modulo-vki 2.1.1

MODULO (MODal mULtiscale pOd) is a software developed at the von Karman Institute to perform Multiscale Modal Analysis of numerical and experimental data.

MODULO: a python toolbox for data-driven modal decomposition

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MODULO is a modal decomposition package developed at the von Karman Institute for Fluid Dynamics (VKI). It offers a wide range of decomposition techniques, allowing users to choose the most appropriate method for their specific problem. MODULO can efficiently handle large datasets natively, thanks to a memory-saving feature that partitions the data and processes the decomposition in chunks (ninni2020modulo). Moreover, it supports non-uniform meshes through a weighted inner product formulation. Currently, MODULO heavily relies on NumPy routines and does not offer additional parallel computing capabilities beyond those naturally provided by NumPy.

While the discontinued MATLAB version of MODULO (ninni2020modulo) is accessible in the “Old_Matlab_Implementation” branch, it is no longer maintained. The latest decomposition techniques are exclusively available in the current Python version.

As a part of the MODULO project, we provide a series of lectures on data-driven modal decomposition, and its applications. These are available at the MODULO YouTube channel <https://www.youtube.com/@modulompod5682>_.

.. contents:: Table of contents

Modal decompositions

Modal decompositions aim to describe the data as a linear combination of modes, obtained by projecting the data onto a suitable set of basis. Different decompositions employ different bases, such as prescribed Fourier basis for the Discrete Fourier Transform (DFT), or data-driven basis, i.e. tailored on the dataset at hand, for the Proper Orthogonal Decomposition (POD). We refer to Mendez (2022) <https://doi.org/10.48550/arXiv.2201.03847>, Mendez (2023) <https://doi.org/10.1017/9781108896214.013>, and Mendez et al. (2023) <https://arxiv.org/abs/2208.07746>_, for an introduction to the topic. MODULO currently features the following decompositions:

• Discrete Fourier Transform (DFT) (Briggs et al. (1997) <https://epubs.siam.org/doi/book/10.1137/1.9781611971514>_)
• Proper Orthogonal Decomposition (POD) (Sirovich et al. (1987) <https://www.ams.org/journals/qam/1987-45-03/S0033-569X-1987-0910464-1/S0033-569X-1987-0910464-1.pdf>_ , Berkooz et al. (1993) <https://doi.org/10.1146/annurev.fl.25.010193.002543>_)
• Multi-Scale Proper Orthogonal Decomposition (mPOD) (Mendez et al. (2019) <https://arxiv.org/abs/1804.09646>_)
• Dynamic Mode Decomposition (DMD) (Schmid (2010) <https://doi.org/10.1017/S0022112010001217>_)
• Spectral Proper Orthogonal Decomposition (SPOD) (Sieber et al. (2016) <https://doi.org/10.48550/arXiv.1508.04642>, Towne et al. (2018) <https://doi.org/10.48550/arXiv.1708.04393>), note that the two are different formulations, and both are available in MODULO.
• Kernel Proper Orthogonal Decomposition (KPOD) (Mika et al. (1998) <https://proceedings.neurips.cc/paper_files/paper/1998/file/226d1f15ecd35f784d2a20c3ecf56d7f-Paper.pdf>_)

We remind the curious reader to the respective references for a detailed description of each decomposition, and to the documentation for a practical guide on how to use them in MODULO.

Release Notes

This version of MODULO includes the following updates:

1. mPOD bug fix: the previous version of mPOD was skipping the last scale of the frequency splitting vector. Fixed in this version.

2. SPOD parallelisation: CSD - SPOD can now be parallelized, leveraging joblib. The user needs just to pass the argument n_processes for the computation to be split between different workers.

3. Simplified decomposition interface: the interface of the decomposition methods has been simplified to improve user experience.

4. Enhanced POD selection: the POD function has been redesigned, allowing users to easily choose between different POD methods.

5. Improved computational efficiency: the code of the decomposition functions has been optimised, resulting in reduced computation time. mPOD now includes two additional optional arguments to enable faster filtering and to avoid recomputing the Sigmas after QR polishing.

6. Extended documentation: the documentation has been significantly enriched, now including theoretical foundations for all the supported modal decomposition techniques.

