On Optimal Pointwise in Time Error Bounds and Difference Quotients for the Proper Orthogonal Decomposition
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.
B. Koc et al., "On Optimal Pointwise in Time Error Bounds and Difference Quotients for the Proper Orthogonal Decomposition," SIAM Journal on Numerical Analysis, vol. 59, no. 4, pp. 2163 - 2196, Society for Industrial and Applied Mathematics (SIAM), Aug 2021.
The definitive version is available at https://doi.org/10.1137/20M1371798
Mathematics and Statistics
Keywords and Phrases
Error analysis; Optimality; Proper orthogonal decomposition; Reduced order model
International Standard Serial Number (ISSN)
Article - Journal
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05 Aug 2021