Multilevel Algorithm for Obtaining the Proper Orthogonal Decomposition

Abstract

A multilevel proper orthogonal decomposition (mPOD) is introduced for obtaining the proper orthogonal decomposition (POD) of data sets with a large number of snapshots without requiring the solution of an equally large eigenvalue problem. mPOD works by producing the PODs for subsets of data first and then applying the POD again to the modes obtained from these subsets. A proof is given that mPOD reproduces identically the modes that would be obtained with the standard POD. mPOD also provides a systematic way to combine POD modes obtained from different simulations and to update POD modes using new data. A numerical algorithm based on mPOD is developed that shows a factor of 10 reduction in computation time compared to snapshot POD. The method is applied to data for fluid flow over a NACA 0015 at with an actuator jet on the top surface of the airfoil, and results are compared to methods such as subsampling, double proper orthogonal decomposition, sequential mode generation methods, recursive singular value decomposition (R-SVD), and partitioned R-SVD. The examples demonstrate that mPOD provides more accurate eigenvalues, requires fewer disk accesses, and generates modes that create more reliable reduced-order models and have better orthogonality properties than any of the other methods.