Error analysis of an incremental proper orthogonal decomposition algorithm for PDE simulation data

Abstract

In our earlier work [16], we proposed an incremental SVD algorithm with respect to a weighted inner product to compute the proper orthogonal decomposition (POD) of a set of simulation data for a partial differential equation (PDE) without storing the data. In this work, we perform an error analysis of the incremental SVD algorithm. We also modify the algorithm to incrementally update both the SVD and an error bound when a new column of data is added. We show the algorithm produces the exact SVD of an approximate data matrix, and the operator norm error between the approximate and exact data matrices is bounded above by the computed error bound. This error bound also allows us to bound the error in the incrementally computed singular values and singular vectors. We illustrate our analysis with numerical results for three simulation data sets from a 1D FitzHugh-Nagumo PDE system with various choices of the algorithm truncation tolerances.

A new approach to proper orthogonal decomposition with difference quotients

“…where the difference quotients are defined by Equation (8). Note that the POD error function in Equation ( 16) does not include the weighted sum of the errors of the regular snapshots; this contrasts with the POD approaches in Section 2, which both include such error terms, see Equation (1) and Equation (7). Furthermore, in the POD approaches in Section 2, we have exact error formulas for these error terms; see Lemma 2.1 and Lemma 2.3.…”

“…This procedure works well and is highly accurate for smaller data sets; for larger data sets one could use an incremental SVD approach or another related algorithm instead, see e.g. [2,4,7,8,16] and the references therein.…”

A note on incremental POD algorithms for continuous time data

“…Therefore, the two different ways of expressing the data did not lead to different incrementally computed POD modes for the data. One benefit of this result is that the error analysis of the discrete time incremental POD algorithm (with one weighted inner product) in [12] is directly applicable to continuous time case, assuming the data is piecewise constant in time and is expressed using the weighted characteristic functions as in (3.9)-(3.10).…”

“…The algorithm is computationally efficient, needs very little storage, and is also easily used in conjunction with an existing time stepping PDE approximation code. We also recently performed an error analysis of the method in [12].…”

“…In both of these earlier works [12,13], we only performed analysis for the discrete time case, i.e., the data set is a finite collection of elements in a Hilbert space. In [13,Section 5], we developed an algorithm for the case of time varying data by approximating the POD integral operator using a Riemann sum and then performing an incremental POD/SVD update with respect to the weighted inner product.…”