then recovered from the transformed data by solving a $\overline{\partial}_{k}$ (D-bar) equation in the transform variable $k$ for solutions known as [*Complex Geometrical Optics (CGO) solutions*]{} to a related partial differential equation. While there are several D-bar methods for EIT, they all have the same basic form:

$$\begin{array}{c} \text{Current/Voltage}\\
\text{data}\end{array} \longrightarrow \begin{array}{c} \text{Scattering}\\ \text{data}\end{array}  \longrightarrow \begin{array}{c} \text{CGO}\\ \text{solutions}\end{array} \longrightarrow \begin{array}{c} \text{Admittivity.}\end{array}$$

Below we provide a brief overview of the D-bar methods for EIT studied here. We compare the results of the D-bar method for real-valued conductivities, which is based on transforming the admittivity equation to the Schrödinger equation in the global uniqueness proof \[[@Nachman1996]\], implemented and further developed in \[[@Siltanen2000], [@Isaacson2004], [@Knudsen2009]\], with the results of two implementations of the D-bar method for complex conductivities \[[@Hamilton2012], [@Herrera2015]\], in which the problem is transformed to a first order elliptic system \[[@Francini2000]\]. Computational details will be discussed in Section \[sec:CompDetails\].

Here and throughout we associate ${{\mathbb R}}^2$ and ${{\mathbb C}}$ via the mapping $z=(x,y)\mapsto x+iy$ and make use of the common $\operatorname{\operatorname{\partial}_z}$ and $\operatorname{{\bar{\partial}}_z}$ derivative operators $$\operatorname{\operatorname{\partial}_z}= \frac12\left(\partial_x -i\partial_y\right),\quad  \operatorname{{\bar{\partial}}_z}= \frac12\left(\partial_x +i\partial_y\right).$$

The D-bar Method for Real-valued Conductivities {#sec:NachDbar}
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Here we review the proof given in \[[@Nachman1996]\], first implemented in \[[@Siltanen2000]\] 