Normalizations of the Proposal Density in Markov

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 α, we set Kn+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 z0, z and parameters λ that preserve the in-herently 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