We present an improved technique for susceptibility artifact correction in echo-planar imaging (EPI), a widely used ultra-fast magnetic resonance imaging (MRI) technique. Our method corrects geometric deformations and intensity modulations present in EPI images. We consider a tailored variational image registration problem incorporating a physical distortion model and aiming at minimizing the distance of two oppositely distorted images subject to invertibility constraints. We derive a novel face-staggered discretization of the variational problem that renders the discretized distance function and constraints separable. Motivated by the presence of a smoothness regularizer, which leads to global coupling, we apply the alternating direction method of multipliers (ADMM) to split the problem into simpler subproblems. We prove the convergence of ADMM for this non-convex optimization problem. We show the superiority of our scheme compared to two state-of-the-art methods both in terms of correction quality and time-to-solution for 13 high-resolution 3D imaging datasets.