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Stabilization of a matrix via a low rank-adaptive ODE

Nicola Guglielmi, Stefano Sicilia

Abstract

Let $A$ be a square matrix with a given structure (e.g. real matrix, sparsity pattern, Toeplitz structure, etc.) and assume that it is unstable, i.e. at least one of its eigenvalues lies in the complex right half-plane. The problem of stabilizing $A$ consists in the computation of a matrix $B$, whose eigenvalues have negative real part and such that the perturbation $Δ=B-A$ has minimal norm. The structured stabilization further requires that the perturbation preserves the structural pattern of $A$. We solve this non-convex problem by a two-level procedure which involves the computation of the stationary points of a matrix ODE. We exploit the low rank underlying features of the problem by using an adaptive-rank integrator that follows slavishly the rank of the solution. We show the benefits derived from the low rank setting in several numerical examples, which also allow to deal with high dimensional problems.

Stabilization of a matrix via a low rank-adaptive ODE

Abstract

Let be a square matrix with a given structure (e.g. real matrix, sparsity pattern, Toeplitz structure, etc.) and assume that it is unstable, i.e. at least one of its eigenvalues lies in the complex right half-plane. The problem of stabilizing consists in the computation of a matrix , whose eigenvalues have negative real part and such that the perturbation has minimal norm. The structured stabilization further requires that the perturbation preserves the structural pattern of . We solve this non-convex problem by a two-level procedure which involves the computation of the stationary points of a matrix ODE. We exploit the low rank underlying features of the problem by using an adaptive-rank integrator that follows slavishly the rank of the solution. We show the benefits derived from the low rank setting in several numerical examples, which also allow to deal with high dimensional problems.
Paper Structure (15 sections, 6 theorems, 49 equations, 5 figures, 7 tables)

This paper contains 15 sections, 6 theorems, 49 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Gradient system
  3. A rank-adaptive integrator for the gradient system
  4. Rank adaptivity for fixed perturbation size
  5. An illustrative example
  6. Grcar matrix
  7. Smoke matrix
  8. Outer iteration: tuning the perturbation size
  9. Another functional
  10. Smoke matrix
  11. Structured distance via a low rank adaptive ODE
  12. Numerical examples for the structured case
  13. Pentadiagonal Toeplitz matrix
  14. Brusselator matrix
  15. Fidap matrix

Key Result

Lemma 2.1

Let $\lambda(t)$ be a simple eigenvalue of a differentiable matrix path $A(t)$ in a neighborhood of $t_0$ and let $x(t)$ and $y(t)$ be, respectively, the left and right unit eigenvectors associated. Then $x(t_0)^*y(t_0)\neq 0$ and

Figures (5)

  • Figure 4.1: Smoke matrix: functional and ranks in the inner iteration for $\tau=10^{-2}$ (up), $\tau=10^{-4}$ (middle) and $\tau=10^{-6}$ (down)
  • Figure 5.1: Smoke-like matrix \ref{['mat_smokelike']}: original eigenvalues (black circles) and stabilized ones (an eigenvalue $\lambda$ is green if $\textnormal{Re}(\lambda)<-\delta$, while orange if $-\delta\leq \textnormal{Re}(\lambda)\leq 0$). On the left the functional considered is $F_\varepsilon$, while on the right $\Phi_\varepsilon$.
  • Figure 7.1: Pentadiagonal Toeplitz matrix: original eigenvalues (black circles) and stabilized ones (an eigenvalue $\lambda$ is green if $\textnormal{Re}(\lambda)<-\delta$, while orange if $-\delta\leq \textnormal{Re}(\lambda)\leq 0$).
  • Figure 7.2: Brusselator matrix: original eigenvalues (black circles) and stabilized ones (an eigenvalue $\lambda$ is green if $\textnormal{Re}(\lambda)<-\delta$, while orange if $-\delta\leq \textnormal{Re}(\lambda)\leq 0$). On the right a zoom of the image of all the eigenvalues in the left.
  • Figure 7.3: Fidap matrix: on the left its structural pattern, on the right its original eigenvalues (black circles) and the stabilized ones (an eigenvalue $\lambda$ is green if $\textnormal{Re}(\lambda)<-\delta$, while orange if $-\delta\leq \textnormal{Re}(\lambda)\leq 0$). On the right a zoom of the image of all the eigenvalues in the left.

Theorems & Definitions (11)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • ...and 1 more