Documentation

The documentation of MODULO is available here <https://lorenzoschena.github.io/MODULO/intro.html>. It contains a comprehensive guide on how to install and use the package, as well as a detailed description of the decompositions required inputs and outputs. A list of YouTube videos <https://www.youtube.com/@modulompod5682> is also available to guide the introduce the user to modal decomposition and MODULO.

Installation

Installation via pip ^^^^^^^^^^^^^^^^^^^^

You can access the latest update of the modulo python package on PyPI using the command line:

.. code-block:: bash

$ pip install modulo_vki

Installation from source ^^^^^^^^^^^^^^^^^^^^^^^^

Alternatively, you can clone the repository and install the package locally:

.. code-block:: bash

$ git clone https://github.com/mendezVKI/MODULO.git

$ cd MODULO

$ python setup.py install

or, if you have pip installed in your environment,

.. code-block:: bash

$ pip install .

Example

Example 1: POD decomposition ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The following example illustrates how to decompose a data set (D) using the POD decomposition.

.. code-block:: python

from modulo_vki import ModuloVKI 
import numpy as np

# Create a random dataset
D = np.random.rand(100, 1000)

# Initialize the ModuloVKI object
m = ModuloVKI(D) 

# Compute the POD decomposition
phi_POD, Sigma_POD, psi_POD = m.POD()

which returns the spatial basis ($\phi$), the temporal basis ($\psi$), and the modal amplitudes ($\Sigma$) of the POD decomposition.

Example 2: Memory Saving option ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

For the Memory Saving option, MODULO decomposes $D$ in N_partitions, defined by the user (refer to examples/ex_04_Memory_Saving.py).

.. code-block:: python

from modulo_vki import ModuloVKI 
import numpy as np

# Create a random dataset
D = np.random.rand(100, 1000)

# Initialize the ModuloVKI object
m = ModuloVKI(D, N_PARTITIONS=10) 

# Compute the POD decomposition
phi_POD, Sigma_POD, psi_POD = m.POD()

Example 3: non-uniform grid ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

If you are dealing with non-uniform grid (e.g. output of a Computational Fluid Dynamic (CFD) simulation), you can use the weighted inner product formulation (refer to examples/ex_05_nonUniform_POD.py).

.. code-block:: python

from modulo_vki import ModuloVKI 
import numpy as np

# Create a random dataset
D = np.random.rand(100, 1000)

# Get the area of the grid
a_dataSet = gridData.compute_cell_sizes()
area = a_dataSet['Area']

# Compute weights
areaTot = np.sum(area)
weights = area/areaTot # sum should be equal to 1

# Initialize the ModuloVKI object
m = ModuloVKI(D, weights=weights) 

# Compute the POD decomposition
phi_POD, Sigma_POD, psi_POD = m.POD()

Community guidelines

Contributing to MODULO ^^^^^^^^^^^^^^^^^^^^^^^ We welcome contributions to MODULO.

It is recommended to perform a shallow clone of the repository to avoid downloading the entire history of the project:

.. code-block:: bash

$ git clone --depth 1 https://github.com/mendezVKI/MODULO.git

This will download only the latest version of the repository, which is sufficient for contributing to the project, and will save you time and disk space.

To create a new feature, please submit a pull request, specifying the proposed changes and providing an example of how to use the new feature (that will be included in the examples/ folder).

The pull request will be reviewed by the MODULO team before being merged into the main branch, and your contribution duly acknowledged.

Report bugs ^^^^^^^^^^^^ If you find a bug, or you encounter unexpected behaviour, please open an issue on the MODULO GitHub repository.

Ask for help ^^^^^^^^^^^^ If you have troubles using MODULO, or you need help with a specific decomposition, please open an issue on the MODULO GitHub repository.

