APS-DFD 2015 Meeting

Screen Shot 2015-12-01 at 2.02.30 PMLast week I traveled to Boston to attend the American Physical Society Division of Fluid Mechanics annual meeting. It was, as usual, a great experience – it’s always a pleasure to see what new things people have thought of since we last talked.

My talk was on the first day, at 8.52a, which made it difficult for some to attend. The slides to the presentation are below, along with updated references. Many thanks to Nick Ouellette whose laptop saved my talk from the claws of Macbook freeze.

In addition, I managed to visit a couple of friends in the area, and saw what MERL and MIT ENDLab look like from inside. Thank you Piyush and Margaux for hosting me.

Scientific software workflow

We know the big differences between trying out a calculation on a back of an envelope and writing a rigorous proof. Likewise, there are differences between prototyping in a Matlab script and writing a reliable pre that supports a reproducible research project. In a 15-minute presentation for a introductory workshop on scientific software at UW Madison math department, I showcased my own workflow on a simple example of plotting several solutions to an ODE in Matlab.

Repository for the workshop can be obtained by

git clone https://bitbucket.org/mbudisic/workshopworkflow.git

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FTBE paper published in Chaos and braidlab v.3.2 released

I’m happy to report that AIP Chaos published our paper on Finite-Time Braiding Exponents (co-authored with Jean-Luc Thiffeault):
MB. and Thiffeault, J.-L. Finite-time braiding exponents. Chaos: An Interdisciplinary Journal of Nonlinear Science 25, 087407 (2015).

In this paper, we use braid group generators to represent the entangling of trajectories of a dynamical system, and compute the Finite-Time Braiding Exponent (FTBE). FTBE should represent a finite-time, finite-information version of the topological entropy of the flow, and we show evidence that this is really the case.

A graph from the AIP Chaos paper on FTBE.

Mean FTBE and top. entropy correlate, and almost-match as number of strands is increased (Fig. 10 from the paper)

In addition to publishing a paper, we have also released the new version of our MATLAB toolbox braidlab (v.3.2) which was used for all computations in the paper. You can see the release notes on our GitHub repository, as well as download source and compiled versions of the toolbox. Please let us know if you’re using braidlab and, especially, submit any problems with it to our issues page.

SIAM DS15 Snowbird meeting

I am spending this week at my all-time favorite SIAM DS Snowbird meeting. As this is my 5th straight DS meeting, it really feels like coming home. I’ll be speaking about my work with Jean-Luc Thiffeault on Finite-Time Braiding Exponent (arXiv) on Wednesday at 3p (in Ballroom 2). My talk is in the MS92: Topological Fluid and Mass Dynamics, organized by Stefanella Boatto and Mark Stremler. The slides of the presentation are embedded below. Additionally, I am co-organizing a two-session minisymposium (MS111 and MS124) with Jean-Luc Thiffeault and Sanjeeva Balasuriya on Thursday. Come by and see what our speakers have done on control of fluids and things inside fluids.
Update: The video of the talk is available on SIAM website.

Compressive sensing

Today I gave a short introduction to compressive sensing, following the article:

Bryan, K., and T. Leise. “Making Do with Less: An Introduction to Compressed Sensing.” SIAM Review 55, no. 3 (January 1, 2013): 547–66. doi:10.1137/110837681.

The examples I’ve shown can be found below:

Two mode decompositions: POD and DMD

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).

UPDATE: I have now posted my own implementation of several algorithms for Koopman mode decompositions. [Software]

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Finite-Time Braiding Exponents

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.