In our previous work [J. R. Singler, SIAM J. Numer. Anal., 52 (2014), pp. 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 nonorthogonal 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.
S. Locke and J. R. Singler, "New Proper Orthogonal Decomposition Approximation Theory for PDE Solution Data," SIAM Journal on Numerical Analysis, vol. 58, no. 6, pp. 3251-3285, Society for Industrial and Applied Mathematics, Nov 2020.
The definitive version is available at https://doi.org/10.1137/19M1297002
Mathematics and Statistics
Center for High Performance Computing Research
Keywords and Phrases
Approximation theory; Projections; Proper orthogonal decomposition
International Standard Serial Number (ISSN)
Article - Journal
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10 Nov 2020