An efficient Markov chain Monte Carlo simulation of a stochastic inverse radiation problem

A novel methodology that combines recent advances in computational statistics and reduced-order modeling is presented to explore the application of Bayesian statistical inference to a stochastic inverse problem in radiative heat transfer. The underlying objective of this work is to reveal the potential of using statistical approaches, mainly Bayesian computational statistics and spatial statistics, to solve data-driven stochastic optimization and uncertainty quantification problems raised in various complex continuum system design and control applications when the robustness and reliability requirements are critical. In this work, an unknown transient heat source in a three-dimensional participating medium is reconstructed from the temperature measurements. The heat source is modeled as a stochastic process, of which the joint posterior probability density function (PPDF) is computed using the Bayes’ formula. Random errors in thermocouple readings are modeled as Gauss random variables. ‘Maximum A Posteriori’ (MAP) and posterior mean estimates of the heat source are then computed. A Markov chain Monte Carlo (MCMC) sampler composed of a cycle of symmetric MCMC kernels is designed to explore the posterior state space numerically. To expedite the sampling speed, a proper orthogonal decomposition (POD) based reduced order modeling technique is used in the likelihood computation. Typical heat source profiles are reconstructed using the simulated data to demonstrate the presented methodology. The results indicate that the Bayesian inference method can provide accurate point estimates as well as uncertainty quantification to the solution of the inverse radiation problem.