Feasibility models are a powerful approach to many real-world problems where simply finding a point that comes close enough to meeting many, sometimes contradictory demands is enough. In this talk I will outline the theoretical foundations for the convergence analysis of fixed point iterations of expansive mappings, and show how this specializes to fundamental algorithms for nonconvex, inconsistent feasibility problems.  As a concrete instance, I will demonstrate the ideas and results on an orbital tomography problem of reconstructing 3 dimensional electron orbitals of small molecules directly from electron scattering measurements. Not only is this a nice demonstration of beautiful mathematics, but the resulting algorithms have enabled us to observe for first time electronic orbitals at several Angstrom resolutions directly from measurements.

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