Dear Sir/Madam
I am working in turbulence flows, spray and atomization, swirling flow. I read some papers based on POD and DMD , but I am not able to understand the full physics behind this.
Tapan K. Sengupta
Thank you sir
I will read the mentioned authors work first.
In my opinion, these decompositions carry no 'physical meaning' on their own.
These are data-driven methods to break your dataset (e.g., velocity fields, pressure fields, etc.) or the solution of a PDE into a linear combination of 'elementary' or 'rank-1' information which we call modes. The number of possible modes in your data depends on the rank (dimensionality) of the information contained in it.
Different decompositions use different criteria to compute the modes. The POD seeks the optimal approximation for a given number of modes smaller than the rank. Therefore, if you want to construct a rank 4 approximation, using the first 4 POD modes gives you the least L2 error. The DMD tries to retrieve the best linear dynamical system describing your data (which is usually not linear) and build its modes from the eigendecomposition of this system.
What you do with these modes is then up to your expertise, your interest, and your goal. These are tools which lay the foundation for many areas of applied mathematics.
For example, some of these modes could represent some specific pattern in your data (e.g., turbulent coherent structures). By analyzing these or looking for these in other datasets, you do pattern recognition or classification.
Otherwise, you might be interested in the fact that few modes describe most of the physical phenomenon that your data was aimed at capturing, while the remaining ones describe irrelevant noise. By distilling this information, you do data compression or filtering.
Or you might be interested in the fact that few of the modes you obtain could be used to simplify your problem. For instance, you could use them as a low-dimensional basis onto which you project a large dynamical system and hence significantly reduce the computational burdens of a simulation. By solving the problem in this projected space, you do Reduced Order Modeling (ROM).
The energy optimality of the POD or the spectral purity of the DMD might be too extreme for your problem, and you might be interested in some hybrid approach. If that is the case, we have recently proposed a multiscale extension of the POD:
Article Multi-scale proper orthogonal decomposition of complex fluid flows
Article Multiscale Modal Analysis of an Oscillating Impinging Gas Jet
I think it will be fair to state that no decomposition can have a physical meaning - it is not logical that some theory would depend on the way we formulate it mathematically or divide into parts by some projection. It could in some sense resemble something else that we would like to address as a physical process (e.g. von Karmat vortex street) but a priori the decomposition does have anything to do with the field of vorticity we try to visualize.
A decent explanation on the subject can be found in the following articles by Prof. Steve L. Brunton and co. You can watch his lectures on his youtube channel. A talk in AIAA Conference is also available on AIAA Forum youtube channel. Also, you may consult the video lectures by Prof. Nathan Kutz available on his youtube channel.
Taira et al. "Modal Analysis of Fluid Flows: An Overview", AIAA Journal, vol 55 (12), 2017.
Taira et al. "Modal Analysis of Fluid Flows: Applications and Outlook", arXiv:1903.05750v1 [physics.flu-dyn] , 2019.
Dear Raju
One starting point to thinking about your question is to note that POD is the same thing mathematically as principal components analysis in statistics, used to simplify multivariate data into fewer modes. This is, I believe the sense in which Miguel Mendez and Alex Liberzon have answered your question. If, you represent a PDE by an infinite number of ODEs and impose some criterion to do with energy then POD drops out as an effective method (I think this is the sense that Tapan Sengupta in referring to). Presumably a similar criterion in terms of periodicity of solutions will lead you to DMD. To see that one is not limited to these techniques, take a look at ICA, which uses an entropy ecriterion to produce inedpendet modes, in contrast to POD which uses a variance-covariance approach to produce linearly uncorrelated modes.n
Hi Raju,
You can take a look at Sections 4 and 5 in this article where both POD and DMD (OPD) are applied in a microfluidics application.
Article Modal analysis of magnetic microconvection
Let me know if you can access the paper, otherwise just let me know.
Dear Raju,
You already very skillful comments from the above experts, I would just suggest you some useful readings related to your query:Preprint On the Physical Interpretation of Proper Orthogonal Decompos...
https://arxiv.org/pdf/1708.04393.pdf
Article Proper orthogonal decomposition and dynamic mode decompositi...
- Badges
- Science topic
- Similar topics
- Biological Science
- Bioecology
- Systems Ecology
- Landscape Ecology
- Connectivity