The ability to visualize the propagation of action potential waves throughout the heart would be invaluable not only in the clinical setting, but in electrophysiological research as well. Here we consider the possibility of reconstructing action potentials throughout the heart using mechanical deformation data as input. This type of problem is normally underdetermined, but is solvable for the case of the heart when we take advantage of the fact that active contraction takes place only along the local fiber direction. Memory and processing time become important considerations when attempting this reconstruction in a fully 3D system. We discuss how we were able to overcome these problems by first projecting the problem onto a lower-dimensional Krylov subspace using Lanczos bidiagonalization, before then performing Tikhonov regularization. We demonstrate how our methods take into account the singularity of the elasticity matrix while simultaneously taking advantage of its symmetry. We present examples of the reconstruction, all of which show the location and form of the action potentials within the cardiac system with reasonable clarity. These results were obtained using a very low-dimensional Krylov subspace (~100 dimensions), which significantly enhanced overall computational speed.