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.

中文翻译：

---

线性随机Galerkin系统的保稳模型降阶。

数学建模通常在科学和工程学中产生线性动力学系统。我们将系统的物理参数更改为随机变量以执行不确定性量化。随机Galerkin方法产生较大的线性动力学系统，其解表示随机过程的近似值。Galerkin系统的模型降阶（MOR）由于具有较高的维度而具有优势。但是，在某些MOR技术中可能会失去渐近稳定性。在Galerkin型MOR方法中，可以通过转换为耗散形式来保证稳定性。原始动力系统或随机Galerkin系统均可进行转换。我们研究了这种保持稳定性方法的两个变体。两种技术都是可行的，同时在数值方法中具有不同的特性。