Inverse problem in anomalous diffusion with uncertainty propagation
Recently, Bevilacqua, Galeão and co-workers have developed a new analytical formulation for the simulation of diffusion with retention phenomena. This new formulation aims at the reduction of all diffusion processes with retention to a unifying model that can adequately simulate the retention effect. This model may have relevant applications in a number of different areas such as population spreading with partial hold up of the population to guarantee territorial domain chemical reactions inducing adsorption processes and multiphase flow through porous media. In this new formulation a discrete approach is firstly formulated taking into account a control parameter which represents the fraction of particles that are able to diffuse. The resulting governing equation for the modelling of diffusion with retention in a continuum medium requires a fourth-order differential term. Specific experimental techniques, together with an appropriate inverse analysis, need to be determined to characterize complementary parameters. The present work investigates an inverse problem which does not allow for simultaneous estimation of all model parameter. In addition a two-step characterization procedure is proposed: in the first step the diffusion coefficient is estimated and in the second one the complementary parameters are estimated. In this paper, it is assumed that the first step is already completed and the diffusion coefficient is known with a certain degree of reliability. Therefore, this work is aimed at investigating the confidence intervals of the complementary parameters estimates considering both the uncertainties due to measurement errors in the experimental data and due to the uncertainty propagation of the estimated value of the diffusion coefficient. The inverse problem solution is carried out through the maximum likelihood approach, with the minimization problem solved with the Levenberg-Marquardt method, and the estimation of the confidence intervals is carried out through the Monte Carlo analysis.