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 $math\tilde{H}$ where multiplication with the inverse of $I + \alpha {math\tilde{H}^T} math\tilde{H}$ can be efficiently performed. We discuss how to take advantage of this property when the Douglas-Rachford algorithm is utilized.