Sequential predictor-corrector methods for the variable regularization of Volterra inverse problems

Abstract

We analyse the convergence of a class of discrete predictor-corrector methods for the sequential regularization of first-kind Volterra integral equations. In contrast to classical methods such as Tikhonov regularization, this class of methods preserves the Volterra (causal) structure of the original problem. The result is a discretized regularization method for which the number of arithmetic operations is (N 2 ) (where N is the dimension of the approximating space) in contrast to standard Tikhonov regularization which requires (N 3 ) operations. In addition, the method considered here is defined using functional regularization parameters so that the possibility for more or less smoothing at different points in the domain of the solution is allowed. We establish a convergence theory for these methods and present relevant numerical examples, illustrating how one functional regularization parameter may be adaptively selected as part of the sequential regularization process. This work generalizes earlier results by the first author to the case of a penalized predictor-corrector formulation, functional regularization parameters, and nonconvolution Volterra equations.