Numerical Solution of First-Kind Volterra Equations by Sequential Tikhonov Regularization

Abstract

We consider the problem of finding regularized solutions to ill-posed Volterra integral equations. The method we consider is a sequential form of Tikhonov regularization that is particularly suited to problems of Volterra type. We prove that when this sequential regularization method is coupled with several standard discretizations of the integral equation (collocation, rectangular and midpoint quadrature), one obtains convergence of the method at an optimal rate with respect to noise in the data. In addition we describe a fast algorithm for the implementation of sequential Tikhonov regularization and show that for small values of the regularization parameter, the method is only slightly more expensive computationally than the numerical solution of the original unregularized integral equation. Finally, numerical results are presented to show that the performance of sequential Tikhonov regularization nearly matches that of standard Tikhonov regularization in practice but considerable savings in cost are realized.