Hierarchical Bayesian models for inverse problems in heat conduction

Stochastic inverse problems in heat conduction with consideration of uncertainties in measured temperature data, thermal sensor location and material thermal properties are addressed using a Bayesian statistical inference method. Both parameter estimation and thermal history reconstruction problems, including boundary heat flux and heat source reconstruction, are studied. The probabilistic specifications of these unknown variables are deduced from temperature measurements. A joint probability distribution approach is taken to specify the conditional (on data) state space of random unknown quantities by multiplying the likelihood and prior distribution functions. Hierarchical Bayesian models are adopted to relax the prior assumptions on the unknowns. The use of a hierarchical Bayesian method for automatic selection of regularization parameter for function estimation inverse problems is discussed. This methodology explores the multi-scale spatial prior models in estimation of temporalspatially varying thermal quantities. Due to the high dimensionality and implicit form of the posterior distribution, Markov chain Monte Carlo (MCMC) simulation is conducted to explore the posterior state space. The methodologies presented are general and applicable to a number of data-driven engineering inverse problems.