Thanks, Larissa! I'm humbled to hear the positive feedback from the community =)

Thanks, Tom! I'm really happy I'm able to help and spread knowledge on this amazing technique!

Julian Roth, May I have your email id. Actually I'm also working on ROM techniques.

I'm glad you found it useful!! Hope I can find more time to do these =]

I'm happy to hear it was useful to you!

Damn I missed your comment, this is a really good take! Thanks for checking out my work! I also feel there is a big gap in content of this kind of advanced math where it doesn't quite reach enough people for creators to justify economically building a whole video on it. My hope is that more people get interested in these amazing techniques, as they're so powerful!

Hey Robin, thanks for the comment! I'm happy that it was helpful!

@@fzigunov I work in a very similar field (PIV analysis of boundary layer transition) and I also use POD. I had sent a connection request to you on Linkedin. Please have a look.

Thanks, Antoine, I'm glad you're interested! I'm not covering the computation of POD because there's many algorithms out there with varying efficiency. If you are using a library such as Matlab or Python, just use SVD (singular value decomposition, it's another name for POD): [U, S, V] = svd(A,0); Where U is the topos matrix and V is the chronos matrix if A is structured as shown in the video. If you really want to dive into building your own SVD algorithm, then I'm afraid you'll have to look into other resources. I know Prof. Nathan Kutz has a really nice video on SVD and the naive algorithm for its computation, might be worth checking it out. Thanks!

@@fzigunov Thank you, with the SVD i managed to do what i wanted !

Thanks for the encouragement! I actually forgot to remove the mean and only noticed in editing (these animations take several days to render!). It doesn't impact anything, though, because the POD will separate the mean in the first mode; provided the mean norm is sufficiently large (mostly always the case in fluid dynamics data sets). Glad this helped you! Edit: The mean of the data set is actually zero; it's just that I'm only showing three of its dimensions (i.e., the cylinder data set that is shown later in the video). If you had an n-dimensional view, the point cloud would be centered in the origin.

@@fzigunov Thanks for the response! I'm doing PIV recently, these animations are far more understandable than JL Lumley's POD papers with tons of equations. Looking forward to more of your videos!

If I'm understanding correctly, you want to perform POD with a single snapshot? I'm afraid that is not possible, as it doesn't have enough statistics to show anything meaningful. In fact the only mode you get will, by definition, be exactly the snapshot given (divided by its norm).

@@fzigunov Thank you for your reply. So does that mean that the number of modes is determined by the number of snapshots (i.e., the number of time data) provided?

@@uchennaogunka You're correct for most applications. The number of modes is min(N_grids,N_snapshots); where N_grids is the number of grid points of the mesh examined times the number of variables (say vx,vy,vz,p).

Very good question. If the correlation between some variable X and Y is high (close to +1), then X and Y move together (i.e., when X is up, Y is also up). If correlation is low, (close to -1), we get the opposite effect and when X is high, Y is low. For POD, it calculates from the data given how they are most correlated (i.e., instead of X and Y, you have many X variables, all contained in the matrix A). So if in a mode a hot spot shows in two different regions of an image (on a high rank mode), it is safe to say they have a high correlation (i.e. , they vary in unison). Hope this helps understand a little better.

@@fzigunov Yes thank you!!

Thanks André, your feedback is really appreciated! Regarding your question, the number of columns of U and A match as you pointed out. As to why they have the same number of columns, it is a very good question. Here's a technical description, which may not be 100% accurate (I'm a mere fluid dynamicist, not a mathematician!): In a rectangular matrix where n_rows>n_columns, the column space of the rectangular data matrix I displayed in the video (A) does not span the entire n_rows-dimensional space. What we are seeking with the SVD (and POD) is to find a set of singular vectors that is mutually orthogonal *AND spans the column space of the original data matrix*. Since the original data matrix only spans n_columns dimensions (it does not have more than n_columns vectors in it), then it stands that the mode matrix U only needs n_columns singular vectors to completely span the column space of A. If that sounds too complicated, maybe a simpler example with a 3 row by 2 column A matrix may help: If A has 2 columns, each with 3 rows, then you can build two vectors in 3D space, one for each column. These two vectors span a plane in 3D space, but do not span the whole 3D space (you cannot build points outside the plane by linearly combining the two vectors from the columns). Thus, the singular vectors of U only need to span this 2D plane. Thus, you only need 2 singular vectors, which by construction will be unitary and orthogonal. Hopefully this helps!

@@fzigunov I will have to read about 20 times to start understanding, but your answer certainly is a great intro to the question. You rock! Hope your channel achieve what it seeks.

Check out eigensteve's channel

@@pinakibhattacharyya7853 thanks

Great question, @angtrinh6495. It doesn't reduce dimensions, it only classifies the data by ranking the most energetic directions the point cloud formed by the data spreads. The dimensionality reduction step is done by the human as a processing step (say, keep X% of the total energy, or the X most energetic modes, etc.).

I'm glad you liked them! Here's the github repo! Codes are a little janky though =) github.com/3dfernando/POD_Intro_Animations

@@fzigunov Could you please briefly explain how to use your virtual environment?

Still on my video production learning curve! Thanks for the feedback!

@@fzigunov no worries! But that said, your video was extremely useful! Thanks a ton! Keep it up!