On Optimal Pointwise in Time Error Bounds and Difference Quotients for the Proper Orthogonal Decomposition
On Optimal Pointwise in Time Error Bounds and Difference Quotients for the Proper Orthogonal Decomposition
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DOI：
10.1137/20m1371798
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发表时间：
2020-10
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通讯作者：
Birgul Koc;S. Rubino;M. Schneier;J. Singler;T. Iliescu
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作者：
Birgul Koc;S. Rubino;M. Schneier;J. Singler;T. Iliescu
在本文中，我们解决了几个长期存在的问题，涉及热方程的适当正交分解（POD）降阶建模的时间误差范围内的最佳点。特别是，我们研究了差商 (DQ) 在获得针对时间离散化误差和 ROM 离散化误差而言最佳的降阶模型 (ROM) 误差界限方面所发挥的作用。当不使用 DQ 时，我们证明 ROM 投影误差和 ROM 误差都不是最优的。当使用 DQ 时，我们证明 ROM 投影误差和 ROM 误差都是最优的。热方程的数值结果支持了理论结果。
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 ROM projection error and the ROM error are suboptimal. When the DQs are used, we prove that both the ROM projection error and the ROM error are optimal. The numerical results for the heat equation support the theoretical results.