Balanced Truncation Model Reduction for a Class of Descriptor Systems with Application to the Oseen Equations

TLDR

The paper aims to extend balanced truncation model reduction to descriptor systems such as semidiscrete Oseen equations and to show how ADI/Smith‑type methods can be used without explicit projection. The method applies balanced truncation to descriptor systems by eliminating algebraic constraints via projection and alternatively by solving saddle‑point problems to avoid explicit projection. The technique effectively produces reduced‑order models for semidiscrete Oseen equations.

Abstract

We discuss the computation of balanced truncation model reduction for a class of descriptor systems which include the semidiscrete Oseen equations with time-independent advection and the linearized Navier–Stokes equations, linearized around a steady state. The purpose of this paper is twofold. First, we show how to apply standard balanced truncation model reduction techniques, which apply to dynamical systems given by ordinary differential equations, to this class of descriptor systems. This is accomplished by eliminating the algebraic equation using a projection. The second objective of this paper is to demonstrate how the important class of ADI/Smith-type methods for the approximate computation of reduced order models using balanced truncation can be applied without explicitly computing the aforementioned projection. Instead, we utilize the solution of saddle point problems. We demonstrate the effectiveness of the technique in the computation of reduced order models for semidiscrete Oseen equations.