  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.
