On explicit expression of the solution to the regularizing by Tikhonov optimization problem in terms of the regularization parameter in the finite-dimensional case

Abstract

It is well known that using the Tikhonov regularization method for solving operator equations of the first kind one has to minimize a regularized residual functional. The minimizer is determined from so called Euler equation which in finite-dimensional case and at its discretization is written as a one-parametric (depending on the regularization parameter) system of linear algebraic equations of special form. Here, there exist various ways of choosing the regularization parameter. In particular, in the frame of principle of generalized residual, it is necessary to solve the corresponding equation of generalized residual with respect to the regularization parameter. And it implies (when solving this equation numerically), in turn, multifold solving a one-parametric system of linear algebraic equations for arbitrary value of the parameter. In this paper we obtain an explicit simple and effective formula of solution to a one-parametric system for an arbitrary value of the parameter. We give an example of computations by above-mentioned formula and also an example of numerical solution of the Fredholm integral equation of the first kind under usage of this formula which substantiates its effectiveness.