Many inverse problems are concerned with the estimation of non-negative parameter functions. In this paper, in order to obtain non-negative stable approximate solutions to ill-posed linear operator equations in a Hilbert space setting, we develop two novel non-negativity preserving iterative regularization methods. They are based on fixed point iterations in combination with preconditioning ideas. In contrast to the projected Landweber iteration, for which only weak convergence can be shown for the regularized solution when the noise level tends to zero, the introduced regularization methods exhibit strong convergence. There are presented convergence results, even for a combination of noisy right-hand side and imperfect forward operators, and for one of the approaches there are also convergence rates results. Specifically adapted discrepancy principles are used as a posteriori stopping rules of the established iterative regularization algorithms. For an application of the suggested new approaches, we consider a biosensor problem, which is modelled as a two dimensional linear Fredholm integral equation of the first kind. Several numerical examples, as well as a comparison with the projected Landweber method, are presented to show the accuracy and the acceleration effect of the novel methods. Case studies of a real data problem indicate that the developed methods can produce meaningful featured regularized solutions.

中文翻译：

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不适定逆问题的两种新的非负保留迭代正则化方法

许多反问题与非负参数函数的估计有关。在本文中，为了在希尔伯特空间设置中获得不适定线性算子方程的非负稳定近似解，我们开发了两种新颖的保留非负性的迭代正则化方法。它们基于定点迭代并结合了预处理思想。与预测的Landweber迭代相反，当噪声水平趋于零时，对于正则化解决方案仅显示弱收敛，引入的正则化方法表现出强收敛性。给出了收敛结果，即使对于嘈杂的右侧和不完善的前向算子的组合，对于其中一种方法，也存在收敛率结果。建立的迭代正则化算法的后验停止规则。对于建议的新方法的应用，我们考虑一个生物传感器问题，该问题被建模为第一类二维线性Fredholm积分方程。给出了几个数值例子，并与投影的Landweber方法进行了比较，以显示新方法的准确性和加速效果。实际数据问题的案例研究表明，所开发的方法可以产生有意义的特征化正则化解决方案。