Homogenization of metric Hamilton-Jacobi Equations
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Homogenization of metric Hamilton-Jacobi equations
Here we outline some homogenization problems. This is joint work by Adam Oberman (Simon Fraser), Ryo Takei (UCLA), and Alex Vladimirsky (Cornell)
Abstract
The objective of this work is the effective numerical solution of front propagation problems in multiscale media. We present a new approach which relates the cell problem for Hamitlon-Jacobi equation, the variational formulation for the Lagrangian, and the variational formulation for the metric. The main advantage of our approach is that we solve just one auxiliary equation to recover the homogenized Hamiltonian H(p). Previous methods require the solution of the cell problem (or a variational problem for each value of p. Computational results are presented in the periodic case for the checkerboard pattern, and several other patterns. Exact solutions are recovered numerically. We also present calculations in the random case.
For more details, see the following article which can be found with citations at Publications
Example Paths
What follows are examples of optimal paths in a periodic, and in a random medium. Click on the image for higher resolution views
Sample MATLAB code
The .zip file below contains a Matlab code that implements the metric homogenization algorithm described in the article. The fast marching code is externally implemented in C; thus, all *.c, *.h, and *.mex files need to be in the same directory as the Matlab script.
The Matlab script for the checkerboard and random cases can be executed by typing "metricHomoAlg('checker');" and "metricHomoAlg('random');", respectively. For further details on its syntax, type "help metricHomoAlg".
Related Work
Application to Cloaking
In addition to the homogenization of metrics, this work has applications to cloaking, rending objects invisible to electromagnetic fields. See Notes on Cloaking
Connections with Percolation
After completing this project we saw some connections between what we compute, and First Passage Percolation. For more information First Passage Percolation
Possible improvements in extracting the homogenized metric
The strength and weakness of taking a set of sample directions and interpolating depends on how the directions are chosen. This is difficult if the periodic medium has no obvious symmetry. Taking too few points may overlook important characteristics, while taking too many will "feel" the oscillation caused by the epsilon-periodic cells.
A more robust approach may be to view the problem as denoising, say, of the 1-level set of the epsilon periodic value function. This takes advantage of all directions (rather than a finite sampled set of directions), yet will automatically remove the ripples created by the epsilon-periodic cells. It will also be robust to rotations. Recent TV minimization denoising algorithm allows one to denoise contours fast.