Balanced truncation with conformal maps

TL;DR

This paper introduces a novel balanced truncation method using conformal maps to create reduced models for large systems with poles in arbitrary complex domains, extending classical techniques.

Contribution

It develops a new balanced truncation framework based on conformal maps, allowing model reduction for systems with poles in general complex domains, with practical algorithms and numerical validation.

Findings

Successfully applied to heat, Schrödinger, and wave equations.

Able to handle poles on the imaginary axis.

Provides a computational method for new Gramians.

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 Schr\"odinger equation, and the undamped linear wave equation, the latter two having spectra on the imaginary axis.