See also

For more detailed discussions on MRPOD, its derivations and applications, please see the publication [MRPOD]. We also kindly ask you to reference this paper if you use mrpod for your publications.

Greetings

Welcome to the documentation of the Python module mrpod for performing Multiresolution Proper Orthogonal Decomposition (MRPOD) of multi-dimensional time series. Although mrpod was created to tackle problems in turbulent flows, this module is equally applicable to various data series to achieve frequency-filtering, classification of dynamics, identification of discontinuities, etc. The built-in wavelet sub-module can be easily applied to 1-D and 2-D dataset for wavelet decomposition and reconstruction.

Why mrpod

Data-driven, modal-decomposition techniques such as POD (also known as the Principle Component Analysis, PCA) and Dynamic Mode Decomposition (DMD) have been widely implemented to extract periodic, coherent structures in turbulent flows. In reacting flows such as confined turbulent flames (in gas turbines), however, we are often confronted with both periodic (e.g., hydrodynamic and acoustic instabilities) and non-periodic dynamics (e.g., flame lift-off, flashback, and bistability), which can coexist over a wide range of time scales and may even interact with each other.

In their most rudimentary forms, POD and DMD have proven insufficient at separating these dynamics while resolving their temporal behaviors (such as discontinuities) at the same time. On the other hand, by marrying the concept of wavelet-based Multiresolution Analysis (MRA) with standard modal decompositions, multiresolution DMD ([MRDMD]) and multi-scale POD ([mPOD]) have demonstrated robust capabilities at identifying unsteady dynamics and discontinuities in time series.

Based on a similar concept, multiresolution POD ([MRPOD]) has been developed by combining Maximum-Overlap Discrete Wavelet Transform (MODWT) with conventional snapshot POD. MRPOD has been successfully applied to time series of velocity (vector) and scalar fields obtained by kHz-rate laser diagnostics (e.g., Particle Image Velocimetry or PIV, Planar Laser-induced Fluorescence or PLIF) in the so-called bistable turbulent swirl flame, a common phenomenon encountered in gas turbines.

Why MODWT

mrpod was developed to satisfy primarily the following two criteria:

  • Dynamics with various frequencies can be identified and adequately isolated.
  • Discontinuities in temporal behaviors can be properly resolved and align perfectly with the original data series for appropriate comparison.

Unlike the classical DWT, shift-invariant DWT such as MODWT is well-defined for arbitrary sample sizes and is not sensitive to the “break-in” point in the time series. Additionally, MODWT affords a more “square-looking” gain response and hence a higher spectral isolation especially for cases with dynamics densely packed in the frequency domain. Although MODWT sacrifices orthonormality, it can still carry out an exact analysis of variance as well as a perfect reconstruction of the time series. With the two aforementioned criteria in mind, MODWT has demonstrated overall better performance than DWT at a (manageable) cost of computational time.

Why not pywt

Instead of using the existing Python library pywt to carry out wavelet transform, a matrix-operation based routine was written from the ground up specifically for more efficient 1-D and 2-D wavelet decomposition/reconstruction of multi-dimensional data series stored in ndarrays. Although several commonly used wavelet filters are built into mrpod, the vast library of wavelet filters in pywt should be taken advantage of when constructing custom composite filters using the filter-cascading method in mrpod.


[MRDMD]Kutz, J., Fu, X., Brunton, S. Multiresolution dynamic mode decomposition. SIAM Journal on Applied Dynamical Systems 15 (2), 713-735, 2016.
[mPOD]Mendez, M. A., Balabane, M., Buchlin, J. M. Multi-scale proper orthogonal decomposition of complex fluid flows. Journal of Fluid Mechanics 870, 988-1036, 2019.
[MRPOD](1, 2) Yin, Z., Stöhr, M. Time–Frequency Localisation of Intermittent Dynamics in a Bistable Turbulent Swirl Flame. Journal of Fluid Mechanics 882, A30, 2020.