This paper presents a theoretical and numerical investigation of the following practical question: Should the time difference quotients of the snapshots be used to generate the proper orthogonal decomposition basis functions? The answer to this question is important, since some published numerical studies use the time difference quotients, whereas other numerical studies do not. The criterion used in this paper to answer this question is the rate of convergence of the error of the reduced order model with respect to the number of proper orthogonal decomposition basis functions. Two cases are considered: the no_DQ case, in which the snapshot difference quotients are not used, and the DQ case, in which the snapshot difference quotients are used. The error estimates suggest that the convergence rates in the $C^0(L^2)$-norm and in the $C^0(H^1)$-norm are optimal for the DQ case, but suboptimal for the no_DQ case. The convergence rates in the $L^2(H^1)$-norm are optimal for both the DQ case and the no_DQ case. Numerical tests are conducted on the heat equation and on the Burgers equation. The numerical results support the conclusions drawn from the theoretical error estimates. Overall, the theoretical and numerical results strongly suggest that, in order to achieve optimal pointwise in time rates of convergence with respect to the number of proper orthogonal decomposition basis functions, one should use the snapshot difference quotients.
A new approach to proper orthogonal decomposition with difference quotients
在最近的工作[B。Koc等人，arXiv：2010.03750，SIAM J. Numer.肛门，出现]，作者表明，包括差异系数（DQ）是必要的，以证明最佳逐点的时间误差界适当的正交分解（POD）降阶模型的热方程。在这项工作中，我们介绍了一种新的方法，包括DQ的POD过程。代替使用所有快照数据和DQ来计算POD模式，我们仅使用第一快照以及所有DQ和特殊POD权重。我们表明，这种方法保留了标准POD DQ方法的所有数值分析优点，同时使用具有标准POD DQ方法一半快照数量的POD数据集，即，新方法在计算上更有效。我们用数值实验来说明我们的理论结果。
Shape Optimization by means of Proper Orthogonal Decomposition and Dynamic Mode Decomposition
SISSA, International School of Advanced Studies（国际高等研究院）；Fincantieri - Divisione Navi Mercantili e Passeggeri, Cantieri Navali Italiani SpA（芬坎蒂尼集团-商船与客船部门，意大利造船厂股份公司）
A non-intrusive reduced-order modelling for uncertainty propagation of time-dependent problems using a B-splines Bézier elements-based method and Proper Orthogonal Decomposition: application to dam-break flows
