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.


Mathematics and Statistics

Research Center/Lab(s)

Center for High Performance Computing Research


J. Singler and Y. Zhang were supported in part by National Science Foundation grant DMS-1217122 . J. Singler and Y. Zhang thank the Institute for Mathematics and its Applications at the University of Minnesota for funding research visits, during which some of this work was completed. The authors thank the referees for their comments, which helped to improve the manuscript.

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

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Document Type

Article - Journal

Document Version


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© 2018 Elsevier, All rights reserved.

Publication Date

01 Mar 2018