This work establishes a framework for solving inverse boundary problems with the geodesic based quadratic Wasserstein distance ($W_{2}$). A general form of the Fr\'echet gradient is systematically derived by optimal transportation (OT) theory. In addition, a fast algorithm based on the new formulation of OT on $\mathbb{S}^{1}$ is developed to solve the corresponding optimal transport problem. The computational complexity of the algorithm is reduced to $O(N)$ from $O(N^{3})$ of the traditional method. Combining with the adjoint-state method, this framework provides a new computational approach for solving the challenging electrical impedance tomography (EIT) problem. Numerical examples are presented to illustrate the effectiveness of our method.
翻译：这项工作建立了一个框架,以解决基于大地测量的二次瓦斯泰因距离(W ⁇ 2}$)的反边界问题。Fr\'echet梯度的一般形式是通过最佳运输理论系统地产生的。此外,还根据以$mathbb{S ⁇ 1}$为单位的OT新配方开发了快速算法,以解决相应的最佳运输问题。算法的计算复杂性从传统方法的$O(N ⁇ 3}$)减少到$O(N)$(N)。这一框架结合了联合状态方法,为解决挑战性电气阻力学问题提供了一种新的计算方法。提供了数字实例,以说明我们方法的有效性。
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