Abstract
We consider the problem of solving of a linear, first kind Volterra convolution equation with finitely smoothing kernel. Deeming that the problem is ill-posed, we approximate its solution using local regularization. In [22], sufficient conditions were established for local regularization of the problem under which regularized approximations based on exact data in C[0, T] were shown to converge uniformly to the problem's true solution. An a priori strategy was also provided for choosing the regularization parameter for which uniform convergence of approximations made with perturbed data in C[0, T] was also guaranteed. However, until now, no a posteriori regularization parameter selection criteria existed to be paired with local regularization and convergence of the resulting method proved. We supply this missing piece by defining a new discrepancy principle for selecting the regularization parameter constant-valued based on measured data and the known level of noise. We establish sufficient conditions for the local regularization scheme, based on those in [22], so that when paired with our discrepancy principle, we are able to prove uniform convergence of approximations made with perturbed data in C[0,1] to the true solution in C[0, 1] as the noise level shrinks to zero. We also extend the theory of local regularization to address the case when the linear Volterra convolution operator with finitely smoothing kernel is defined on the space L super(p)0, 1], 1 < p < !. We amend our conditions slightly and prove them sufficient for L super(p)convergence of approximations based on exact data in L super(p)0, 1] and provide an a priori rule for selecting the regularization parameter given perturbed data in L super(p)0, 1]. We redefine our principle and again establish sufficient conditions on the local regularization scheme so that when paired with the principle, approximations based on perturbed data in L super(p)0, 1] converge to the true solution in L super(p)0, 1] as the noise level shrinks to zero. For both the C[0, 1] and L super(p)0, 1] cases, we provide a rate of convergence. Numerical examples are provided to illustrate the method's effectiveness. Our principle is found to be a natural complement to the existing theory in C[0, 1] as well as its extension to L super(p) 0, 1], 1 < p < !. This is an initial, yet fundamental step in the development of a posteriori principles for use with local regularization in solving linear and eventually non-linear Volterra equations.