# Bayesian inversion benchmark: Matlab version
 
This code implements a Matlab version of the benchmark in D. Aristoff and W. Bangerth, "A benchmark for the Bayesian inversion of coefficients in partial differential equations," arXiv:2102.07263. The article is available on the web at https://arxiv.org/abs/2102.07263.

The benchmark is intended for comparing advanced methods for Bayesian inverse problems. The benchmark is designed to be more complex than simple testcases like Gaussian mixtures, but not so complex as to be intractable or unrepeatable. The benchmark concerns the determination of a spatially variable coefficient, discretized by 64 values, in a Poisson equation, based on point measurements of the solution. 

This code runs a basic Metropolis-Hastings sampler for the posterior distribution. To run it, put all the .m and the .mat files from this repository in your current Matlab directory, and type "main" into the Matlab command window. This starts the main program, main.m, which lets you can choose a number of independent Markov chains, the length of each chain, the lag time between recording samples, and the number of parallel workers to use. Other parameters for the simulations can be found within precomputations.m.

The posterior distribution has argument theta, which is a vector of length 64 representing the spatially variable coefficient in the Poisson equation. This vector represents the values of the coefficient in a 8x8 grid. In this code, theta is represented by an 8x8 matrix. To convert a vector, theta, of length 64 -- with entries associated to coefficients as described in the article above -- into an equivalent 8x8 matrix suitable for this code, use the Matlab command "theta = reshape(theta,[8 8])';". The matrix of exact coefficients can be obtained using the command "theta = ones(8,8); theta(2:3,2:3) = 0.1; theta(6:7,6:7) = 10;".