Cauchy problem of non-homogenous stochastic heat equation and application to inverse random source problem

In this paper, a Cauchy problem of non-homogenous stochastic heat equation is considered together with its inverse source problem, where the source term is assumed to be driven by an additive white noise. The Cauchy problem (direct problem) is to determine the displacement of random temperature field, while the considered inverse problem is to reconstruct the statistical properties of the random source, i.e. the mean and variance of the random source. It is proved constructively that the Cauchy problem has a unique mild solution, which is expressed an integral form. Then the inverse random source problem is formulated into two Fredholm integral equations of the first kind, which are typically ill-posed. To obtain stable inverse solutions, the regularized block Kaczmarz method is introduced to solve the two Fredholm integral equations. Finally, numerical experiments are given to show that the proposed method is efficient and robust for reconstructing the statistical properties of the random source.