This is a short overview of the Proper Orthogonal and Dynamic/Koopman Mode Decompositions, which are commonly used in analysis of velocity fields of fluid flows. While I worked with the theoretical side of Koopman modes, I never implemented the numerical code myself; I wrote these notes up a I was teaching myself the basics of numerics of these decompositions, and consequently used the notes for two lectures. The notes are based References at the end of the post. Caveat lector: Notes may contain gross oversimplifications — the emphasis was on understanding and not on precision. I welcome your corrections and comments below. (You can always stop by my Van Vleck office if you’re in Madison to discuss any part of this).
Jean-Luc Thiffeault and I have just uploaded our paper on Finite-Time Braiding Exponents (FTBE) to arXiv. In the paper we study how closely topological entropy of a mixing dynamical system can be approximated by sampling only finitely many trajectories from the flow.
Based on numerical results (obtained using braidlab 3.1), we find that FTBEs of finitely-many trajectories converge to topological entropy as number of trajectories is increased. The paper further explores robustness and dependence of FTBEs on time step, number of trajectories, and their length.
The latest full release of braidlab is out! Braidlab is a MATLAB toolbox which incorporates algorithms for analyzing braid groups of punctured disks in both theoretical and applied contexts. It was primarily written by Jean-Luc Thiffeault, but Michael Allshouse and I have contributed code to it as well. Feel free to direct your questions either at JLT or at me.
You can find a good summary of release updates on the main release-3.0 announcement, but here’s my list of favorites:
- We moved the repository to GitHub. This means that you can (and should) use our Issue Tracker to let us know what went wrong or what you would like to see included in future releases. We are also present on MATLAB Central.
- Installation from source should now work on Matlab 2014b without any special configuration. There are two known installation issues which are out of our reach: conflict of “mex” command with a LaTeX command, and lack of GMP libraries on your system. Make sure you read the installation guidelines in the manual first to see how to resolve these (and other) known problems.
- “Data braids” now have a broader support. This type of braids is useful if you are trying to represent physical trajectories, which have an independent variable, e.g., time, attached to them.
- More functions are implemented as MEX C++ code, which means that they ultimately run faster (some of them have even been parallelized!)
To install, go to the release page, scroll down, download the pre-packaged binaries. If you want to build from source, you can either download the source or even clone our git repository to stay up to date with the latest developments.
In all cases: let us know if braidlab works on your end and if you find it useful.
Last night I had an amazing time at Madison Math Circle, hanging out with a bunch of bright school kids, who were undaunted by either diseases or mathematics. Since I had quite a few questions after the lecture, I decided to post an overview so anyone interested can go over it themselves. For those of you who attended and have further questions, or even wish to explore mathematical dynamics on your own, feel free to leave a comment below or e-mail me at firstname.lastname@example.org and I’ll try to answer all your questions and provide some guidance for future.
I am currently attending 10th AIMS Conference on dynamics in Madrid, where I participated in the session on transport barriers in unsteady fluid flows. One of the talks in the session suggested that the ergodic quotient techniques for detecting coherent sets would not apply to finite-time setting, i.e., when ergodic averages are replaced by finite-time averages. This is, of course, of interest in cases when ergodic averages do not exist, so finite-time average is a potential resolution of this problem. I decided to investigate the potentially-problematic example given (2d linear saddle) using the code I made available at [github]. I found that in that case, ergodic quotient algorithm still works, even away from the asymptotic averaging limit.
Yesterday, I have given a presentation at AIMS Conference in Madrid, Spain. Braid Dynamics of Non-Periodic Trajectories [pdf]. Click on “Read more” in this post for the abstract.
The annotated reading list is below. I have copies of all the books listed so come and talk to me if you’re interested.
- An example of a common use of graph Laplacians as a stand-in for combinatorial optimization:
Shi, J., & Malik, J. (2000). Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8), 888–905.
- On structure of eigenvectors:
Biyikouglu, T., Leydold, J., & Stadler, P. F. (2007). Laplacian eigenvectors of graphs (Vol. 1915). Berlin: Springer.
- THE monograph on spectral graph theory, a lot of material, but very readable.
Chung, F. R. K. (1997). Spectral graph theory (Vol. 92). Published for the Conference Board of the Mathematical Sciences, Washington, DC. Sample chapters on Fan Chung’s page.
- The random walk picture:
Lovász, L. (1996). Random walks on graphs: a survey. In Combinatorics, Paul Erdős is eighty, Vol.\ 2 (Keszthely, 1993) (Vol. 2, pp. 353–397). Budapest: János Bolyai Math. Soc.
- A good cross-section through applications of spectral graph theory:
Mohar, B. (1997). Some applications of Laplace eigenvalues of graphs. In Graph symmetry (Montreal, PQ, 1996) (Vol. 497, pp. 225–275). Dordrecht: Kluwer Acad. Publ.