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Interior Eigensolver Based on Rational Filter with Composite rule

Yuer Chen, Yingzhou Li

TL;DR

This paper proves the asymptotic optimality of the trapezoidal quadrature of the contour integral in terms of the rational function separation and develops two interior eigensolvers based on the composite rule.

Abstract

Contour-integral-based rational filter leads to interior eigensolvers for non-Hermitian generalized eigenvalue problems. Based on Zolotarev's third problem, this paper proves the asymptotic optimality of the trapezoidal quadrature of the contour integral in terms of the spectrum separation. A composite rule of the trapezoidal quadrature is derived, and two interior eigensolvers are proposed based on it. Both eigensolvers adopt direct factorization and multi-shift generalized minimal residual method for the inner and outer rational functions, respectively. The first eigensolver fixes the order of the outer rational function and applies the subspace iterations to achieve convergence, whereas the second eigensolver doubles the order of the outer rational function every iteration to achieve convergence without subspace iteration. The efficiency and stability of proposed eigensolvers are demonstrated on synthetic and practical sparse matrix pencils.

Interior Eigensolver Based on Rational Filter with Composite rule

TL;DR

This paper proves the asymptotic optimality of the trapezoidal quadrature of the contour integral in terms of the rational function separation and develops two interior eigensolvers based on the composite rule.

Abstract

Contour-integral-based rational filter leads to interior eigensolvers for non-Hermitian generalized eigenvalue problems. Based on Zolotarev's third problem, this paper proves the asymptotic optimality of the trapezoidal quadrature of the contour integral in terms of the spectrum separation. A composite rule of the trapezoidal quadrature is derived, and two interior eigensolvers are proposed based on it. Both eigensolvers adopt direct factorization and multi-shift generalized minimal residual method for the inner and outer rational functions, respectively. The first eigensolver fixes the order of the outer rational function and applies the subspace iterations to achieve convergence, whereas the second eigensolver doubles the order of the outer rational function every iteration to achieve convergence without subspace iteration. The efficiency and stability of proposed eigensolvers are demonstrated on synthetic and practical sparse matrix pencils.
Paper Structure (25 sections, 6 theorems, 55 equations, 7 figures, 5 tables, 3 algorithms)

This paper contains 25 sections, 6 theorems, 55 equations, 7 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Contribution
  4. Organization
  5. Subspace iteration with rational filter
  6. Subspace iteration
  7. Contour-based filter and discretization
  8. Cost of implementation
  9. Two eigensolvers based on the composite rule
  10. The composite rule
  11. Interior eigensolver with subspace iteration
  12. Composite rule eigensolver without subspace iteration
  13. Asymptotically optimal contour discretization
  14. Spectrum separation
  15. Zolotarev's third problem
  16. ...and 10 more sections

Key Result

Theorem 4.2

\newlabelthm:explicitzolo30 Let $\mathcal{S} = \{z \in \mathbb{C}: |z - \frac{1 + \ell}{2}| \le \frac{1 - \ell}{2}\}, 0 < \ell < 1$. Then the rational function attains the infimum of the Zolotarev's third problem $Z_k(\mathcal{S},-\mathcal{S})$ and the infimum equals to $(\frac{1 + \sqrt{\ell\,}}{1 - \sqrt{\ell\,}})^{-2k}$.

Figures (7)

  • Figure 1: Regions in Zolotarev's third problem when $E$ and $G$ are symmetric disks.
  • Figure 1: We plot the mapping on $[-2, 2] + [-2, 2] * \imath$. There are $201$ equally spaced points in the direction of the real part and the imaginary part, $40401$ points in total. The outer points are those $|z|>h=1.1$ and the inner points are those $|z|\le 1$ where the contour is $|z|=1$. We fix the figure window at $[-3, 3]+ [-3, 3]*\imath$ except for top right figure which is shown at $[-2.5, 3.5]+ [-3, 3]*\imath$. We let $k_1=k_2=8$ and the poles in all figures are the poles of $R_{k_1k_2}(z)$. The original eigengap is almost invisible, see the top left figure. From the top right figure, $R_8(z)$ maps the inner part to be close to 1 while the outer part to be close to 0 and the poles are mapped to the line $\mathrm{Real}(z)=0.5$. A more clear comparison of pre and post-mapping eigengaps are shown as the difference between the top left figure and bottom left figure. The composite mapping successfully maps the outer part close to 0 and the inner part close to 1 or modulus greater than 1, see bottom right figure.
  • Figure 1: The separation ratio \ref{['eq: ratio']} for various quadrature rules and numbers of poles. The number of poles $k$ ranges from 2 to 128. The trapezoidal quadrature shows the same slope as the optimal ratio, while the Gauss quadrature behaves differently.
  • Figure 1: Normalized solving cost in the composite rule. In each test matrix, the bars show the number of triangular solves in each subspace iteration, which are normalized by the number of triangular solves in the first subspace iteration.
  • Figure 2: (a) Patterns of $G$ and $C$ when $n_x = 10$; (b) Eigenvalues distribution.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Definition 4.1
  • Theorem 4.2
  • Lemma 4.3
  • Proof 1
  • Theorem 4.4
  • Proof 2
  • Corollary 4.5
  • Theorem 5.1
  • Proof 3
  • Proposition 5.2
  • ...and 2 more