Abstract: New experimental technologies have the potential to capture information from important biological objects and new materials at unprecedented detail and scales. However, the accelerating size, rate, complexity, and sensitivity of these new measurements are far outpacing the capability of traditional inversion methods to efficiently and robustly reconstruct the desired information from the data.

In this talk, I will present a new general mathematical framework capable of overcoming many of these challenges in inversion. This framework, called Multi- Tiered Iterative Projections (M-TIP), is based on exploiting the mathematical structure of the inverse problem via an iterative application of projection operators that optimally target the physics of the experiment to maximize speed, accuracy, and robustness. I will demonstrate the use of M-TIP in solving important open problems in inversion from fluctuation scattering, single-particle diffraction, coherent surface scattering imaging, multiple-scattering electron crystallography, and more.