Enabling Equation-Free Modeling via Diffusion Maps

Equation-free modeling estimates macro-level behavior through a coarse time stepper in three steps: (1) lift: build the microstate from the macrostate, (2), evolve: simulate the microsystem for short bursts, and (3) restrict: estimate the macrostate from the evolved microstate.

In this work, we use diffuson maps to identify macroscopic variables in a traffic model and define appropriate lifting and restriction operators. We then use these operators to compute and continue traffic jam solutions.

Folder navivgation

1) src contains the source code

2) data contains raw data in .csv files

3) results contains .csv files of results

Generate traffic data

Data used in the diffusion maps is created with genTrafficData.m and saved in the /data directory.

Compute a diffusion map

Diffusion maps for the traffic data are computed with and explored through createDiffMaps.m.

Test the lifting and restriction operators

We test the accuracy of the lifting and restriction operators with the script testOperators.m.

Computing the bifurcation diagram

1) Using the microsystem: microBifurcation.m

2) Using the standard deviation as a marco variable: eqFreeBifurcation.m

3) Using a 1D diffusion map applied to phase-shifted traffic profiles: eqFreeDiffBifurcation.m

4) Using a 2D diffusion map applied to traffic profiles: eqFreeDiffBifurcation2D.m

Visualizing results

Plot all results in plot_eq_free.ipynb

BibTex Citation:

@article {eq-free-traffic,  
	author = {Chin, Tracy and Ruth, Jacob and Sanford, Clayton and Santorella, Rebecca and Carter, Paul and Sandstede, Bjorn},  
	title = {Enabling Equation-Free Modeling via Diffusion Maps},  
	year = {2021},  
}