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On Optimal Pointwise in Time Error Bounds and Difference Quotients for the Proper Orthogonal Decomposition
SIAM Journal on Numerical Analysis ( IF 2.8 ) Pub Date : 2021-08-05 , DOI: 10.1137/20m1371798 Birgul Koc , Samuele Rubino , Michael Schneier , John Singler , Traian Iliescu
SIAM Journal on Numerical Analysis ( IF 2.8 ) Pub Date : 2021-08-05 , DOI: 10.1137/20m1371798 Birgul Koc , Samuele Rubino , Michael Schneier , John Singler , Traian Iliescu
SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2163-2196, January 2021.
In this paper, we resolve several long-standing issues dealing with optimal pointwise in time error bounds for proper orthogonal decomposition (POD) reduced order modeling of the heat equation. In particular, we study the role played by difference quotients (DQs) in obtaining reduced order model (ROM) error bounds that are optimal with respect to both the time discretization error and the ROM discretization error. When the DQs are not used, we prove that both the POD projection error and the ROM error are suboptimal. When the DQs are used, we prove that both the POD projection error and the ROM error are optimal. The numerical results for the heat equation support the theoretical results.
中文翻译：
关于正正交分解的时间误差界和差商的最优逐点
SIAM 数值分析杂志，第 59 卷，第 4 期，第 2163-2196 页，2021
年1 月。在本文中，我们解决了几个长期存在的问题，这些问题涉及最佳的逐点时间误差界限，以便正确正交分解 (POD) 降阶建模热方程。特别是，我们研究了差商 (DQ) 在获得关于时间离散化误差和 ROM 离散化误差最佳的降阶模型 (ROM) 误差界限中所起的作用。当不使用 DQ 时，我们证明 POD 投影误差和 ROM 误差都是次优的。当使用 DQ 时，我们证明 POD 投影误差和 ROM 误差都是最优的。热方程的数值结果支持理论结果。
更新日期：2021-08-07
In this paper, we resolve several long-standing issues dealing with optimal pointwise in time error bounds for proper orthogonal decomposition (POD) reduced order modeling of the heat equation. In particular, we study the role played by difference quotients (DQs) in obtaining reduced order model (ROM) error bounds that are optimal with respect to both the time discretization error and the ROM discretization error. When the DQs are not used, we prove that both the POD projection error and the ROM error are suboptimal. When the DQs are used, we prove that both the POD projection error and the ROM error are optimal. The numerical results for the heat equation support the theoretical results.
中文翻译：
关于正正交分解的时间误差界和差商的最优逐点
SIAM 数值分析杂志，第 59 卷，第 4 期，第 2163-2196 页，2021
年1 月。在本文中，我们解决了几个长期存在的问题，这些问题涉及最佳的逐点时间误差界限，以便正确正交分解 (POD) 降阶建模热方程。特别是，我们研究了差商 (DQ) 在获得关于时间离散化误差和 ROM 离散化误差最佳的降阶模型 (ROM) 误差界限中所起的作用。当不使用 DQ 时，我们证明 POD 投影误差和 ROM 误差都是次优的。当使用 DQ 时，我们证明 POD 投影误差和 ROM 误差都是最优的。热方程的数值结果支持理论结果。
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