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Efficient solution of sequences of parametrized Lyapunov equations with applications

Davide Palitta, Zoran Tomljanović, Ivica Nakić, Jens Saak

TL;DR

Two novel numerical procedures are proposed that address problems where the parameter dependency of the coefficient matrix is encoded as a low‐rank modification to a seed, fixed matrix and are superior to state‐of‐the‐art techniques as they are able to remarkably speed up the computation of accurate solutions.

Abstract

Sequences of parametrized Lyapunov equations can be encountered in many application settings. Moreover, solutions of such equations are often intermediate steps of an overall procedure whose main goal is the computation of $\text{trace}(EX)$ where $X$ denotes the solution of a Lyapunov equation and $E$ is a given matrix. We are interested in addressing problems where the parameter dependency of the coefficient matrix is encoded as a low-rank modification to a \emph{seed}, fixed matrix. We propose two novel numerical procedures that fully exploit such a common structure. The first one builds upon the Sherman-Morrison-Woodbury (SMW) formula and recycling Krylov techniques, and it is well-suited for small dimensional problems as it makes use of dense numerical linear algebra tools. The second algorithm can instead address large-scale problems by relying on state-of-the-art projection techniques based on the extended Krylov subspace. We test the new algorithms on several problems arising in the study of damped vibrational systems and the analyses of output synchronization problems for multi-agent systems. Our results show that the algorithms we propose are superior to state-of-the-art techniques as they are able to remarkably speed up the computation of accurate solutions.

Efficient solution of sequences of parametrized Lyapunov equations with applications

TL;DR

Two novel numerical procedures are proposed that address problems where the parameter dependency of the coefficient matrix is encoded as a low‐rank modification to a seed, fixed matrix and are superior to state‐of‐the‐art techniques as they are able to remarkably speed up the computation of accurate solutions.

Abstract

Sequences of parametrized Lyapunov equations can be encountered in many application settings. Moreover, solutions of such equations are often intermediate steps of an overall procedure whose main goal is the computation of where denotes the solution of a Lyapunov equation and is a given matrix. We are interested in addressing problems where the parameter dependency of the coefficient matrix is encoded as a low-rank modification to a \emph{seed}, fixed matrix. We propose two novel numerical procedures that fully exploit such a common structure. The first one builds upon the Sherman-Morrison-Woodbury (SMW) formula and recycling Krylov techniques, and it is well-suited for small dimensional problems as it makes use of dense numerical linear algebra tools. The second algorithm can instead address large-scale problems by relying on state-of-the-art projection techniques based on the extended Krylov subspace. We test the new algorithms on several problems arising in the study of damped vibrational systems and the analyses of output synchronization problems for multi-agent systems. Our results show that the algorithms we propose are superior to state-of-the-art techniques as they are able to remarkably speed up the computation of accurate solutions.
Paper Structure (10 sections, 1 theorem, 86 equations, 7 figures, 3 tables, 2 algorithms)

This paper contains 10 sections, 1 theorem, 86 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction and setting
  2. Synopsis of the paper:
  3. The novel solution methods
  4. Small-scale setting: SMW and recycling Krylov methods
  5. Large-scale setting: a projection framework
  6. Applications
  7. Damped vibrational systems
  8. Multi-agent systems
  9. Numerical examples
  10. Conclusions

Key Result

Proposition 2.1

The backward error $\Delta_{m,\mathsf{v}}=\frac{\|R_{m}(\mathsf{v})\|_F}{2\|A(\mathsf{v})\|_F\|X_{\delta,m}(\mathsf{v})\|_F+\|P(\mathcal{I}_2\otimes D(\mathsf{v}))P^{\mkern-1.5mu\mathsf{T}}\|_F}$ provided by the approximate solution $X_{\delta,m}(\mathsf{v})=V_{m}Y_{m}(\mathsf{v})V_{m}^{\mkern-1.5mu where $a=\mathop{\mathrm{diag}}(B_{r}^{\mkern-1.5mu\mathsf{T}}A_0B_{l})$.

Figures (7)

  • Figure 1: Example \ref{['ex1']}, illustration of the mechanical system.
  • Figure 2: Example \ref{['ex1']}, the case a). Relative errors in the total average energy (squares) and in the viscosity (circles) at optimal gains for the SMW+recycling Krylov method.
  • Figure 3: Example \ref{['ex1']}, case a). Acceleration factors in the overall minimization procedure attained by employing our novel recycling Krylov approach.
  • Figure 4: Example \ref{['ex1']}, case b). Relative errors in the total average energy (squares) and in the viscosity (circles) at optimal gains for the projection framework.
  • Figure 5: Example \ref{['ex1']}, case b). Acceleration factors in the overall minimization procedure attained by employing our new projection framework.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Proposition 2.1
  • proof
  • Example 4.1
  • Example 4.2
  • Example 4.3