Projected finite dimensional iteratively regularized Gauss–Newton method with a posteriori stopping for the ionospheric radiotomography problem

Abstract

We investigate a class of finite dimensional iteratively regularized Gauss–Newton methods for solving nonlinear irregular operator equations in a Hilbert space. The developed technique allows to investigate in a uniform style various discretization methods such as projection, quadrature and collocation schemes and to take into account available restrictions on the solution. We propose an a posteriori stopping rule for the iterative process and establish an accuracy estimate for obtained approximation. The regularized Gauss–Newton method combined with the quadrature discretization and the a posteriori iteration stopping is applied to a model ionospheric radiotomography problem. The problem is reduced to a nonlinear integral equation describing the phase shift of a sounding radio signal in dependence of the free electron concentration in the ionospheric plasma. We establish the unique solvability of the inverse problem in the class of analytic functions.