Citation

If you use MODULO in your research, please cite it as follows:

Poletti, R., Schena, L., Ninni, D., Mendez, M. A. (2024).
MODULO: A Python toolbox for data-driven modal decomposition.
Journal of Open Source Software, 9(102), 6753. https://doi.org/10.21105/joss.06753

.. code-block:: text

@article{Poletti2024, 
  doi = {10.21105/joss.06753}, 
  url = {https://doi.org/10.21105/joss.06753}, 
  year = {2024},
  publisher = {The Open Journal}, 
  volume = {9}, 
  number = {102}, 
  pages = {6753},
  author = {R. Poletti and L. Schena and D. Ninni and M. A. Mendez}, 
  title = {MODULO: A Python toolbox for data-driven modal decomposition},
  journal = {Journal of Open Source Software}
}

and

Ninni, D., & Mendez, M. A. (2020).
MODULO: A software for Multiscale Proper Orthogonal Decomposition of data.
SoftwareX, 12, 100622. https://doi.org/10.1016/j.softx.2020.100622

.. code-block:: text

@article{ninni2020modulo,
    title={MODULO: A software for Multiscale Proper Orthogonal Decomposition of data},
    author={Ninni, Davide and Mendez, Miguel A},
    journal={SoftwareX},
    volume={12},
    pages={100622},
    year={2020},
    publisher={Elsevier}
}

References

• Mendez, Miguel Alfonso. "Statistical Treatment, Fourier and Modal Decomposition." arXiv preprint arXiv:2201.03847 (2022).
• Mendez, M. A. (2023) "Generalized and Multiscale Modal Analysis". In : Mendez M.A., Ianiro, A., Noack, B.R., Brunton, S. L. (Eds), "Data-Driven Fluid Mechanics: Combining First Principles and Machine Learning". Cambridge University Press, 2023:153-181. https://doi.org/10.1017/9781108896214.013. The pre-print is available at https://arxiv.org/abs/2208.12630.
• Ninni, Davide, and Miguel A. Mendez. "MODULO: A software for Multiscale Proper Orthogonal Decomposition of data." SoftwareX 12 (2020): 100622.
• Mendez, Miguel A. "Linear and nonlinear dimensionality reduction from fluid mechanics to machine learning." Measurement Science and Technology 34.4 (2023): 042001.
• Briggs, William L., and Van Emden Henson. The DFT: an owner's manual for the discrete Fourier transform. Society for Industrial and Applied Mathematics, 1995.
• Berkooz, Gal, Philip Holmes, and John L. Lumley. "The proper orthogonal decomposition in the analysis of turbulent flows." Annual review of fluid mechanics 25.1 (1993): 539-575.
• Sirovich, Lawrence. "Turbulence and the dynamics of coherent structures. III. Dynamics and scaling." Quarterly of Applied mathematics 45.3 (1987): 583-590.
• Mendez, M. A., M. Balabane, and J-M. Buchlin. "Multi-scale proper orthogonal decomposition of complex fluid flows." Journal of Fluid Mechanics 870 (2019): 988-1036.
• Schmid, Peter J. "Dynamic mode decomposition of numerical and experimental data." Journal of fluid mechanics 656 (2010): 5-28.
• Sieber, Moritz, C. Oliver Paschereit, and Kilian Oberleithner. "Spectral proper orthogonal decomposition." Journal of Fluid Mechanics 792 (2016): 798-828.
• Towne, Aaron, Oliver T. Schmidt, and Tim Colonius. "Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis." Journal of Fluid Mechanics 847 (2018): 821-867.
• Mika, Sebastian, et al. "Kernel PCA and de-noising in feature spaces." Advances in neural information processing systems 11 (1998).

Related projects

MODULO encapsulates a wide range of decomposition techniques, but not all of them. We refer to the project below for an additional set of decomposition techniques:

• ModRed, https://github.com/belson17/modred

There are also decomposition-specific projects, some of which are listed below:

• Rogowski, Marcin, Brandon CY Yeung, Oliver T. Schmidt, Romit Maulik, Lisandro Dalcin, Matteo Parsani, and Gianmarco Mengaldo. "Unlocking massively parallel spectral proper orthogonal decompositions in the PySPOD package." Computer Physics Communications 302 (2024): 109246.
• Lario, A., Maulik, R., Schmidt, O.T., Rozza, G. and Mengaldo, G., 2022. Neural-network learning of SPOD latent dynamics. Journal of Computational Physics, 468, p.111475.
• Ichinaga, Andreuzzi, Demo, Tezzele, Lapo, Rozza, Brunton, Kutz. PyDMD: A Python package for robust dynamic mode decomposition. arXiv preprint, 2024.
• Rogowski, Marcin, et al. "Unlocking massively parallel spectral proper orthogonal decompositions in the PySPOD package." Computer Physics Communications 302 (2024): 109246.