This work investigates the elasticity imaging inverse problem of tumor identification in a fully incompressible medium through a family of inverse problems in a nearly incompressible medium. We develop an inversion framework for saddle point problems that goes far beyond the elasticity imaging inverse problem and applies to a wide variety of inverse problems. We introduce a family of convex optimization problems with regularized saddle point problems as the constraint and prove its convergence. We discretize the inverse problem by using the finite element approach and prove the convergence of the discrete problems. We offer formulas for the gradient and the Hessian computation. The outcome of detailed numerical computations, carried out using the tissue phantom data, shows the efficacy of the developed framework.

中文翻译：

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用于识别鞍点问题中参数的凸反演框架及其在逆不可压缩弹性中的应用

这项工作通过一系列在几乎不可压缩的介质中的反问题来研究在完全不可压缩的介质中肿瘤识别的弹性成像反问题。我们为鞍点问题开发了一个反演框架，该框架远远超出了弹性成像反演问题，并且适用于各种反演问题。我们引入了以正则化鞍点问题为约束的一系列凸优化问题，并证明了其收敛性。我们通过有限元方法离散化反问题，并证明离散问题的收敛性。我们提供了用于梯度和Hessian计算的公式。使用组织体模数据进行的详细数值计算的结果表明了开发框架的有效性。