The paper is devoted to an investigation of a number of issues related to a multilevel continuation approach for the rapid solution of a nonlinear inverse problem. These include the use of grids for the chosen model and forward solution, as well as the regularization matrix and regularization parameter. The proposed solution includes a search for the regularization parameter in a coarse grid, and then a gradual refinement technique for finding both the forward and inverse solutions on finer grids. The grid spacing and question of weighting of the entire regularization term is studied. Correct scaling, nonuniform grid usage and interpolation for grid refinement are considered. Finally, important remarks and conclusions about the proposed solution are provided.