Full convergence of sequential local regularization methods for Volterra inverse problems

Local regularization methods for ill-posed linear Volterra equations have been shown to be efficient regularization procedures preserving the causal structure of the Volterra problem and allowing for sequential solution methods. However questions posed recently in Ring and Prix (2000 Inverse Problems 16 619–34) raise doubts as to whether such methods are convergent for problems which are more than just mildly ill-posed. In this paper we address these questions by reformulating the local regularization method via the use of signed Borel measures instead of the positive Borel measures used in earlier approaches. The result is a new theory for the local regularization of ν-smoothing Volterra problems for which stability and convergence is assured for all ν = 1, 2, . . . . In this paper we discuss this new local regularization theory for general finitely smoothing Volterra problems and demonstrate convergence and stability of the resulting method. In addition we indicate why using signed Borel measures instead of positive measures makes sense in the context of the Volterra problem and also has connections to the theory of mollification and approximate inverses (Louis A K 1996 Inverse Problems 12 175–90). Finally we include numerical examples which illustrate the improvement which comes from using signed measures instead of positive measures and which facilitates an examination of the role played by the regularization parameter.