Current Research

Frozen Gaussian Approximation: Elastic Wave Equation in an isotropic media

The Frozen Gaussian approximation (FGA) performs better than the Gaussian Beam Method and overcomes difficulty of other methods from geometric optics for high frequency wave propagation. Currently I'm programming the FGA to test the smoothed marmousi model and interface conditions for the crust over mantel model using fortran and mpi to run on a cluster. The code for 2d in MatLab can be found at can be found on bitbucket in a repository under my username jhateley.

CVT/ODT Smoothing

There are several way to go about mesh smoothing. Considering the energy functional for an ODT and a CVT, minimizing both simultaneously leads to the optimization method(s) "fighting" with eachother most of the time. A local minimum for the ODT energy functional is not necessarily a minimum for the CVT functional. Decomposing these two meshes and cosidering quadrilaterals formed. One can consider this energy funcitonal and minimize it, effectively finding a good balance between both meshes.

Projects and Research

Submitted Research

fast methods for CVT smoothing: We've accelerated the run time of a Voronoi tessellations to a centroidal Voronoi tessellation (CVT), using the graph Laplacian as a preconditioner in a BFGS scheme. Code for Scalar quantization. The Code using the BFGS and NCLG optimization methods for finding a 2D CVT with our graph Laplacian approximation as a preconditioner can be found on bitbucket in a repository under my username jhateley.

CVT Paper: DOI: 10.1007/s10915-014-9894-1

Local Copy

CVT Presentation

On going projects:

Hamiltonian path: I'm trying developing an algorithm to reconstruct a polyhedra given a set of vertices. Like the development of many algorithms, the goal is to have a speedy run time. Ricci flow fails on a certain class of polyhedra. Mollifing the polyhedra and assuming positive curvature, I'm tring to construct a "barrier" operator that deforms a sphere into my polyhedra.

Various Matlab files

Derivative using the Fast Fourier Transform.
Integration using the Fast Fourier Transform.
Cubic Spline interpolation.

1D V-cycle multigrid for Poissons equation using recursion.
Various Spectral/(Psuedo Spectral) Methods for the Advection Diffusion equation .
chebyshev interpolation with Gauss Lobatto points /w examples .

Gaussian elimination with full pivoting.
Linear solver using Jacobian iterations or Gauss-Seidel interations.

Papers / Special Lectures

The Lorenz system

Notes about the Lorentz system, an introduction to Chaos.

An overview of static Hamilton-Jacobi equations

ABSTRACT:There is a voluminous amount of literature on Hamilton-Jacobi equations. This paper reviews some of the existence and uniqueness techniques for the time independent cases with a particular interest in Eikonal-like Equations. In addition, key features of these equations, that have helped develop numerical schemes are noted.

A Review of Rational Tangles

ABSTRACT: The idea of a rational tangle has aided with the classification of knots. This paper reviews the general definitions of knots and tangles, combinatorial proofs of the classifications for rational tangles and poses an operation to transform knots into tangles.

Minimum path in a Euclidean Space

ABSTRACT: Given a finite number of points and a fixed starting point in Euclidean Space 2 or 3 space, a minimum or minimal distance between a finite number of points is desired for many applications in the sciences. Using the property that a respective sphere is classified by its surface, linear maps and geodesic curves can be used to order a set of points to contain the minimum or a minimal distance.