Using Bayesian statistics in the estimation of heat source in radiation Jingbo Wang, Nicholas Zabaras * Materials Process Design and Control Laboratory, Sibley School of Mechanical and Aerospace Engineering, 188 Frank H.T. Rhodes Hall, Cornell University, Ithaca, NY 14853-3801, United States Received 5 January 2004; received in revised form 10 August 2004 Available online 14 October 2004 Abstract An unknown transient heat source in a three-dimensional participating medium is reconstructed from temperature measurements using a Bayesian inference method. The heat source is modeled as a stochastic process. The joint pos- terior probability density function (PPDF) of heat source values at consecutive time points is computed using the BayesÕ formula. The errors in thermocouple readings are modeled as independent identically distributed (i.i.d.) Gauss random variables. ÔMaximum A PosterioriÕ (MAP) and posterior mean estimates of the heat source are then computed using a Markov chain Monte Carlo (MCMC) simulation method. The designed MCMC sampler is composed of a cycle of sym- metric MCMC kernels. To increase the sampling speed, a model-reduction technique is used in the direct computation of temperatures at thermocouple locations given a guessed heat source, i.e. in the likelihood computation. Two typical heat source profiles are reconstructed using simulated data to demonstrate the presented methodologies. 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. Ó 2004 Elsevier Ltd. All rights reserved. 1. Introduction Study of thermal radiation has been stimulated by a wide range of applications including thermal control in space technology, combustion, high temperature form- ing and coating technology, solar energy utilization, high temperature engine, furnace technology and other [1]. In participating media, radiation is accompanied by heat conduction and convection. To simulate such proc- esses, a coupled system of partial differential equations (PDEs) governing temperature and radiation intensity evolution needs to be solved iteratively. Difficulties arise in the solution of such systems because the heat flux con- tributed by radiation varies nonlinearly with the temper- ature, the radiation intensity varies in space and in direction, and the radiation intensity equation is an inte- gro-differential equation [2]. The direct radiation prob- lem, in which the temperature distribution is computed with prescribed thermal properties, source generation and initial/boundary conditions, is often solved using a combination of spatial discretization methods such as finite volume or finite element methods (FEM) and ordinate approximation such as P N and S N methods [2]. The inverse radiation problem in a participating 0017-9310/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2004.08.009 * Corresponding author. Tel.: +1 607 255 9104; fax: +1 607 255 9410. E-mail address: zabaras@cornell.edu (N. Zabaras). URL: http://www.mae.cornell.edu/zabaras/. International Journal of Heat and Mass Transfer 48 (2005) 15–29 www.elsevier.com/locate/ijhmt