Bayesian inference for solving a class of heat conduction problems

Complex thermal systems are usually disturbed by random factors and need to be modeled by random differential equations. Scholars have grown interests in numerical methods for stochastic differential equations. Milstein[1] used Taylor’s expansion to get a numerical solution of a stochastic differential equation, Higham [2] studied the mean-square stability of the Euler-Maruyama method, Higham et al.[3] proved the convergence of the Euler-Maruyama method under the condition of non-total Lipschitz, Hutzenthaler [4] proposed an explicit numerical method for solving stochastic differential equation with non-global Lipschitz continuous coefficients and proved its strong convergence. Wang and Gan [5] suggested an explicit strongly convergent numerical scheme for stochastic differential equations with commutative noise. Various numerical methods have been developed to estimate the parameters of stochastic differential equations, for example, the Gibbs algorithm proposed by Gemans [6], and the resampling algorithm given by Gordon et al. [7]. Eraker[8] used the Bayesian method to discuss estimation of the parameter in model with a stochastic volatility component. Golightly and Wilknson discussed the parameter estimation of the nonlinear multivariate diffusion models based on the missing data [9]. Miguez, et al. [10] discussed the sequential Monte Carlo method in general state-space models. Many studies have been conducted in the field of heat conduction. The interdisciplinary study has also developed rapidly and has penetrated to many disciplines. Various methods for solving the thermal conductivity of materials have been proposed, such as the spirit sensitivity method, the least squares method, the regularization method, and the conjugate gradient method. Martín-Fernández