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. 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问题。为了在实践中解决这一反面问题,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。这是标准规范化技术的一大优势和重大改进。更详细地说,我们根据单一度方法,对将残留问题最小化加以限制,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,我们还比较了基于单一度的正规化的包容性数字重组和标准方法(与Tikhonov相似的正规化为一步线化),后者也显示了我们有关噪音方法的稳健性。