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Estimating the condition number of Chebyshev filtered vectors with application to the ChASE library

Edoardo Di Napoli, Xinzhe Wu

TL;DR

This work shows how a mechanism for the choice of QR-factorization in the ChASE library can be bound from above with precise and inexpensive estimates, and shows how such mechanism enhance the performance of the library without compromising on its accuracy.

Abstract

Chebyshev filtered subspace iteration is a well-known algorithm for the solution of (symmetric/Hermitian) algebraic eigenproblems which has been implemented in several application codes~\cite{Kronik:2006ff, abinit:2020} or in stand alone libraries~\cite{ChASE}. An essential part of the algorithm is the QR-factorization of the array of vectors spanning the active subspace that have been filtered by the Chebyshev filter. Typically such an array has an a-priori unknown high condition number that directly influences the choice of QR-factorization algorithm. In this work we show how such condition number can be bound from above with precise and inexpensive estimates. We then proceed to use these estimates to implement a mechanism for the choice of QR-factorization in the ChASE library. We show how such mechanism enhance the performance of the library without compromising on its accuracy.

Estimating the condition number of Chebyshev filtered vectors with application to the ChASE library

TL;DR

This work shows how a mechanism for the choice of QR-factorization in the ChASE library can be bound from above with precise and inexpensive estimates, and shows how such mechanism enhance the performance of the library without compromising on its accuracy.

Abstract

Chebyshev filtered subspace iteration is a well-known algorithm for the solution of (symmetric/Hermitian) algebraic eigenproblems which has been implemented in several application codes~\cite{Kronik:2006ff, abinit:2020} or in stand alone libraries~\cite{ChASE}. An essential part of the algorithm is the QR-factorization of the array of vectors spanning the active subspace that have been filtered by the Chebyshev filter. Typically such an array has an a-priori unknown high condition number that directly influences the choice of QR-factorization algorithm. In this work we show how such condition number can be bound from above with precise and inexpensive estimates. We then proceed to use these estimates to implement a mechanism for the choice of QR-factorization in the ChASE library. We show how such mechanism enhance the performance of the library without compromising on its accuracy.
Paper Structure (21 sections, 7 theorems, 56 equations, 3 figures, 3 tables, 4 algorithms)

This paper contains 21 sections, 7 theorems, 56 equations, 3 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Filtered subspace iteration: its inception and modern revival
  3. A modern eigensolver based on Chebyshev Accelerated Subspace iteration
  4. Replacing Householder QR with Communication-Avoiding CholeskyQR Variants
  5. Condition number estimates and dynamic CAQR
  6. Spectral decomposition of the subspace filtered by Chebyshev polynomials
  7. Chebyshev polynomials and their asymptotic properties
  8. Filtering vectors with Chebyshev polynomials
  9. Spectral decomposition of the filtered vectors
  10. The case of locked and deflated vectors
  11. Bounding the condition number of an array of filtered vectors
  12. Single polynomial degree and no locking mechanism
  13. Optimized polynomial degree
  14. Locking and deflating the vectors
  15. Numerical experiments
  16. ...and 6 more sections

Key Result

Theorem 2.2

Let $\left|\gamma\right|>1$ and $\mathbb{P}_m$ denote the set of polynomials of degree smaller or equal to $m$. Then the extremum is reached by

Figures (3)

  • Figure 1: Schematic use of the Chebyshev polynomial $C_m(\lambda)$ to enhance the components $V$ aligned to the eigenvectors corresponding to the desired portion of the spectrum $[\lambda_1, \lambda_{\sf nev}]$ (in green) of an Hermitian matrix $A$. Conversely the components of $V$ aligned with eigenpairs in the interval $[\lambda_{\sf nev}, \lambda_N]$ (in red) are suppressed. Here $\lambda_N$ indicates the largest eigenvalue of $A$.
  • Figure 1: Condition estimates per iteration for various materials with and without optimization. Subplots (a)--(i) show results for different material systems.
  • Figure 2: Strong-scaling performance and parallel efficiency comparison between CholeskyQR and Householder QR across configurations. Each measurement was repeated five times. Runtime plots show the average execution time, with shaded bands indicating the minimum–maximum range across runs. Parallel efficiency is computed from the averaged runtimes.

Theorems & Definitions (15)

  • Definition 2.1: Chebyshev Polynomials
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proof 1
  • Lemma 3.1
  • Proof 2
  • Proposition 3.2
  • Proof 3
  • Lemma 3.3
  • ...and 5 more