Proximal Mappings Involving Almost Structured Matrices

We consider a minimization problem where the cost function consists of the sum of a quadratic data fidelity term and a penalty term. The quadratic involves a matrix H that can be embedded into a larger matrix H̃ where multiplication with the inverse of I + αH̃
^T
H̃ can be efficiently performed. We discuss how to take advantage of this property when the Douglas-Rachford algorithm is utilized.