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A preconditioned inverse iteration with an improved convergence guarantee

Foivos Alimisis, Daniel Kressner, Nian Shao, Bart Vandereycken

TL;DR

This work provides a non-asymptotic convergence guarantee for a PINVIT-like method by recasting preconditioned inverse iteration as Riemannian steepest descent on the sphere. Convergence is shown under a weaker initial-vector condition expressed via a distortion angle $\varphi$, with the rate nearly matching classical PINVIT and two practical preconditioner classes analyzed. The results yield global convergence when the preconditioner yields small $\varphi$, as demonstrated for Additive Schwarz and mixed-precision preconditioners, and are supported by extensive numerical experiments on Laplace eigenproblems and kernel matrices. The approach offers a framework for accelerated and robust preconditioned eigenvalue solvers with potential extensions to other preconditioned eigensolvers and problem classes.

Abstract

Preconditioned eigenvalue solvers offer the possibility to incorporate preconditioners for the solution of large-scale eigenvalue problems, as they arise from the discretization of partial differential equations. The convergence analysis of such methods is intricate. Even for the relatively simple preconditioned inverse iteration (PINVIT), which targets the smallest eigenvalue of a symmetric positive definite matrix, the celebrated analysis by Neymeyr is highly nontrivial and only yields convergence if the starting vector is fairly close to the desired eigenvector. In this work, we prove a new non-asymptotic convergence result for a variant of PINVIT. Our proof proceeds by analyzing an equivalent Riemannian steepest descent method and leveraging convexity-like properties. We show a convergence rate that nearly matches the one of PINVIT. As a major benefit, we require a condition on the starting vector that tends to be less stringent. This improved global convergence property is demonstrated for two classes of preconditioners with theoretical bounds and a range of numerical experiments.

A preconditioned inverse iteration with an improved convergence guarantee

TL;DR

This work provides a non-asymptotic convergence guarantee for a PINVIT-like method by recasting preconditioned inverse iteration as Riemannian steepest descent on the sphere. Convergence is shown under a weaker initial-vector condition expressed via a distortion angle , with the rate nearly matching classical PINVIT and two practical preconditioner classes analyzed. The results yield global convergence when the preconditioner yields small , as demonstrated for Additive Schwarz and mixed-precision preconditioners, and are supported by extensive numerical experiments on Laplace eigenproblems and kernel matrices. The approach offers a framework for accelerated and robust preconditioned eigenvalue solvers with potential extensions to other preconditioned eigensolvers and problem classes.

Abstract

Preconditioned eigenvalue solvers offer the possibility to incorporate preconditioners for the solution of large-scale eigenvalue problems, as they arise from the discretization of partial differential equations. The convergence analysis of such methods is intricate. Even for the relatively simple preconditioned inverse iteration (PINVIT), which targets the smallest eigenvalue of a symmetric positive definite matrix, the celebrated analysis by Neymeyr is highly nontrivial and only yields convergence if the starting vector is fairly close to the desired eigenvector. In this work, we prove a new non-asymptotic convergence result for a variant of PINVIT. Our proof proceeds by analyzing an equivalent Riemannian steepest descent method and leveraging convexity-like properties. We show a convergence rate that nearly matches the one of PINVIT. As a major benefit, we require a condition on the starting vector that tends to be less stringent. This improved global convergence property is demonstrated for two classes of preconditioners with theoretical bounds and a range of numerical experiments.

Paper Structure

This paper contains 20 sections, 12 theorems, 104 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. PINVIT as steepest descent
  3. Quality of preconditioner
  4. Global: spectral equivalence
  5. Local: angle of distortion
  6. Convergence analysis
  7. Smoothness-type property
  8. Quadratic growth
  9. Weak-quasi-convexity
  10. Weak-quasi-strong-convexity
  11. Convergence analysis
  12. Distortion angle for specific preconditioners
  13. Additive Schwarz preconditioners
  14. Mixed-precision preconditioners
  15. Numerical experiments
  16. ...and 5 more sections

Key Result

Proposition 2

Consider the iterates $x_t$ produced by the recurrence algoSD with a step size satisfying eq:stepsizen. Then the transformed vectors $u_{t}:=B^{-1/2}x_{t}$ satisfy the recurrence with a certain step size $\eta_{t}^{*}>0$, a normalization $\beta_{t+1}>0$ chosen such that $\lVert u_{t+1}\rVert_{B}=1$, and the residual $r_{t} = Au_{t}-\lambda(u_{t})u_{t}$.

Figures (2)

  • Figure 1: Angle of distortion $\varphi$. Vectors $x$ in the white region satisfy $\mathop{\mathrm{dist}}\nolimits(x,x_{*})<\varphi$.
  • Figure 2: Construction of an overlapping domain decomposition. Example and figure taken from shao2023riemannian.

Theorems & Definitions (27)

  • Remark 1
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Remark 5
  • Proposition 6
  • proof
  • ...and 17 more