We deal with the shape reconstruction of inclusions in elastic bodies. Therefore, we review the monotonicity methods introduced in a former work of the authors. These monotonicity methods build the basis for the improvement of the standard one-step linearization method. The one-step linearization method consists of solving a minimization problem with standard regularization techniques, which is commonly used in practice but builds only a heuristical approach since there is no proven theory of its convergence. In contrary, the monotonicity-based regularization, where we introduce constraints for the minimization problem via the monotonicity methods, has a rigorously proven theory, i.e., we prove the existence and uniqueness of a minimizer as well as the convergence of the method for noisy data. Finally, we present numerical experiments.

翻译：我们处理的是弹性体融合的形状重建。 因此, 我们审查在作者以前的工作中引入的单一度方法。 这些单一度方法为改进标准的一步线性方法奠定了基础。 单步线性方法包括解决标准正规化技术的最小化问题, 标准正规化技术在实践中通常使用, 但由于没有证据证明其趋同理论, 仅建立超自然法则。 相反, 单调法的正规化, 通过单一度方法对最小化问题引入限制, 具有严格证明的理论, 也就是说, 我们证明最小化器的存在和独特性, 以及杂乱数据方法的趋同。 最后, 我们提出了数字实验。