AI 中文总结
本文针对参数化时变偏微分方程的POD-ROM，对比了不同投影场景的性能，采用DEIM降低了非线性项计算成本，还通过二维布鲁塞尔子模型分析了多参数模型的相关方法。
AI 中文摘要
本文研究了演化型参数化时变偏微分方程（PDEs）的本征正交分解降阶模型（POD-ROMs）的计算实现。对于单参数模型，文献中许多论文通过投影到$L^2(\boldsymbol{\beta})$构建相关矩阵，即便在投影到$H^1_0(\boldsymbol{\beta})$时可得到最优逐点误差估计，我们对比了两种场景，发现实际表现相近。此外，获取POD近似需计算降阶方程中的非线性项，这会产生高计算成本，且问题越复杂成本越高。本文的数值结果展示了解决该问题的不同方法，离散经验插值方法（DEIM）是最高效的方法，与通过张量构造计算完整有限元（FEM）格式相比，它将计算时间减少了约一半，同时保持相同的精度。对于多参数模型，我们使用二维布鲁塞尔子（Brusselator）模型分析了文献[\text{newmethod}]中提出的新方法与标准方法，以补充结果并为更复杂系统的误差分析提供支撑。
英文摘要
This paper studies the computational implementation of proper orthogonal decomposition reduced-order models (POD-ROMs) for evolutionary parametric time-dependent partial differential equations (PDEs). For a one-parameter model, many papers in the literature build the correlation matrix by projecting onto $L^2(Ω)$ even though optimal pointwise error estimates are proved when projecting onto $H^1_0(Ω)$. We compare both scenarios and observe the similar performance in practice. Additionally, to get the POD approximation it is necessary to compute the nonlinear term in the reduced equations. This requires a high computational cost that increases when the problem gets more complex. Numerical results in this paper show different approaches to address this issue. Discrete Empirical Interpolation Method (DEIM) is the most efficient approach. It reduces computational time by approximately half compared to computing the whole FEM formulation by a tensor construction while maintaining the same accuracy. For a multiparameter model, we analyze the new and standard method proposed in \cite{newmethod} using a two-dimensional Brusselator model to complement the results and support the error analysis with a more complex system.
