# Stability-preserving model order reduction for linear stochastic Galerkin systems

## TL;DR

This paper investigates methods to preserve stability in model order reduction of linear stochastic Galerkin systems, ensuring reliable reduced models for high-dimensional uncertainty quantification in engineering applications.

## Contribution

It compares two stability-preserving transformations for Galerkin systems, analyzing their properties and demonstrating their effectiveness through numerical examples.

## Key findings

- Both transformation techniques effectively preserve stability.
- Numerical results show accurate reduced models for mechanical and electrical systems.
- Different properties of the two methods influence their numerical performance.

## Abstract

Mathematical modeling often yields linear dynamical systems in science and engineering. We change physical parameters of the system into random variables to perform an uncertainty quantification. The stochastic Galerkin method yields a larger linear dynamical system, whose solution represents an approximation of random processes. A model order reduction (MOR) of the Galerkin system is advantageous due to the high dimensionality. However, asymptotic stability may be lost in some MOR techniques. In Galerkin-type MOR methods, the stability can be guaranteed by a transformation to a dissipative form. Either the original dynamical system or the stochastic Galerkin system can be transformed. We investigate the two variants of this stability-preserving approach. Both techniques are feasible, while featuring different properties in numerical methods. Results of numerical computations are demonstrated for two test examples modeling a mechanical application and an electric circuit, respectively.