On local regularization methods for linear Volterra equations and nonlinear equations of Hammerstein type

Abstract

Local regularization methods allow for the application of sequential solution techniques for the solution of Volterra problems, retaining the causal structure of the original Volterra problem and leading to fast solution techniques. Stability and convergence of these methods was shown to hold on a large class of linear Volterra problems, i.e., the class of ν-smoothing problems for ν = 1, 2, … in Lamm (2005 Inverse Problems 21 785–803). In this paper, we enlarge the family of convergent local regularization methods to include sequential versions of classical regularization methods such as sequential Tikhonov regularization. In fact, sequential Tikhonov regularization was considered earlier by Lamm and Eldén (1997 SIAM J. Numer. Anal. 34 1432–50) but there the theory was limited to the class of discretized one-smoothing Volterra problems. An interesting feature of sequential classical regularization methods is that they involve two regularization parameters: the usual local regularization parameter r controls the size of the local problem while a second parameter α controls the amount of regularization to be applied in each subproblem. This approach suggests a wavelet type of regularization method with the parameter r controlling spatial resolution and α controlling frequency resolution. In this paper, we also show how the ‘future polynomial regularization’ method of Cinzori (2004 Inverse Problems 20 1791–806) can be viewed as a special case of the general framework of Lamm (2005) in the 1-smoothing case. In addition, we extend the results of Lamm (2005) to nonlinear Volterra problems of Hammerstein type and give numerical results to illustrate the effectiveness of the method in this case.