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.

翻译：本文研究针对极点位于复平面一般区域（而非仅限于开左半平面或开单位圆盘等特定区域）的大规模系统构建降阶模型的问题。我们的目标是基于已有理论方法设计一种模型降阶方案，使其能够涵盖此类新型模型。为此，我们通过共形映射建立平衡截断框架来处理一般区域的极点问题。该方法与经典平衡截断的主要区别在于Gramian矩阵的构造形式。我们证明这些新型Gramian矩阵仍可通过求解特定共形映射下的修正Lyapunov方程来计算。本文开发了基于共形映射的平衡截断数值算法，并在三个数值算例中进行了验证：热传导模型、薛定谔方程以及无阻尼线性波动方程，其中后两者的谱位于虚轴上。