Error Analysis of an Incremental Proper Orthogonal Decomposition Algorithm for PDE Simulation Data
In our earlier work Fareed et al. (2018), 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.
H. Fareed and J. R. Singler, "Error Analysis of an Incremental Proper Orthogonal Decomposition Algorithm for PDE Simulation Data," Journal of Computational and Applied Mathematics, vol. 368, Elsevier B.V., Apr 2020.
The definitive version is available at https://doi.org/10.1016/j.cam.2019.112525
Mathematics and Statistics
Keywords and Phrases
Error Analysis; Incremental Algorithm; Proper Orthogonal Decomposition; Singular Value Decomposition; Weighted Norm
International Standard Serial Number (ISSN)
Article - Journal
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