“Proper orthogonal decomposition (POD) projection errors and error bounds for POD reduced order models of partial differential equations have been studied by many. In this research we obtain new results regarding POD data approximation theory and present a new difference quotient (DQ) approach for computing the POD modes of the data.
First, we improve on earlier results concerning POD projection errors by extending to a more general framework that allows for non-orthogonal 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 within this generalized framework. We also apply our results to several example problems, and show how the new results improve on previous work.
Next, we consider the relationship between POD, difference quotients (DQs), and pointwise ROM error bounds. It is known that including DQs is necessary in order to prove optimal pointwise in time error bounds for POD reduced order models of the heat equation. We introduce a new approach to including DQs in the POD procedure to further investigate the role DQs play in POD numerical analysis. Instead of computing the POD modes using all of the snapshot data and DQs, we only use the first snapshot along with all of the DQs and special POD weights. We show that this approach retains all of the numerical analysis benefits of the standard POD DQ approach, while using a POD data set that has half the number of snapshots as the standard POD DQ approach, i.e., the new approach is more computationally efficient. We illustrate our theoretical results with numerical experiments”--Abstract, page iv.
Singler, John R.
Duan, Lian, 1983-
Mathematics and Statistics
Ph. D. in Mathematics
Missouri University of Science and Technology
viii, 95 pages
© 2021 Sarah Katherine Locke, All rights reserved.
Dissertation - Open Access
Locke, Sarah Katherine, "Proper orthogonal decomposition: New approximation theory and a new computational approach" (2021). Doctoral Dissertations. 3012.