We consider linear first-order systems of ordinary differential equations (ODEs) in port-Hamiltonian (pH) form. Physical parameters are remodeled as random variables to conduct an uncertainty quantification. A stochastic Galerkin projection yields a larger deterministic system of ODEs, which does not exhibit a pH form in general. We apply transformations of the original systems such that the stochastic Galerkin projection becomes structure-preserving. Furthermore, we investigate meaning and properties of the Hamiltonian function belonging to the stochastic Galerkin system. A large number of random variables implies a high-dimensional stochastic Galerkin system, which suggests itself to apply model order reduction (MOR) generating a low-dimensional system of ODEs. We discuss structure preservation in projection-based MOR, where the smaller systems of ODEs feature pH form again. Results of numerical computations are presented using two test examples.

中文翻译：

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线性一阶常微分方程的随机伽略金法和波特哈密顿形式

我们考虑端口哈密尔顿 (pH) 形式的常微分方程 (ODE) 的线性一阶系统。物理参数被重新建模为随机变量以进行不确定性量化。随机伽辽金投影产生更大的确定性 ODE 系统，该系统通常不表现出 pH 形式。我们应用原始系统的变换，使得随机伽辽金投影变得结构保持。此外，我们研究了属于随机伽辽金系统的哈密顿函数的含义和性质。大量随机变量意味着高维随机伽辽金系统，这意味着应用模型降阶 (MOR) 生成低维 ODE 系统。我们讨论基于投影的 MOR 中的结构保存，其中较小的 ODE 系统再次以 pH 形式为特征。使用两个测试示例给出了数值计算的结果。