Approaches to iterative algorithms for solving nonlinear equations with an application in tomographic absorption spectroscopy

TL;DR

This paper introduces a derivative-free iterative method for solving nonlinear equations, specifically tailored for tomographic absorption spectroscopy, avoiding derivative calculations and ensuring convergence through a novel fixed point approach.

Contribution

It proposes an alternating fixed points algorithm and a superiorized derivative-free method tailored for complex nonlinear problems in tomography.

Findings

The new method effectively solves nonlinear systems without derivatives.

Experimental results show improved convergence and robustness.

The approach is applicable to high-dimensional, coupled nonlinear problems.

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.