About some methods of solution regularization of the first kind nonliner operator equation in hilbert space

Abstract

Many applied problems of physics and geophysics are reduced to operator equations of the first kind. In cases when the expression for the Green’s function is unknown, inverse problems of mathphysics are also reduced to these equations. The inverse problem of wells’ electric logging which defines fields and calculates the reserves of mineral resources is the exact example of such problems. The applied problems written above are urgent problems of modern science, the solution of these problems can discover new verges of current state of mankind’s development. In this regard, it stresses the importance of research of ill-posed problems. In the article [1], M.M. Lavrent’ev offered the regularization method for the solution of first kind linear operator equation in Hilbert space with the replacement of the original equation that is close to it, for which the problem of finding a solution is resistant to small changes in the right part and solvable for any right part, i.e. the equation replaces by another equation z + Az = u, where A- linear operator > 0 - regularization parameter. In the article [2] Newton’s method of аpproximate solution of equations was distributed by L.V. Kantorovich for functional equations K(z) = 0, where K(z) - nonlinear, twice differentiable operator by Frechet, acting from the one Banach space into another. In this article the combined method of the new type regularization is offered, combining the ideas of M. M. Lavrent’ev’s method [1], Newton-Kantorovich method [2] for the solution regularization of first kind nonlinear operator equation in Hilbert space.