A Bayesian inference approach to the ill‐posed Cauchy problem of steady‐state heat conduction

This paper studies a Bayesian inference approach to the Cauchy problem in steady‐state heat conduction of probabilistically calibrating the boundary temperature. The prior modeling is achieved via the Markov random field, and its regularizing property is investigated. A hierarchical Bayesian model is adopted for selecting the regularization parameter and detecting the noise level automatically. The posterior state space is explored using the Markov chain Monte Carlo for obtaining relevant statistics. Two augmented Tikhonov regularization methods that could determine the regularization parameter and the noise level are proposed and analyzed. Numerical results indicate that the Bayesian inference approach could yield an accurate estimate of the solution with its uncertainties quantified, and the augmented Tikhonov regularization methods are accurate and flexible.