Orbital tomography (OT) via angle-resolved photoelectron spectroscopy is a novel and powerful method to directly visualize ‘molecular orbitals’, the quantum-mechanical wave functions of electrons in molecules. In our project, we will develop the indispensable inverse reconstruction algorithms for OT and benchmark these using high-quality data acquired using the newest experimental technologies. This work will provide a firm basis for OT, allowing for its application to a wide-range of systems. To achieve this goal, we will extend OT to non-flat and non-static orbitals, eventually providing access to three-dimensional femtosecond dynamics of molecular orbitals. The mathematical challenge here consists of finding a practical balance between precise nonlinear mathematical models for the physical experiment and efficient computational approximations to those models.