# Error Analysis of an Incremental POD Algorithm for PDE Simulation Data

## TL;DR

This paper analyzes the error bounds of an incremental SVD algorithm used for computing POD of PDE simulation data, providing theoretical guarantees and numerical validation for the algorithm's accuracy.

## Contribution

It offers a detailed error analysis and modifications to an incremental SVD algorithm for PDE data, including bounds on errors in singular values and vectors.

## Key findings

- The algorithm produces the exact SVD of an approximate data matrix.
- The operator norm error is bounded by the computed error bound.
- Numerical results validate the theoretical error bounds.

## Abstract

In our earlier work [Fareed et al., Comput. Math. Appl. 75 (2018), no. 6, 1942-1960], 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.