On the connection and equivalence of two methods for solving an ill-posed inverse problem based on FRAP data

Abstract

Two methods for solving an ill-posed inverse problem based on Fickian diffusion equation and spatio-temporal data from FRAP measurements are presented. The most usual method is the Tikhonov regularization. Nevertheless, in our specific problem we have detected difficulties residing in determination of the optimal regularization parameter α. Hence, an equivalent method based on least squares with a quadratic constraint regularization is proposed. This latter approach naturally takes into account the noise level in the data and corresponds to Morozov’s discrepancy principle as well. The equivalence of both methods is rigorously proven and on a simple numerical example with synthetic input data practically documented.