Linear inverse problems in viscoelastic continua and a minimax method for Fredholm equations of the first kind

A new theory based on an extensively modified version of the minimax method is proposed to estimate the cause from the result, that is, the characteristic functions of viscoelastic media from experimentally obtained material functions through the solution of Fredholm integral equations of the first kind. The method takes into account the non-Gaussian outliers, and does not require the assumption of a priori error bounds as in other smoothing techniques which may lead to instability or to a stable solution not representative of the true solution. The algorithm is applied to several hypothetical test problems to show the excellent performance of the method in extreme severe conditions. The shortcomings of the Tikhonov's regularization and other smoothing techniques are discussed. It is shown that the solution via these methods may not represent the real solution in any norm. The new method is applied to linear viscoelasticity to obtain the relaxation spectrum from experimental material functions. The relaxation spectra of some materials obtained via the proposed adaptive-robust minimax algorithm and experiments run in a rotary viscometer are presented.