Title: Bayesian Inference for Estimating Heat Sources through Temperature Assimilation

Abstract: This paper introduces a Bayesian inference framework for two-dimensional steady-state heat conduction, focusing on the estimation of unknown distributed heat sources in a thermally-conducting medium with uniform conductivity. The goal is to infer heater locations, strengths, and shapes using temperature assimilation in the Euclidean space, employing a Fourier series to represent each heater's shape. The Markov Chain Monte Carlo (MCMC) method, incorporating the random-walk Metropolis-Hasting algorithm and parallel tempering, is utilized for posterior distribution exploration in both unbounded and wall-bounded domains. Strong correlations between heat strength and heater area prompt caution against simultaneously estimating these two quantities. It is found that multiple solutions arise in cases where the number of temperature sensors is less than the number of unknown states. Moreover, smaller heaters introduce greater uncertainty in estimated strength. The diffusive nature of heat conduction smooths out any deformations in the temperature contours, especially in the presence of multiple heaters positioned near each other, impacting convergence. In wall-bounded domains with Neumann boundary conditions, the inference of heater parameters tends to be more accurate than in unbounded domains.