  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.
