Electrical Impedance Tomography (EIT) calculates the internal conductivity distribution within a body using electrical contact measurements. Previous investigation has shown that optimizing electrode placement can give better information about the stroke volume and better separation between the ventricles and atria than with the electrodes attached in the transverse plane. In our investigation we are developing fast three dimensional imaging of the heart (two planes of 16 electrodes) including internal electrodes in esophagus. The reconstruction speed in EIT is one of the main limitations for real time imaging when using a detailed three dimensional finite element mesh. For that reason we investigated new iterative algorithms for solving large scale L1 regularization. In this research we compare these algorithms on noise reliability and speed for 2D cardiac models. The four methods were as follows: (1) an interior point method for solving L1-regularized least squares problems (L1-LS); (2) total variation using a Lagrangian multiplier (TVAL3); (3) a two-step iterative shrinkage / thresholding method (TWIST) for solving the L0-regularized least squares problem; (4) The Least Absolute Shrinkage and Selection Operator (LASSO). In our investigation, using 1600 elements, we found all four algorithms provided an improvement over the best conventional EIT reconstruction method, Total Variation, in three important areas: robustness to noise, increased computational speed of at least 40× and a visually apparent improvement in spatial resolution. Out of the four algorithms we found TWIST was the fastest with at least a 100× speed increase.