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An increasing rank Riemannian method for generalized Lyapunov equations

Zhenwei Huang, Wen Huang

Abstract

In this paper, we consider finding a low-rank approximation to the solution of a large-scale generalized Lyapunov matrix equation in the form of $A X M + M X A = C$, where $A$ and $M$ are symmetric positive definite matrices. An algorithm called an Increasing Rank Riemannian Method for Generalized Lyapunov Equation (IRRLyap) is proposed by merging the increasing rank technique and Riemannian optimization techniques on the quotient manifold $\mathbb{R}_*^{n \times p} / \mathcal{O}_p$. To efficiently solve the optimization problem on $\mathbb{R}_*^{n \times p} / \mathcal{O}_p$, a line-search-based Riemannian inexact Newton's method is developed with its global convergence and local superlinear convergence rate guaranteed. Moreover, we derive a preconditioner which takes $M \neq I$ into consideration. Numerical experiments show that the proposed Riemannian inexact Newton's method and preconditioner has superior performance and IRRLyap is preferable compared to the tested state-of-the-art methods when the lowest rank solution is desired.

An increasing rank Riemannian method for generalized Lyapunov equations

Abstract

In this paper, we consider finding a low-rank approximation to the solution of a large-scale generalized Lyapunov matrix equation in the form of , where and are symmetric positive definite matrices. An algorithm called an Increasing Rank Riemannian Method for Generalized Lyapunov Equation (IRRLyap) is proposed by merging the increasing rank technique and Riemannian optimization techniques on the quotient manifold . To efficiently solve the optimization problem on , a line-search-based Riemannian inexact Newton's method is developed with its global convergence and local superlinear convergence rate guaranteed. Moreover, we derive a preconditioner which takes into consideration. Numerical experiments show that the proposed Riemannian inexact Newton's method and preconditioner has superior performance and IRRLyap is preferable compared to the tested state-of-the-art methods when the lowest rank solution is desired.
Paper Structure (47 sections, 8 theorems, 99 equations, 2 figures, 4 tables, 6 algorithms)

This paper contains 47 sections, 8 theorems, 99 equations, 2 figures, 4 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Low-rank methods for Lyapunov equations.
  5. Riemannian Newton's methods.
  6. Outline
  7. Preliminaries and Notation
  8. Riemannian geometry.
  9. The considered manifold.
  10. Function classes.
  11. Notations.
  12. Problem Statement
  13. A Riemannian Truncated Newton's Method
  14. Algorithm Statement
  15. Convergence Analysis
  16. ...and 32 more sections

Key Result

Theorem 4.1

If $g_x(\mathrm{Hess}f(x)[d^{(i)}], d^{(i)})>\epsilon \delta^{(i)}$, $i=0,1,\dots,k$, then

Figures (2)

  • Figure 1: Compare RNewton, RTRNewton, RCG, and LRBFGS methods. For RCG, VT1 and VT2 respectively means that the vector transport is given vector transport by parallelization and by projection. For LRBFGS, $m$ means memory size.
  • Figure 2: The relative residual for one-rank right-hand-side RAIL benchmark with $n=1357$.

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Theorem 4.1
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • Proposition 5.1
  • ...and 5 more