Multi-scale Proper Orthogonal Decomposition (mPOD)

We propose a novel data-driven decomposition referred to as Multi-scale Proper Orthogonal Decomposition (mPOD). This decomposition combines Multi-resolution Analysis (MRA) via Discrete Wavelet Transform (DWT) and Proper Orthogonal Decomposition (POD). Using MRA analysis, the mPOD decomposes the correlation matrix into the contribution of different scales, each retaining a non-overlapping portion of the correlation spectra and each leading to a POD constrained to a limited frequency bandwidth. The MRA analysis of the correlation matrix is presented both in terms of frequency response and eigenvalue spectra, in order to link the POD to the Fourier spectra. Finally, the decomposition is tested on a nonlinear advection-diffusion problem and compared to standard Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD) and Spectral Proper Orthogonal Decomposition (SPOD) in terms of feature extraction capabilities, residual decay versus the number of modes, and time-frequency localization.