A multigrid–homotopy method for nonlinear inverse problems

Abstract

In the present contribution, we develop a novel method combining the multigrid idea and the homotopy technique for nonlinear inverse problems, in which the forward problems are modeled by some forms of partial differential equations. The method first attempts to use the multigrid method to decompose the original inverse problem into a sequence of sub-inverse problems which depend on the grid variables and are solved in proper order according to the grid size from the coarsest to the finest, and then carries out the inversion on the coarsest grid by the homotopy method. The strategy may give a rapidly and globally convergent method. As a practical application, this method is used to solve the nonlinear inverse problem of a nonlinear convection–diffusion equation, which is the saturation equation within the two-phase porous media flow. We demonstrate the effectiveness and merits of the multigrid–homotopy method on two actual model problems.