  Data-driven decompositions are becoming essential tools in fluid dynamics,
allowing for tracking the evolution of coherent patterns in large datasets, and
for constructing low order models of complex phenomena. In this work, we
analyze the main limits of two popular decompositions, namely the Proper
Orthogonal Decomposition (POD) and the Dynamic Mode Decomposition (DMD), and we
propose a novel decomposition which allows for enhanced feature detection
capabilities. This novel decomposition is referred to as Multiscale Proper
Orthogonal Decomposition (mPOD) and combines Multiresolution Analysis (MRA)
with a standard POD. Using MRA, the mPOD splits the correlation matrix into the
contribution of different scales, retaining non-overlapping portions of the
correlation spectra; using the standard POD, the mPOD extracts the optimal
basis from each scale. After introducing a matrix factorization framework for
data-driven decompositions, the MRA is formulated via 1D and 2D filter banks
for the dataset and the correlation matrix respectively. The validation of the
mPOD, and a comparison with the Discrete Fourier Transform (DFT), DMD and POD
are provided in three test cases. These include a synthetic test case, a
numerical simulation of a nonlinear advection-diffusion problem, and an
experimental dataset obtained by the Time-Resolved Particle Image Velocimetry
(TR-PIV) of an impinging gas jet. For each of these examples, the
decompositions are compared in terms of convergence, feature detection
capabilities, and time-frequency localization.
