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In this paper, the adaptive multigrid conjugate gradient method is applied to the inversion of the space-dependent permeability function in a nonlinear diffusion equation, which plays an important role in promoting the permeability estimation within multiphase porous media flow. The forward problem is discretized using finite-difference methods and the estimation is formulated as a least-square minimization problem with a regularization term. Firstly, the conjugate gradient method for optimization is used to solve this parameter identification problem. Secondly, to reduce the amount of calculation, the idea of multigrid inversion and the necessary condition of “the optimal solution should be the fixed point of multigrid inversion” is considered. Then, by combining the multigrid multiscale method with the conjugate gradient method, a new inversion method called the adaptive multigrid conjugate gradient method is designed. The results of numerical simulations indicate that the proposed method can effectively reduce the required computation burden and have good ability in anti-noise.

An adaptive multigrid conjugate gradient method for permeability identification of nonlinear diffusion equation