List of publications:
A. Tcheng & J.-C. Nave. A low-complexity algorithm for non-monotonically evolving fronts. Submitted on Sept. 12, 2014. pdf
A. Tcheng & J.-C. Nave. A fast-marching algorithm for non-monotonically evolving fronts. Submitted on April 16, 2015. pdf
A. Tcheng, B. Keith & J.-C. Nave. General embedding methods for solving PDEs on Riemannian manifolds. In preparation.

Click on the title of a project to learn more about it.

Tracking curve evolution using fast finite-differences schemes. Ph.D. project. September 2011 - Present.
Ph.D. work on the development of a numerical scheme to model curve motion.

The goal of this project is to design an algorithm of low complexity describing the evolution of co-dimension one manifolds. eg: A curve in xy-space evolving in time. We assume that the front gets propagated in the normal direction with a prescribed speed function F which may depend on space as well as on time, and may vanish and change sign. To achieve our goal, we make use of the level-set method, the fast marching method, and borrow some ideas from control theory, and the rich theory on Hamilton-Jacobi equations.
The resulting first-order algorithm is an augmented version of the traditional fast marching method: In regions where F is close to 0, a different formalism is used to accurately capture the evolution of the front. The predicted complexity of this method is comparable to that of the fast marching method.
Under the supervision of Prof.J.-C. Nave.

Here is a video from the second part of this project: A circle is evolved under the time-dependent speed F(t)=1-2t.
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Numerically solving PDEs on Riemannian manifolds. July 2012 - Present.
Development of a general method to numerically solve PDEs on manifolds.

Let a PDE be given on a manifold M of dimension n-1 that embeds in R^n. Consider a companion PDE, say PDE*, defined on a neighbourhood of M. The first goal of this project is to investigate the conditions under which PDE* is well-posed, and such that the restriction of its solution to M yields the solution to the original PDE given on M. Secondly, a numerical algorithm is proposed to solve PDE* on a Cartesian grid. It is run on various examples, such as the heat equation on a circle, and the traffic flow equation on a race track.
In collaboration with Brendan Keith, under the supervision of Prof.J.-C. Nave.

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The Einstein's constraint equations. Master's project. September 2009 - November 2011
The Einstein constraint equations.

In this project, the questions of existence & uniqueness of solutions to the Einstein constraint equations on compact three-dimensional manifolds was addressed. A toy-model was investigated following the work of Prof.D.Maxwell, with explicit results on the existence & uniqueness of solutions on a Yamabe-positive manifold. Here is my Master's thesis.
Under the supervision of Prof.G.Tsogtgerel.

Summer project: Cosmology, hybrid inflation. May 2008 - August 2008
Group project in cosmology on hybrid inflation.

This project started with lectures on general relativity between upper-year undergraduates. It then turned into a research project on cosmology. Four of us studied the influence of second order perturbations in entropy at the end of hybrid inflation, using both a perturbative expansion and an analytical approach. My main contribution was to implement the numerical simulations of those models, which were then compared to the expected theoretical results.
Under the supervision of Prof.R.H.Brandenberger.

Summer project: Introduction to quantum field theory. May 2008 - August 2008
Introduction to Quantum Field Theory. Introduction to Lie Algebra and applications to particle physics. Monopole solutions.

This project was an informal reading course on field theory and its applications to magnetic monopole solutions. At the end of the summer, two presentations were given to final year students.
Under the supervision of Prof.K.Dasgupta.