Dongyang Kuang

Dongyang Kuang

Postdoc @ Oden Institute，UT-Austin

geodesics on 2d surfaces

1 minute read

Published:

Geometries can be fun, though it will require a deep and strict mathematical formulation. Physics and geometry has nature and deep connections with each other. It is well known that einstein’s theory of general relaltivity will need the support from non-Euclidean Geometry. One of the interesting topics in geometry is about finding geodesics. You can image the geodesic as a curve that minimizes certain metrics bewteen two fixed points in the considered space. One natural way of formulating the problem formally is by calculus of variations where you write your target metric as a functional and the Euler - Langrange Equation of this functional will govern the behavior of the geodesic. Usually, if the configuration space will have dimension 2N where N is the dimension of the manifold with geodesics on. There is a beatiful theorm: Noether’s theorem that can help deduct the dimension to N by solving a corresponding equation on the Lie Algebra. I think Prof. Terrence Tao’s post gives a very good explanation about this techique.

If the manifold is a smooth 3d graph or a smooth 2d surface where things are not too complicated there. We can directly work on the configuration space, write down the patch map, the corresponding EL-equation and uses certain numerical schemes to integration out the solution. It should be aware that systems like this usually will have intrinsic invariance or symmetries, so it is better to use a symplectic integrator for long geodesics.

The following .mp4 files illustrates some geodesics on selected surfaces. You can download .py files here for the revolutional ellipzoid animation and here for the Gaussian-density-liked surface.

Q: For general ellipzoid, i.e.

$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$,

where $a$, $b$, $c$ are all different. The program can give very wired curves. I did not find many useful contents about geodesics on general ellipzoid from google, is it still an open question? or at least part of it?

  • One geodesic on a revolutional ellipzoid

  • Another geodesic on the same revolutional ellipzoid

  • Different geodesics on the surface $ f(x, y) = exp(-\frac{x^2}{2}-y^2)$

You May Also Enjoy

Solving ODE/PDE with Neural Networks

5 minute read

Published:

Differential equations and neural networks are naturally bonded. The best paper “Neural Ordinary Differential Equations” in NeurIPS 2018 caused a lot of attentions by utilizing ODE mechanisms when updating layer weights. On the other direction, there are also many research using neural network approaches to help investigate differential equations such as “Deep learning for universal linear embeddings of nonlinear dynamics”, “DGM: A deep learning algorithm for solving partial differential equations” or “Solving Irregular and Data-enriched Differential Equations using Deep Neural Networks”. In this post, I will make two toy examples to show the very the basic idea of using deep learning method for solving differential equations.

Matching-algorithm-for-Raman-Spectra

4 minute read

Published:

Some intro

Recently I attended a workshop helping solve industrial problem hosted by the Fields Institute. One of the problems presented is developing accurate/efficient methods for matching Raman spectra from test sample to samples recorded in the library so that different chemicals can be detected effectively. This is a quite chanllenging problem. Theoretically speaking, though Raman spectroscopy is commonly used in chemistry to provide a structural fingerprint by which molecules can be identified, there are a huge amount of chemicals out in the nature among which many have quite similar Raman spectra. The fact that test samples are usually a mixture of different molecules make the problem even more difficult. Pratically speaking, spectra data recorded is not perfect. There will be noises of different kinds and background/baseline signal flooding the useful information.

Uottawa VPN connection via wifi

less than 1 minute read

Published:

It seems that the bash script provided by the university does not work for my machine with Ubuntu 16.04 LTS. For a wireless connection through VPN to be able to be “on campus”, you can follow the easy steps listed below.

  • Download the configuration at here. You must be on campus to download it.
  • Run the following commends in the location where you saved the configuration file:
    pcf2vpnc empl.pcf  empl.conf  
sudo cp empl.conf /etc/vpnc/  
sudo vpnc empl
  • If previous commands go well, you will be asked to provide username and account. (You can also hard code username and password in empl.conf file by uncommenting Xauth username Xauth password)

Booting from grub

1 minute read

Published:

The Problem

I have a dual system window10/Ubuntu16.04 installed in my laptop. Today I can not access window files from Ubuntu and tried one command line from youtube which seems to mess things up :< The system did not boot like before but entering into the grub prompt instead. It seems that system does not know where/what to boot now and may need a manual configuration.