A note on incremental POD algorithms for continuous time data

TL;DR

This paper analyzes continuous time data for proper orthogonal decomposition (POD), demonstrating that incremental algorithms for approximating SVDs with weighted inner products yield equivalent results, thus advancing POD computational methods.

Contribution

It develops and compares incremental algorithms for SVD in continuous time POD, showing their equivalence and accuracy in this context.

Findings

Incremental algorithms produce equivalent SVD results.

Both algorithms accurately approximate the continuous time POD.

Analysis applies to piecewise constant data snapshots in Hilbert spaces.

Abstract

In our earlier work [Fareed et al., Comput. Math. Appl. 75 (2018), no. 6, 1942-1960], we developed an incremental approach to compute the proper orthogonal decomposition (POD) of PDE simulation data. Specifically, we developed an incremental algorithm for the SVD with respect to a weighted inner product for the discrete time POD computations. For continuous time data, we used an approximate approach to arrive at a discrete time POD problem and then applied the incremental SVD algorithm. In this note, we analyze the continuous time case with simulation data that is piecewise constant in time such that each data snapshot is expanded in a finite collection of basis elements of a Hilbert space. We first show that the POD is determined by the SVD of two different data matrices with respect to weighted inner products. Next, we develop incremental algorithms for approximating the two matrix SVDs with respect to the different weighted inner products. Finally, we show neither approximate SVD is more accurate than the other; specifically, we show the incremental algorithms return equivalent results.