Normalizations of the Proposal Density in Markov Chain Monte Carlo Algorithms

We explore the effects of normalizing the proposal density in Markov Chain Monte Carlo algorithms in the context of reconstructing the conductivity term $K$ in the $2$-dimensional heat equation, given temperatures at the boundary points, $d$. We approach this nonlinear inverse problem by implementing a Metropolis-Hastings Markov Chain Monte Carlo algorithm. Markov Chains produce a probability distribution of possible solutions conditional on the observed data. We generate a candidate solution $K'$ and solve the forward problem, obtaining $d'$. At step $n$, with some probability $\alpha$, we set $K_{n+1}=K'$. We identify certain issues with this construction, stemming from large and fluctuating values of our data terms. Using this framework, we develop normalization terms $z_0,z$ and parameters $\lambda$ that preserve the inherently sparse information at our disposal. We examine the results of this variant of the MCMC algorithm on the reconstructions of several $2$-dimensional conductivity functions.