Inverse Problem for the Nonlinear Convection–Diffusion Equation by Using the Multigrid Method and Constraint Data

In the article, we propose a combination method based on the multigrid method and constraint data to solve the inverse problem in the context of the nonlinear convection–diffusion equation in the multiphase porous media flow. The inverse problem consists of a data-fitting term involving the discretization of a direct problem, a constraint term concerning the incorporation of constraint data, and a regularization term dealing with the improvement of stability. A multigrid method, which is specialized for large-scale problems and works by keeping the consistence of objective functionals between different grids, is applied in the process of inversion. Based on the numerical results, the proposed combination method has the advantages of fast calculation, high precision, good stability, and strong anti-noise ability in computation. It obtains good performance under various noise levels, as well as outperforming any one method used alone.