Approaches to iterative algorithms for solving nonlinear equations with an application in tomographic absorption spectroscopy
F. J. Aragón-Artacho, W. Cai, Y. Censor, A. Gibali, C. Shui, D. Torregrosa-Belén
TL;DR
This work addresses solving nonlinear systems without derivative information in settings where operators map between different spaces, motivated by tomographic absorption spectroscopy (TAS). It develops an alternating common fixed points framework and a derivative-free descent-pairs algorithm, further enhanced by the superiorization methodology to inject priors without sacrificing convergence. The approach is validated experimentally on TAS problems, showing competitive performance against standard nonlinear fitting methods and traditional TAS techniques, with SUP-DPA often achieving better accuracy and reduced computation time. Overall, the paper contributes a practical derivative-free pathway for nonlinear TAS inversion and enriches fixed-point theory for non-self mappings in multivariable settings.
Abstract
In this paper we propose an approach for solving systems of nonlinear equations without computing function derivatives. Motivated by the application area of tomographic absorption spectroscopy, which is a highly-nonlinear problem with variables coupling, we consider a situation where straightforward translation to a fixed point problem is not possible because the operators that represent the relevant systems of nonlinear equations are not self-mappings, i.e., they operate between spaces of different dimensions. To overcome this difficulty we suggest an "alternating common fixed points algorithm" that acts alternatingly on the different vector variables. This approach translates the original problem to a common fixed point problem for which iterative algorithms are abound and exhibits a viable alternative to translation to an optimization problem, which usually requires derivatives information. However, to apply any of these iterative algorithms requires to ascertain the conditions that appear in their convergence theorems. To circumvent the need to verify conditions for convergence, we propose and motivate a derivative-free algorithm that better suits the tomographic absorption spectroscopy problem at hand and is even further improved by applying to it the superiorization approach. This is presented along with experimental results that demonstrate our approach.