In this paper we investigate the non-linear and ill-posed inverse problem of simultaneously identifying the conductivity and the reaction in diffuse optical tomography with noisy measurement data available on an accessible part of the boundary. We propose an energy functional method and the total variational regularization combining with the quadratic stabilizing term to formulate the identification problem to a PDEs constrained optimization problem. We show the stability of the proposed regularization method and the convergence of the finite element regularized solutions to the identification in the Lebesgue norms and in the sense of the Bregman distance with respect to the total variation semi-norm. To illustrate the theoretical results, a numerical case study is presented which supports our analytical findings.

中文翻译：

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用不完整和嘈杂的柯西数据确定漫射光学断层扫描中的两个系数

在本文中，我们研究了非线性和不适定逆问题，即在漫射光学断层扫描中同时识别电导率和反应，并在边界的可访问部分提供噪声测量数据。我们提出了一种能量泛函方法和全变分正则化，结合二次稳定项，将识别问题公式化为 PDE 约束优化问题。我们展示了所提出的正则化方法的稳定性以及有限元正则化解在 Lebesgue 范数和 Bregman 距离相对于全变半范数的意义上的收敛性。为了说明理论结果，提供了一个数值案例研究来支持我们的分析结果。