We present a powerful and easy-to-implement iterative algorithm for solving large-scale optimization problems that involve L1/total-variation (TV) regularization. The method is based on combining the Alternating Directions Method of Multipliers (ADMM) with a Conjugate Directions technique in a way that allows reusing conjugate search directions constructed by the algorithm across multiple iterations of the ADMM. The new method achieves fast convergence by trading off multiple applications of the modeling operator for the increased memory requirement of storing previous conjugate directions. We illustrate the new method with a series of imaging and inversion applications.

Maharramov, Levin. Compressive Conjugate Directions: Linear Theory (preprint in PDF)