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APPROACHES TO ITERATIVE ALGORITHMS FOR SOLVING NONLINEAR EQUATIONS WITH AN APPLICATION IN TOMOGRAPHIC ABSORPTION

Francisco J Aragón-Artacho¹, Weiwei Cai², Yair Censor³

¹Department of Mathematics, University of Alicante, Alicante, Spain.

|May 27, 2024

Summary

This study introduces a novel derivative-free algorithm for solving complex nonlinear equations, particularly for tomographic absorption spectroscopy. The method avoids derivative calculations and enhances convergence for challenging inverse problems.

Keywords:

35Q99 47J05 65K05 90C30 Nonlinear equations alternating fixed points algorithm common fixed point cyclic sequential algorithm descent pairs algorithm superiorization tomographic absorption spectroscopy

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Area of Science:

Numerical Analysis
Applied Mathematics
Spectroscopy

Background:

Solving systems of nonlinear equations is crucial in many scientific fields.
Tomographic absorption spectroscopy presents a highly nonlinear problem with coupled variables.
Standard methods often require derivative information, which can be difficult to obtain.

Approach:

Proposes an "alternating common fixed points algorithm" for systems where operators are not self-mappings.
Translates the problem into a common fixed point problem solvable with iterative algorithms.
Introduces a derivative-free algorithm, enhanced by the superiorization approach, to circumvent convergence condition verification.

Key Points:

The novel algorithm effectively solves nonlinear systems without computing function derivatives.
It addresses challenges in tomographic absorption spectroscopy where direct fixed-point translation is not feasible.
The derivative-free nature and superiorization approach improve applicability and convergence for inverse problems.

Conclusions:

The presented derivative-free approach offers a viable alternative to derivative-based or optimization methods.
Experimental results demonstrate the efficacy of the proposed algorithm for tomographic absorption spectroscopy.
This work advances numerical methods for complex, high-dimensional nonlinear problems.