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Balanced truncation with conformal maps

Alessandro Borghi, Tobias Breiten, Serkan Gugercin

TL;DR

The paper addresses reduced-order modeling for large-scale LTI systems whose poles lie in general domains $\mathbb{A}\subset\mathbb{C}$, not restricted to traditional regions. It develops a conformal-mapping balanced truncation framework that leverages a conformal map $\psi$ to transform the problem and defines Gramians in the conformal domain via the transformed operator $\mathfrak{H}_{\mathbf{G}}$, enabling projection-based reduction in this generalized setting. For Möbius transformations, the Gramians satisfy modified Lyapunov equations $m^{-1}(\mathbf{A})\mathbf{X}_c+\mathbf{X}_c m^{-1}(\mathbf{A})^* = -\mathbf{Q}_c$ and $\mathbf{X}_o m^{-1}(\mathbf{A}) + m^{-1}(\mathbf{A})^*\mathbf{X}_o = -\mathbf{Q}_o$, yielding a practical conformalBT algorithm with stability guarantees and an $\mathcal{H}_2$-type error bound. The method is validated on heat, Schrödinger, and undamped wave equations, showing accurate reduced models even when spectra lie on the imaginary axis. Overall, the work extends BT to pole domains beyond classical regions, enabling stable, high-fidelity reductions for PDE-like systems with nonstandard spectral domains.

Abstract

We consider the problem of constructing reduced models for large scale systems with poles in general domains in the complex plane (as opposed to, e.g., the open left-half plane or the open unit disk). Our goal is to design a model reduction scheme, building upon theoretically established methodologies, yet encompassing this new class of models. To this aim, we develop a balanced truncation framework through conformal maps to handle poles in general domains. The major difference from classical balanced truncation resides in the formulation of the Gramians. We show that these new Gramians can still be computed by solving modified Lyapunov equations for specific conformal maps. A numerical algorithm to perform balanced truncation with conformal maps is developed and is tested on three numerical examples, namely a heat model, the Schrödinger equation, and the undamped linear wave equation, the latter two having spectra on the imaginary axis.

Balanced truncation with conformal maps

TL;DR

The paper addresses reduced-order modeling for large-scale LTI systems whose poles lie in general domains , not restricted to traditional regions. It develops a conformal-mapping balanced truncation framework that leverages a conformal map to transform the problem and defines Gramians in the conformal domain via the transformed operator , enabling projection-based reduction in this generalized setting. For Möbius transformations, the Gramians satisfy modified Lyapunov equations and , yielding a practical conformalBT algorithm with stability guarantees and an -type error bound. The method is validated on heat, Schrödinger, and undamped wave equations, showing accurate reduced models even when spectra lie on the imaginary axis. Overall, the work extends BT to pole domains beyond classical regions, enabling stable, high-fidelity reductions for PDE-like systems with nonstandard spectral domains.

Abstract

We consider the problem of constructing reduced models for large scale systems with poles in general domains in the complex plane (as opposed to, e.g., the open left-half plane or the open unit disk). Our goal is to design a model reduction scheme, building upon theoretically established methodologies, yet encompassing this new class of models. To this aim, we develop a balanced truncation framework through conformal maps to handle poles in general domains. The major difference from classical balanced truncation resides in the formulation of the Gramians. We show that these new Gramians can still be computed by solving modified Lyapunov equations for specific conformal maps. A numerical algorithm to perform balanced truncation with conformal maps is developed and is tested on three numerical examples, namely a heat model, the Schrödinger equation, and the undamped linear wave equation, the latter two having spectra on the imaginary axis.
Paper Structure (15 sections, 6 theorems, 84 equations, 5 figures, 1 algorithm)

This paper contains 15 sections, 6 theorems, 84 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. A conformal mapping framework
  5. The H_2(barAĉ) space
  6. Balanced truncation
  7. Balanced truncation with conformal maps
  8. Defining the Gramians
  9. Lyapunov equations
  10. Stability preservation and $\mathcal{H}_2$ error bound
  11. Numerical experiments
  12. Heat equation
  13. Schrödinger equation
  14. Undamped linear wave equation
  15. Conclusions

Key Result

Theorem 1

Suppose $\mathbb{X},\mathbb{Y}\subset\mathbb{C}$ are open sets and let $\psi\colon\mathbb{X}\rightarrow\mathbb{Y}$ be Fréchet differentiable as a function of two real variables. The mapping $\psi$ is conformal in $\mathbb{X}$ if and only if it is analytic in $\mathbb{X}$ and $\psi'(z_0)\neq 0$ for e

Figures (5)

  • Figure 1: An illustration of a conformal map satisfying \ref{['assumption:1']}. The arrows between the grey sets $\mathbb{X}$ and $\mathbb{A}$ indicate the bijectivity of $\psi$. The same holds for the dashed arrow lines between the dashed boundaries $\mathrm{i}\mathbb{R}$ and $\partial\mathbb{A}^+$ (in this depiction $\partial\mathbb{A}^+$ coincides with $\partial\mathbb{A}$). The dots $\bullet$ indicate the poles of the transfer function $\mathbf{G}$. In addition, $\psi$ conformally maps the white sets $\tilde{\mathbb{X}}$ and $\bar{\mathbb{A}}^{\mathsf{c}}$.
  • Figure 2: (Top) $\mathcal{H}_2(\bar{\mathbb{A}}^{\mathsf{c}})$ error norm between the full and reduced order systems along with the $\mathcal{H}_2$ error bound in \ref{['eq:H2bound']} for different values of $r$. (Bottom) relative error of the ROM impulse response with $r=10$ computed with conformalBT and BT applied to the Heat equation with $n=200$.
  • Figure 3: (Top) $\mathcal{H}_2(\bar{\mathbb{A}}^{\mathsf{c}})$ error norm between the full and reduced order systems along with the $\mathcal{H}_2$ error bound in \ref{['eq:H2bound']} for different values of $r$. (Middle) relative error of the ROM step response with $r=9$ computed with conformalBT applied to the Schrödinger equation with $n=1000$. (Bottom) Input adopted for evaluating conformalBT. Here, $u^{(1)}$ and $u^{(2)}$ follow the same trajectory equal to $\mathbf{u}$.
  • Figure 4: A depiction of the conformal map in \ref{['eq:joukowskiwave']} centered at the origin ($c=0$). The grey sets on the left and on the right are, respectively, the domain and range of $\psi$. Here we have the scaling being $M\in\mathrm{i}\mathbb{R}$. The thick line in $\tilde{\mathbb{B}}_{M,c}$ indicates the strip $[-1,1]$ after the scaling.
  • Figure 5: (Top) impulse response of the full and reduced order systems, $y^{(i)}$ and $y_r^{(i)}$, respectively, with $i=1,2$. Here conformalBT computed an $r=40$ reduced order model on a discretized wave equation with $n=5000$. (Bottom) the output error.

Theorems & Definitions (14)

  • Theorem 1: Weg12, Theorem 6.1.2
  • Definition 1: $\mathcal{H}_2(\bar{\mathbb{A}}^{\mathsf{c}})$ space, BorBre23
  • Lemma 1: Unique solution
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • ...and 4 more