We study the inverse problem of parameter identification in general saddle point problems. For saddle point problems, the use of elliptic regularization is an essential component. Saddle point problems, after discretization, lead to a non-invertible system, whereas the regularized saddle point problems result in an invertible system. Regularization methods, in the context of saddle point problems, have also been used to mitigate the role of the Inf-Sup condition, synonymously, also called the Babuska-Brezzi condition. This work aims to analyze the impact of regularizing the saddle point problem on the inverse problem. We investigate the inverse problem by using the output least-squares objective. To exploit the use of regularization fully, we work under the assumption that the solution map is nonempty. We regularize the saddle point problem and consider a family of optimization problems using the output least-squares objective for the regularized saddle point problem where some noise contaminates the whole data set. We give a complete convergence analysis showing that the optimization problems, given for the regularized output least-squares, approximate the original problem suitably. We also provide the first-order and the second-order adjoint method for the computation of the first-order and the second-order derivatives of the output least-squares objective. We present some heuristic numerical results. In the context of the elasticity imaging inverse problem, we conduct detailed numerical experiments on synthetic data (to study the role of the regularization parameter) as well as on phantom data.

中文翻译：

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分析 Inf-Sup 条件在弹性成像应用中鞍点问题中参数识别的作用

我们研究了一般鞍点问题中参数辨识的逆问题。对于鞍点问题，椭圆正则化的使用是必不可少的组成部分。离散化后的鞍点问题导致不可逆系统，而正则化的鞍点问题导致可逆系统。在鞍点问题的背景下，正则化方法也被用来减轻 Inf-Sup 条件的作用，同义词，也称为 Babuska-Brezzi 条件。这项工作旨在分析正则化鞍点问题对逆问题的影响。我们通过使用输出最小二乘目标来研究逆问题。为了充分利用正则化，我们假设解图是非空的。我们正则化鞍点问题并考虑使用输出最小二乘目标的一系列优化问题，用于正则化鞍点问题，其中一些噪声会污染整个数据集。我们给出了一个完整的收敛分析，表明为正则化输出最小二乘法给出的优化问题适当地逼近了原始问题。我们还提供了用于计算输出最小二乘目标的一阶和二阶导数的一阶和二阶伴随方法。我们提出了一些启发式数值结果。在弹性成像逆问题的背景下，我们对合成数据（以研究正则化参数的作用）以及幻像数据进行了详细的数值实验。