In our previous work [Singler, SIAM J. Numer. Anal. 52 (2014), no. 2, 852-876], we considered the proper orthogonal decomposition (POD) of time varying PDE solution data taking values in two different Hilbert spaces. We considered various POD projections of the data and obtained new results concerning POD projection errors and error bounds for POD reduced order models of PDEs. In this work, we improve on our earlier results concerning POD projections by extending to a more general framework that allows for non-orthogonal POD projections and seminorms. We obtain new exact error formulas and convergence results for POD data approximation errors, and also prove new pointwise convergence results and error bounds for POD projections. We consider both the discrete and continuous cases of POD. We also apply our results to several example problems, and show how the new results improve on previous work.

中文翻译：

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偏微分方程解数据的新正交分解逼近理论

在我们之前的工作中 [Singler, SIAM J. Numer. 肛门。52 (2014)，没有。2, 852-876]，我们考虑了在两个不同希尔伯特空间中取值的时变 PDE 解数据的适当正交分解 (POD)。我们考虑了数据的各种 POD 投影，并获得了关于 PDE 的 POD 降阶模型的 POD 投影误差和误差界限的新结果。在这项工作中，我们通过扩展到允许非正交 POD 投影和半范数的更通用框架来改进我们早期关于 POD 投影的结果。我们获得了新的精确误差公式和 POD 数据近似误差的收敛结果，并证明了 POD 投影的新的逐点收敛结果和误差范围。我们同时考虑 POD 的离散和连续情况。我们还将我们的结果应用于几个示例问题，