  This work derives explicit series reversions for the solution of Calder\'on's
problem. The governing elliptic partial differential equation is
$\nabla\cdot(A\nabla u)=0$ in a bounded Lipschitz domain and with a
matrix-valued coefficient. The corresponding forward map sends $A$ to a
projected version of a local Neumann-to-Dirichlet operator, allowing for the
use of partial boundary data and finitely many measurements. It is first shown
that the forward map is analytic, and subsequently reversions of its Taylor
series up to specified orders lead to a family of numerical methods for solving
the inverse problem with increasing accuracy. The convergence of these methods
is shown under conditions that ensure the invertibility of the Fr\'echet
derivative of the forward map. The introduced numerical methods are of the same
computational complexity as solving the linearised inverse problem. The
analogous results are also presented for the smoothened complete electrode
model.

Analysis of PDEs;Numerical Analysis;Numerical Analysis