Approximation of Solution Components for Ill-Posed Problems by the Tikhonov Method with Total Variation

Abstract

An ill-posed problem in the form of a linear operator equation given on a pair of Banach spaces is considered. Its solution is representable as a sum of a smooth and a discontinuous component. A stable approximation of the solution is obtained using a modified Tikhonov method in which the stabilizer is constructed as a sum of the Lebesgue norm and total variation. Each of the functionals involved in the stabilizer depends only on one component and takes into account its properties. Theorems on the componentwise convergence of the regularization method are stated, and a general scheme for the finite-difference approximation of the regularized family of approximate solutions is substantiated in the n-dimensional case.