Application of the generalized multiscale finite element method in an inverse random source problem

Abstract In this work, an inverse random source problem in the fractional diffusion equation with heterogeneous medium is considered. The measurements used are the statistical moments of the realizations of the single point data u ( x 0 , t , ω ) . We build the representation of the solution u in integral sense, then prove that the unknowns can be bounded by the moments theoretically. For the numerical reconstruction, to handle the highly heterogeneous medium, the generalized multiscale finite element method (GMsFEM) will be employed in the forward problem solver. With the simulated data, we establish an iterative algorithm of regularized Levenberg-Marquardt type, and some numerical results generated from this algorithm are displayed.