A wavelet multiscale method for the inverse problem of a nonlinear convection-diffusion equation

This paper is concerned with the problem of identifying the diffusion parameters in a nonlinear convection-diffusion equation, which arises as the saturation equation in the fractional flow formulation of the two-phase porous media flow equations. The forward problem is discretized using finite-difference methods and the inverse problem is formulated as a minimization problem with regularization terms. In order to overcome disturbance of local minimum, a wavelet multiscale method is applied to solve this parameter identification inverse problem. This method works by decomposing the inverse problem into multiple scales with wavelet transform so that the original inverse problem is reformulated to a set of sub-inverse problems relying on scale variables, and successively solving these sub-inverse problems according to the size of scale from the smallest to the largest. The stable and fast regularized Gauss-Newton method is applied to each scale. Numerical simulations show that the proposed algorithm is widely convergent, computationally efficient, and has the anti-noise and de-noising abilities.