Incremental Proper Orthogonal Decomposition for PDE Simulation Data
We propose an incremental algorithm to compute the proper orthogonal decomposition (POD) of simulation data for a partial differential equation. Specifically, we modify an incremental matrix SVD algorithm of Brand to accommodate data arising from Galerkin-type simulation methods for time dependent PDEs. The algorithm is applicable to data generated by many numerical methods for PDEs, including finite element and discontinuous Galerkin methods. The algorithm initializes and efficiently updates the dominant POD eigenvalues and modes during the time stepping in a PDE solver without storing the simulation data. We prove that the algorithm without truncation updates the POD exactly. We demonstrate the effectiveness of the algorithm using finite element computations for a 1D Burgers' equation and a 2D Navier-Stokes problem.
H. Fareed et al., "Incremental Proper Orthogonal Decomposition for PDE Simulation Data," Computers and Mathematics with Applications, vol. 75, no. 6, pp. 1942-1960, Elsevier, Mar 2018.
The definitive version is available at https://doi.org/10.1016/j.camwa.2017.09.012
Mathematics and Statistics
Keywords and Phrases
Eigenvalues and eigenfunctions; Galerkin methods; Navier Stokes equations; Numerical methods; Partial differential equations; Principal component analysis; 2D Navier-Stokes problem; Burgers' equations; Discontinuous Galerkin methods; Finite element computations; Incremental algorithm; Proper orthogonal decompositions; Time dependent PDEs; Weighted norm; Finite element method
International Standard Serial Number (ISSN)
Article - Journal
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