Fast Bayesian inference for inverse heat conduction problem using polynomial chaos and Karhunen–Loeve expansions

We present a fast Bayesian inference framework for solving inverse heat conduction problems (IHCP) for the estimation of boundary heat flux. The framework leverages the polynomial chaos expansions (PCEs) for generating computationally efficient and statistically equivalent surrogate model of the computationally expensive forward model and dimensionality reduction based on Karhunen–Loeve (K–L) expansion. We demonstrate the potential of this approach using three model problems to estimate heat flux in inverse heat conduction models. We represent the heat flux using K–L expansion to reduce the high dimensionality of the heat flux and compute the posterior probability distribution of the heat flux using the temperature measurements. The first inverse problem involves the estimation of time–varying heat flux in a one–dimensional slab using transient temperature measurements. The second problem involves the estimation of the unknown time–dependent heat flux of the disc in a realistic axisymmetric disc brake system. The third problem involves the estimation of the surface heat flux on a sounding rocket having a realistic three–dimensional geometry of the rocket module used in actual flight tests. We further obtain the inverse solution by computing the expectation of the distribution and demonstrate that this approach becomes orders of magnitude more efficient when we use the PCE surrogate model instead of the direct forward model for comparable accuracy. The numerical results show that this framework overcomes the drawback of high computational cost of Markov chain Monte Carlo (MCMC) without compromising on solution accuracy. The combination of PC–based surrogate modeling and K–L based dimensionality reduction leads to over 2000–fold speed up of numerical solution of IHCP.