Fourier velocity encoding (FVE) is a technique capable of delivering clinically treatable data at short acquisition times. FVE resolves the velocity distribution in each voxel of the image with high signal-to-noise ratio. This makes it suitable for the calculation of relevant biomarkers (e.g. wall shear rate and oscillatory shear index). However, it does not provide the blood flow velocity field directly. Techniques to estimate the actual blood flow from FVE velocity distributions have been previously presented. In this work, we present a novel method for velocity map estimation based on a PDE-constrained optimization that provides better results than previous methods.