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Model Reduction of Parametric Differential-Algebraic Systems by Balanced Truncation

Jennifer Przybilla, Matthias Voigt

TL;DR

This work develops a balanced truncation-based model reduction framework for parameter-dependent differential-algebraic equations (DAEs). It tackles the computational bottleneck by coupling projected Lyapunov equations with the reduced basis method, enabling offline construction of low-dimensional Gramian representations and rapid online evaluation for many parameters. Two residual-based error estimators are introduced to quantify the approximation quality, and the method is extended to handle polynomial (algebraic) parts of the transfer function via Markov parameters. Numerical experiments on Stokes-like DAEs and constrained mechanical systems demonstrate substantial offline savings, fast online reductions, and high-fidelity transfer behavior across parameter samples.

Abstract

We deduce a procedure to apply balanced truncation to parameter-dependent differential-algebraic systems. For that we solve multiple projected Lyapunov equations for different parameter values to compute the Gramians that are required for the truncation procedure. As this process would lead to high computational costs if we perform it for a large number of parameters, we combine this approach with the reduced basis method that determines a reduced representation of the Lyapunov equation solutions for the parameters of interest. Residual-based error estimators are then used to evaluate the quality of the approximations. After introducing the procedure for a general class of differential-algebraic systems we turn our focus to systems with a specific structure, for which the method can be applied particularly efficiently. We illustrate the efficiency of our approach on several models from fluid dynamics and mechanics.

Model Reduction of Parametric Differential-Algebraic Systems by Balanced Truncation

TL;DR

This work develops a balanced truncation-based model reduction framework for parameter-dependent differential-algebraic equations (DAEs). It tackles the computational bottleneck by coupling projected Lyapunov equations with the reduced basis method, enabling offline construction of low-dimensional Gramian representations and rapid online evaluation for many parameters. Two residual-based error estimators are introduced to quantify the approximation quality, and the method is extended to handle polynomial (algebraic) parts of the transfer function via Markov parameters. Numerical experiments on Stokes-like DAEs and constrained mechanical systems demonstrate substantial offline savings, fast online reductions, and high-fidelity transfer behavior across parameter samples.

Abstract

We deduce a procedure to apply balanced truncation to parameter-dependent differential-algebraic systems. For that we solve multiple projected Lyapunov equations for different parameter values to compute the Gramians that are required for the truncation procedure. As this process would lead to high computational costs if we perform it for a large number of parameters, we combine this approach with the reduced basis method that determines a reduced representation of the Lyapunov equation solutions for the parameters of interest. Residual-based error estimators are then used to evaluate the quality of the approximations. After introducing the procedure for a general class of differential-algebraic systems we turn our focus to systems with a specific structure, for which the method can be applied particularly efficiently. We illustrate the efficiency of our approach on several models from fluid dynamics and mechanics.

Paper Structure

This paper contains 26 sections, 5 theorems, 99 equations, 13 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model problems
  3. Problem 1: Stokes-like DAEs
  4. Problem 2: Mechanical systems
  5. Preliminaries
  6. Decoupling of differential and algebraic equations
  7. Model reduction of differential-algebraic systems
  8. Balanced truncation
  9. Solving projected Lyapunov equations
  10. Alternative approach to determine the algebraic parts
  11. The index one case
  12. The index two case
  13. The index three case
  14. System realization from Markov parameters
  15. Reduced basis method
  16. ...and 11 more sections

Key Result

Lemma 3.1

Let the matrix pencil $s\mathbf{E}-\mathbf{A}$ with $\mathbf{E},\,\mathbf{A} \in\mathbb{R}^{N, N}$ be regular and asymptotically stable. Let further the matrix $\mathbf{A}$ be nonsingular and $\mathbf{B} \in\mathbb{R}^{N, m}$. Assume that the left and right spectral projectors onto the finite spectr with is equivalent to the projected continuous-time Lyapunov equation eq:propLE_contr, i. e., thei

Figures (13)

  • Figure 1: Error and error estimates of the approximated controllability Gramians for the Stokes system \ref{['Index2']} after the first iteration of the reduced basis method.
  • Figure 2: Results for the reduction of the Stokes system \ref{['Index2']}.
  • Figure 3: Output and output error of the original and reduced Stokes system \ref{['Index2']}
  • Figure 4: Error and error estimates of the approximated Gramians for the Stokes system \ref{['Index2']} with improper parts.
  • Figure 5: Results for the reduction of the Stokes system \ref{['Index2']} with improper parts.
  • ...and 8 more figures

Theorems & Definitions (14)

  • Remark 1
  • Lemma 3.1
  • Proposition 3.2
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 4.2
  • proof
  • Remark 5
  • Definition 4.3
  • ...and 4 more