polynomial filtering step. Numerical results demonstrate the efficiency and robustness of the proposed method compared with classical approaches.

Adaptive Polynomial Filtering for Hermitian Interior Eigenproblems: Convergence Analysis

Abstract

Interior eigenvalue problems for large-scale sparse Hermitian matrices are fundamental in computational science. We propose an adaptive polynomial filtering strategy based on Chebyshev expansion of a step function, integrated into a filtered subspace iteration framework. We establish pointwise convergence bounds in both undamped and damped settings and incorporate an enhanced spurious eigenvalue detection technique to improve efficiency and robustness. At the implementation level, we employ MaSpMM to accelerate the polynomial filtering step. Numerical results demonstrate the efficiency and robustness of the proposed method compared with classical approaches.

Paper Structure

This paper contains 16 sections, 5 theorems, 52 equations, 6 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Filtered Subspace Iteration framework
  4. Polynomial Filtering
  5. FSI with Adaptive Polynomial Filtering
  6. Adaptive Polynomial Filter
  7. Accuracy of the filter
  8. Algorithm and Implementation Details
  9. Efficient SpMM Implementation
  10. Spurious Eigenvalues
  11. Deflation
  12. Other details
  13. Numerical experiments
  14. Sparse Matrices from real applications
  15. Parameter Sensitivity Analysis
  16. ...and 1 more sections

Key Result

Lemma 2.1

\newlabellem:convergence0 jia2023feastsaad2011numerical Assume $|\gamma_e^{(i)}| > |\gamma_{p+1}^{(i)}|$ and that $V_p^* \hat{V}^{(0)}$ is nonsingular. Define Then, the following statements hold: $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: Errors and error bounds for \ref{['eq:damped_fourier_error_bound']}.
  • Figure 1: The relative runtime and SpMVs of AdaPolySI and CJ-FEAST over EVSL.
  • Figure 2: (a) The CSR SpMM; (b) The J-Stream workflow. The column-segment format is used in this work, in which each column-segment within a row block of $\mathbf{A}$ is stored contiguously; (c) The MaSpMM workflow. The row-segment format is further adopted in this work, in which the dense matrix is partitioned into column blocks, and each row-segment within a column block is stored contiguously.
  • Figure 2: Runtime and average degree of AdaPolySI for different values of $m$ and $\tau_a$.
  • Figure 3: Ritz values and residuals for the VCNT4000 with interval $[-0.12,0.01]$.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Lemma 2.1
  • Lemma 3.1
  • Proof 1
  • Theorem 3.2
  • Proof 2
  • Remark 3.3
  • Theorem 3.4
  • Proof 3
  • Theorem 3.5
  • Proof 4