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.

翻译：本文提出了一种无需计算函数导数的非线性方程组求解方法。受断层吸收光谱学这一具有变量强耦合特性的高非线性问题驱动，我们考虑了一种特殊情形：由于表征非线性方程组的算子并非自映射（即其作用空间维度不同），因此无法直接转化为不动点问题。为克服这一困难，我们提出一种"交替公共不动点算法"，该算法交替作用于不同的向量变量。该方法将原始问题转化为公共不动点问题（此类问题已有丰富的迭代算法），从而为常需导数信息的优化问题转化提供了可行替代方案。然而，应用这些迭代算法需要验证其收敛定理中的预设条件。为规避收敛性条件验证，我们提出并论证了一种无需导数的算法，该算法更适合当前断层吸收光谱学问题，并通过引入超优化方法进一步改进。文中还展示了验证本方法的实验结果。