Approximation of Ill-Posed Volterra Problems via Predictor–Corrector Regularization Methods

Abstract

First-kind Volterra problems arise in numerous applications, from inverse problems in mathematical biology to inverse heat conduction problems. Unfortunately, such problems are also ill posed due to lack of continuous dependence of solutions on data. Consequently, numerical methods to solve first-kind Volterra equations are effective only when regularizing features are built into the algorithms or used to control stepsize. Classical methods often combine numerical discretization with Tikhonov regularization, but in doing so the underlying Volterra (or causal) nature of the original problem is often destroyed. Instead, a “predictor–corrector” type of numerical method is proposed which combines at each step “local regularization” ideas with the use of small intervals of future data. The result is a regularized numerical method which retains much of the causal nature of the Volterra problem and may be solved in fast sequential steps, often improving upon the performance of classical algorithms such as those based on standard Tikhonov regularization. In this paper, the discretized local regularization method is described, and proofs are given of convergence of the method, with rate of convergence being “best possible” with regard to the amount of error in the data. Further, by linking the regularization parameter of the stabilizing method (i.e., the length of the “future interval”in the future-sequential method) to the approximation stepsize, great simplification of the resulting numerical algorithm is obtained. Relevant numerical examples are included.