Iteratively regularized Gauss-Newton method in the inverse problem of ionospheric radiosonding (Q2058313)

Over the last decade, numerous original articles in mathematical journals and chapters in monographs have been devoted to the \textit{iteratively regularized Gauss Newton method} as a specific facet of the iterative regularization theory aimed at finding stable approximate solutions to inverse problems. However, it is always of interest to bring such theory in contact with concrete applications in natural sciences and engineering, including the practice check of occurring mathematical conditions for feasibility and convergence. This is the case with the article on applying this iterative regularization method to the inverse problem of ionospheric radiosounding. The authors present the physical and mathematical models for the nonlinear inverse problems of reconstructing the vertical profile function of the electron concentration in parts of the ionosphere based on indirect measurements by moving satellites. In \S 1 of the article, one can moreover find discussions of the mathematical model, which culminate in the formulation of a nonlinear integral equation. More detailed mathematical studies of this equation and its non-unique solution are given in \S 2. A finite dimensional approximation and a corresponding formalization of the result in form of an operator equation \[ F(u)=f,\quad u \in D, \] with a Fréchet differentiable forward operator \(F\) mapping in Hilbert spaces and a convex closed domain \(D\) is presented in \S 3. The Fréchet derivative \(F^\prime(u)\) is Lipschitz continuous, but not necessarily continuously invertible, which indicates the ill-posedness of the inverse problem under consideration. Hence, some kind of regularization seems to be required. The authors choose the iteratively regularized Gauss Newton method with stepwise metric projections from the Hilbert space onto the set \(D\), which expresses the objective a priori information about the expected solution in form of solution constraints. Well-known assertions from regularization theory are applicable and yield accuracy estimates under the discrepancy principle as stopping rule. Numerical experiments in \S 4 illustrate the theoretical results.