We deal with the shape reconstruction of inclusions in elastic bodies. For solving this inverse problem in practice, data fitting functionals are used. Those work better than the rigorous monotonicity methods from [5], but have no rigorously proven convergence theory. Therefore we show how the monotonicity methods can be converted into a regularization method for a data-fitting functional without losing the convergence properties of the monotonicity methods. This is a great advantage and a significant improvement over standard regularization techniques, which do not have a rigorous theory of convergence. In more detail, we introduce constraints on the minimization problem of the residual based on the monotonicity methods and prove the existence and uniqueness of a minimizer as well as the convergence of the method for noisy data. In addition, we compare numerical reconstructions of inclusions based on the monotonicity-based regularization with a standard approach (one-step linearization with Tikhonov-like regularization), which also shows the robustness of our method regarding noise in practice.

翻译：我们处理弹性体融合的形状重建。 为了在实践中解决这个反面问题, 使用了数据匹配功能。 这些功能比[ 5] 的严格单一度方法要好, 但没有严格证明的趋同理论。 因此, 我们展示了如何在不丧失单一度方法的趋同特性的情况下,将单一度方法转换成符合数据功能的正规化方法。 这是与标准规范化技术相比的巨大优势和重大改进,因为标准规范化技术没有严格的趋同理论。 更详细地说, 我们根据单一度方法对残留物的最小化问题提出限制, 并证明最小化器的存在和独特性以及噪音数据方法的趋同。 此外, 我们比较了基于单一度规范法的包容性数字重组与标准方法( 单步线化与Tikhonov相似的正规化)的数值重组, 后者也显示了我们有关噪音的方法在实践中的稳健性。