I was doing a little bit of research on Mulit-stage Markov Chains for a friend when I ran across this nice set of slides on Multi-stage Markov Chains for diffusion through porous media (i.e. oil or water flowing through the ground) by Pereira and Rahunanthan.
The authors first review the multi-scale equations for diffusion under pressure (see Darcy’s law) and mixed finite element analysis. Then, interestingly they introduce a Bayesian Markov chain Monte Carlo method (MCMC) to get approximate solutions to the differential equation. They use Multi-Stage Hastings-Metropolis and prefetching (to parallelize the computation) to improve the speed of MCMC. The slides also display the results of their algorithm applied to aquifer contamination and oil recovery.
It’s really cool how methods used for probabilistic graphical models can be used to approximate the solutions to partial differential equations.
I’m hoping to get back to 2048 in a few weeks. Turns out that it takes a long time to write code, make the code look nice, run the code, analyze the results, and then put together a blog post. It’s much easier and quicker to read papers and summarize them or to try to explain things you already know.
Have a great weekend. – Hein