Deconvolution for oscillatory shear rheometry using the landweber iteration

In the deconvolution of convolution equations of the form pp-hqpxq-» 8-8 ppx-yqhpyqdy-gpxq; where the kernel p is specified explicitly, the goal is to recover from measurements g, estimates of the corresponding solution h. Such situations arise in a wide range of applications including, in rheology, the recovery of estimates of the storage and loss moduli characterization of a linear viscoelastic material from oscillatory shear measurements (Davies and Goulding 2012). Various algorithms have been proposed for performing the deconvolution iteratively. The classical and historic example of the Neumann iteration has been examined in Anderssen et al. 2019, where conditions have been established that guaranteed its theoretical convergence. It is also noted that the corresponding numerical convergence is quite sensitive to the underlying frequencies in the discrete data gn used to model g. Thus, the presence of noise, particularly at high frequencies, can give rise to poor convergence behavior. This leads to the idea that the numerical convergence might be improved by first smoothing the discrete d ata. One way of achieving this is to use the Landweber iteration (Landweber 1951), as it corresponds to generating the iterative solution of the least squares counterpart of p-h-g, namely p-p-h-p-g. It is shown that, though the Neumann iteration converges rapidly for smooth (exact) data, it performs quite poorly for noisy data, whereas the Landweber iteration, though slower, yields useful approximations in the presence of small noise perturbations in the data. Consequently, for iterative schemes, such as that of Landweber, appropriate smoothing of the data must be used when working with experimental data.