A Bayesian inference approach to the inverse heat conduction problem

Abstract

A Bayesian inference approach is presented for the solution of the inverse heat conduction problem. The posterior probability density function (PPDF) of the boundary heat flux is computed given temperature measurements within a conducting solid. Uncertainty in temperature measurements is modeled as stationary zero-mean white noise. The inverse solution is obtained by computing the expectation of the PPDF. The posterior state space is exploited using Markov chain Monte Carlo (MCMC) algorithms in order to obtain estimates of the statistics of the unknown heat flux. The MCMC sampling strategy enables the extension of the Bayesian inference approach to inverse problems having high-dimensional, non-standard distribution, and/or complex PPDFs. The ill-posedness (un-identifiability) of the inverse problem is cured through prior distribution modeling (Bayesian prior regularization) of the unknown heat flux. A special model of Markov random field (MRF) is adopted for prior distribution modeling of the unknown heat flux. An augmented Bayesian model is proposed for estimating the statistics of the measurement noise as well as the unknown heat flux. Two inverse heat conduction examples are presented to demonstrate the potential of the MCMC-based Bayesian approach. The simulation results indicate that MRF provides an effective prior regularization, the estimates using MCMC samples are accurate and the Bayesian approach captures very well the probability distribution of the unknown heat flux.