# An Iterative Model Reduction Scheme for Quadratic-Bilinear Descriptor Systems with an Application to Navier-Stokes Equations

## TL;DR

This paper develops an efficient iterative model reduction method for quadratic-bilinear descriptor systems, extending interpolation-based techniques to these systems and demonstrating effectiveness on Navier-Stokes equations.

## Contribution

It introduces a transformation approach to apply QBODE reduction techniques to descriptor systems, avoiding explicit projector computation for improved efficiency.

## Key findings

- Successfully reduces Navier-Stokes based systems
- Produces near-optimal reduced models
- Enhances model reduction reliability for descriptor systems

## Abstract

We discuss model reduction for a particular class of quadratic-bilinear (QB) descriptor systems. The main goal of this article is to extend the recently studied interpolation-based optimal model reduction framework for QBODEs [Benner et al. '16] to a class of descriptor systems in an efficient and reliable way. Recently, it has been shown in the case of linear or bilinear systems that a direct extension of interpolation-based model reduction techniques to descriptor systems, without any modifications, may lead to poor reduced-order systems. Therefore, for the analysis, we aim at transforming the considered QB descriptor system into an equivalent QBODE system by means of projectors for which standard model reduction techniques for QBODEs can be employed, including aforementioned interpolation scheme. Subsequently, we discuss related computational issues, thus resulting in a modified algorithm that allows us to construct \emph{near}--optimal reduced-order systems without explicitly computing the projectors used in the analysis. The efficiency of the proposed algorithm is illustrated by means of a numerical example, obtained via semi-discretization of the Navier-Stokes equations.