Category Archives: My Work

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

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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.

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. [GitHub]

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Ergodic quotient in a finite-time, expanding context

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.
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My whiteboard scribbles!

Julia Collins, a mathematician at U of Edinburgh, maintains a website that showcases photos of mathematicians’ black- and whiteboards. She published mine! Check it out here:

What’s on my whiteboard (Sep 2012)

From Chuck Akemann's class

From Chuck Akemann’s class homework


Presentation from 65th APS DFD (San Diego)

Below are the presentation slides from my talk “Visualization of invariant sets in incompressible fluid flows from Lagrangian data”. The links to the referenced articles:

Published: Conditioning of measures for entropy optimization

2012-03-16: Dr. Mihai Putinar and I have completed a paper on entropy optimization of singular measures. The pre-print is available on arXiv:
Conditioning moments of singular measures for entropy optimization. I

2012-11-05 UPDATE: The paper has been officially published by Indagationes Mathematicae: DOI: 10.1016/j.indag.2012.05.008