| This thesis explores the application of complexity penalized algorithms to solve a variety of geophysical inverse problems: Hydraulic Fracture Monitoring (HFM), hyper-spectral imaging, and reflection seismology. Through these examples, the thesis examines how the physics of several systems gives rise to sparsity or low-dimensionality when posed in the proper basis. This low complexity can be quantified into several types of convex norms such as the ℓ 1 and nuclear norm. In this work we demonstrate how optimization algorithms which exploit this complexity by penalizing the relevant convex norms can improve inversion. First and second order as well as stochastic algorithms are used to solve these minimization problems and I give details as to how the structure of the problem dictates the best technique to apply. |
