A study of frozen iteratively regularized Gauss–Newton algorithm for nonlinear ill-posed problems under generalized normal solvability condition

A parameter identification inverse problem in the form of nonlinear least squares is considered. In the lack of stability, the frozen iteratively regularized Gauss–Newton (FIRGN) algorithm is proposed and its convergence is justified under what we call a generalized normal solvability condition. The penalty term is constructed based on a semi-norm generated by a linear operator yielding a greater flexibility in the use of qualitative and quantitative a priori information available for each particular model. Unlike previously known theoretical results on the FIRGN method, our convergence analysis does not rely on any nonlinearity conditions and it is applicable to a large class of nonlinear operators. In our study, we leverage the nature of ill-posedness in order to establish convergence in the noise-free case. For noise contaminated data, we show that, at least theoretically, the process does not require a stopping rule and is no longer semi-convergent. Numerical simulations for a parameter estimation problem in epidemiology illustrate the efficiency of the algorithm.

Normal solvability of general linear elliptic problems

The paper is devoted to general elliptic problems in the Douglis-Nirenberg sense. We obtain a necessary and sufficient condition of normal solvability in the case of unbounded domains. Along with the ellipticity condition, proper ellipticity and Lopatinsky condition that determine normal solvability of elliptic problems in bounded domains, one more condition formulated in terms of limiting problems should be imposed in the case of unbounded domains.