Regularized inversion of finitely smoothing Volterra operators: predictor - corrector regularization methods

We present a `predictor - corrector' type of regularization method for inverse problems modelled by first-kind Volterra integral equations and extend the convergence/regularization theory developed previously to the case where the integral kernel satisfies general -smoothing conditions. The theoretical basis for this method comes from replacing the original first-kind equation by a related second-kind equation which is constructed using `future values' of the original kernel and the data on a small interval of length . In practical implementation this method takes the form of a sequential regularization scheme in which one first predicts a rigid (regularized) solution over a small interval and then, before moving forward in the sequential process, one makes a correction of the solution in order to avoid over-regularization and to improve accuracy. In addition to the convergence theory developed for noise-free data, we show how selection of the regularization parameter as a function of the level of error present in the data serves to facilitate convergence in the case of noisy data. Finally, to further examine the extent to which improves stability, we show how an increase in serves to decrease the condition number of the matrices associated with a discretization of the original problem.