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Low-rank-modified Galerkin methods for the Lyapunov equation

Kathryn Lund, Davide Palitta

Abstract

Of all the possible projection methods for solving large-scale Lyapunov matrix equations, Galerkin approaches remain much more popular than minimal-residual ones. This is mainly due to the different nature of the projected problems stemming from these two families of methods. While a Galerkin approach leads to the solution of a low-dimensional matrix equation per iteration, a matrix least-squares problem needs to be solved per iteration in a minimal-residual setting. The significant computational cost of these least-squares problems has steered researchers towards Galerkin methods in spite of the appealing properties of minimal-residual schemes. In this paper we introduce a framework that allows for modifying the Galerkin approach by low-rank, additive corrections to the projected matrix equation problem with the two-fold goal of attaining monotonic convergence rates similar to those of minimal-residual schemes while maintaining essentially the same computational cost of the original Galerkin method. We analyze the well-posedness of our framework and determine possible scenarios where we expect the residual norm attained by two low-rank-modified variants to behave similarly to the one computed by a minimal-residual technique. A panel of diverse numerical examples shows the behavior and potential of our new approach.

Low-rank-modified Galerkin methods for the Lyapunov equation

Abstract

Of all the possible projection methods for solving large-scale Lyapunov matrix equations, Galerkin approaches remain much more popular than minimal-residual ones. This is mainly due to the different nature of the projected problems stemming from these two families of methods. While a Galerkin approach leads to the solution of a low-dimensional matrix equation per iteration, a matrix least-squares problem needs to be solved per iteration in a minimal-residual setting. The significant computational cost of these least-squares problems has steered researchers towards Galerkin methods in spite of the appealing properties of minimal-residual schemes. In this paper we introduce a framework that allows for modifying the Galerkin approach by low-rank, additive corrections to the projected matrix equation problem with the two-fold goal of attaining monotonic convergence rates similar to those of minimal-residual schemes while maintaining essentially the same computational cost of the original Galerkin method. We analyze the well-posedness of our framework and determine possible scenarios where we expect the residual norm attained by two low-rank-modified variants to behave similarly to the one computed by a minimal-residual technique. A panel of diverse numerical examples shows the behavior and potential of our new approach.
Paper Structure (11 sections, 7 theorems, 55 equations, 12 figures, 1 table, 1 algorithm)

This paper contains 11 sections, 7 theorems, 55 equations, 12 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Low-rank modification framework
  3. A Pseudo-Minimal Residual Method
  4. Connections to the minimal residual method
  5. A residual-minimizing Kronecker sum approach
  6. Compress-and-restart strategy
  7. Numerical results
  8. Convergence behavior
  9. Performance relative to block size
  10. Restarting
  11. Conclusions and outlook

Key Result

Proposition 1

Let $Y_m^{\hbox{\normalfont\tiny\sffamily mod}}$ be the solution to the Lyapunov equation eq:lyap_mod_galerkin. Then the residual matrix $R_m^{\hbox{\normalfont\tiny\sffamily mod}} = A X_m^{\hbox{\normalfont\tiny\sffamily mod}} + X_m^{\hbox{\normalfont\tiny\sffamily mod}} A^* + \bm{C} \bm{C}^*$ can where and $\underline{\mathcal{I}} = \in \mathbb{R}^{(m+1)r \times mr}$. Moreover,

Figures (12)

  • Figure 1: Relative residuals with respect to iteration index for various methods and $A$ as a 1D Laplacian. "M-diff" is $\frac{\left\lVert\bm{M}^{\hbox{\normalfont\tiny\sffamily PMR}} - \bm{M}^{\hbox{\normalfont\tiny\sffamily NKS}}\right\rVert_2}{\left\lVert\bm{M}^{\hbox{\normalfont\tiny\sffamily PMR}}\right\rVert_2}$.
  • Figure 2: Convergence results for Example \ref{['ex:log_diag_rand_n1000_r3']}.
  • Figure 3: Convergence results for Example \ref{['ex:bad_cond_diag_rand_n500_r3']}.
  • Figure 4: Convergence results for Example \ref{['ex:laplacian_2d_rand_n100_r3']}.
  • Figure 5: Convergence results for Example \ref{['ex:conv_diff_3d_rand_n25_r3']}.
  • ...and 7 more figures

Theorems & Definitions (24)

  • Proposition 1
  • proof
  • Lemma 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • ...and 14 more