Combined Regularization Method for Solving an Implicit Operator Equation of the First Kind

Many applied problems of physics and biophysics are reduced to the operator equations of the first kind. These equations are reduced as inverse problems of mathematical physics in cases where the expression for the Green’s function is unknown. The inverse problem of electric logging of wells defining deposits and reserve calculation of mineral resources is an example of such problems. The above application tasks are urgent tasks of modern science, these tasks can discover new facets of the current state of development of mankind. In this regard, it stresses the importance of studying ill-posed problems. MM Lavrent’ev proposed regularization method for solving linear operator equations of the first kind in Hilbert space with the replacement of the original equation close to him, in a sense, for which the task of finding a solution is resistant to small changes in the right-hand side and is solvable for any right-hand side. Newton’s method of approximate solution of the equations was distributed L.V. Kantorovich on functional equations. In this paper, we propose a combined method of regularization of a new type, combining the idea of the method of M. M. Lavrent’ev, Newton-Kantorovich method for the regularization of solutions of nonlinear operator equations of the first kind in Hilbert space.