Classical problems in mathematics often expose students to computation of integrals and derivatives of well known functions. Inverse problems on the other hand often involve the recovery of an unknown multidimensional function given many lower dimensional integrals of the underlying function, such methods that benefit by making use of regularity assumptions that involve smoothness ideals based on derivatives. In this talk we will first discuss general methodologies for inverse problems, those particularly involved with 11 optimization commonly categorized as compressed sensing algorithms. We will discuss the practicality of such algorithms and how they are adapted in a practical way to problems such as synthetic aperture radar (SAR) and electron tomography.