Reconstruction of a permeability field with the wavelet multiscale–homotopy method for a nonlinear convection–diffusion equation

This paper deals with the reconstruction of a piecewise constant permeability field for a nonlinear convection–diffusion equation, which arises as the saturation equation in the fractional flow formulation of the two-phase porous media flow equations. This permeability identification problem is solved through the minimization of a cost functional which depends on the discrepancy, in a least-square sense, between some measurements and associated predictions. In order to cope with the local convergence property of the optimizer, the presence of numerous local minima in the cost functional and the large computational cost, a wavelet multiscale–homotopy is developed, implemented, and validated. This method combines the wavelet multiscale inversion idea with the homotopy method. Numerical results illustrate the global convergence, computational efficiency, anti-noise ability of the proposed method.