Future-Sequential Regularization Methods for III-Posed Volterra Equations: Applications to the Inverse Heat Conduction Problem

Abstract

We develop a theoretical context in which to study the future-sequential regularization method developed by Beck for the Inverse Heat Conduction Problem. In the process, we generalize Beck′s ideas and view that method as one in a large class of regularization methods in which the solution of an ill-posed first-kind Volterra equation is seen to be the limit of a sequence of solutions of well-posed second-kind Volterra equations, Such techniques are important because standard regularization methods (such as Tikhonov regularization) tend to transform a naturally sequential Volterra problem into a full-domain Fredholm problem, destroying the underlying causal nature of the Volterra model and leading to inefficient global approximation strategies. In contrast, the ideas we present here preserve the original Volterra structure of the problem and thus can lead to easily implemented localized approximation strategies. Theoretical properties of these methods are discussed and proofs of convergence are given.