On the construction of stable iterative methods for solving ill-posed nonlinear equations with nondifferentiable operators

Abstract

In this paper we present a general scheme of constructing stable iterative methods for solving nonlinear operator equations free of standard differentiability and regularity assumptions on the whole of the solution space. Equations of this type are of frequent occurrence in modern studies of inverse problems in mathematical physics and engineering. In the work we relax the classical smoothness and regularity assumptions and replace them by similar conditions on an appropriate finite dimensional affine subspace of the original solution space. In the framework of the developed scheme, we propose several implementable iterative processes for stable approximation of solutions to nondifferentiable irregular operator equations. One of the methods has been applied to a 3D inverse problem of reconstruction of the interface between two media of different densities from measurements of the gravity field on the plane outside the dividing surface. Numerical experiment confirm stability of the method relative to the noise in gravimetric data.