Normalizations of the Proposal Density in Markov Chain Monte Carlo Algorithms

TL;DR

This paper investigates how normalizing proposal densities in MCMC algorithms affects the reconstruction of conductivity in a 2D heat equation inverse problem, proposing normalization techniques to improve solution stability.

Contribution

It introduces normalization terms and parameters in MCMC algorithms to address issues caused by large data fluctuations in reconstructing conductivities.

Findings

Normalization improves stability of MCMC reconstructions.

Proposed methods effectively handle large data fluctuations.

Enhanced reconstruction accuracy demonstrated on 2D conductivity functions.

Abstract

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.