New proper orthogonal decomposition approximation theory for PDE solution data

TL;DR

This paper advances the theory of proper orthogonal decomposition (POD) for PDE solution data by extending to non-orthogonal projections and seminorms, providing new error formulas and convergence results.

Contribution

It introduces a more general framework for POD, including non-orthogonal projections and seminorms, with new error formulas and convergence theorems for PDE data approximation.

Findings

New exact error formulas for POD projections

Convergence results for both discrete and continuous POD

Improved error bounds demonstrated on example problems

Abstract

In our previous work [Singler, SIAM J. Numer. Anal. 52 (2014), no. 2, 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 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. We also apply our results to several example problems, and show how the new results improve on previous work.