THREE-COMPONENT VERSION OF THE TIKHONOV REGULARIZATION METHOD FOR OPERATOR EQUATIONS OF THE FIRST KIND

Abstract

t An ill-posed problem in the form of a linear operator equation is considered. It is assumed that the solution to the equation in the one-dimensional case can be represented in the form of a sum of three components: the first component contains discontinuities, the second contains discontinuities in the derivative, and the third is continuous. To construct a stable approximate solution, the three-component Tikhonov method is used. In this case, the stabilizer is the sum of three functionals: BVp-norm of the first component, BVp-norm of the derivative for the second component and the norm of the Sobolev space for the third component, and each functional depends on only one component. The convergence of the sum of regularized components to the solution of the original equation is proved. In addition, piecewise uniform convergence of approximate solutions is established. The results of numerical experiments on reconstructing a three-component model solution for the Fredholm equation of the first kind are presented.