Adjoint state method for fractional diffusion: Parameter identification

Fractional partial differential equations provide models for sub-diffusion, among which the fractal Mobile–Immobile Model (fMIM) is often used to represent solute transport in complex media. The fMIM involves four parameters, among which we have the order of an integro-differential operator that accounts for the possibility for solutes to be sequestered during very long times. To guess fMIM parameters from experiments, an accurate method consists in optimizing an objective function that measures how much model solutions deviate from data. We show that solving an adjoint problem helps accurate computing of the gradient of such an objective function, with respect to the parameters. We illustrate the method by applying it on experimental data issued from tracing tests in porous media.