An Alternating Direction Approximate Newton Algorithm for Ill-Conditioned Inverse Problems with Application to Parallel MRI

An alternating direction approximate Newton (ADAN) method is developed for solving inverse problems of the form $$\min \{\phi (Bu) +(1/2)\Vert Au-f\Vert _{2}^{2}\},$$min{ϕ(Bu)+(1/2)‖Au-f‖22}, where $$\phi $$ϕ is convex and possibly nonsmooth, and A and B are matrices. Problems of this form arise in image reconstruction where A is the matrix describing the imaging device, f is the measured data, $$\phi $$ϕ is a regularization term, and B is a derivative operator. The proposed algorithm is designed to handle applications where A is a large dense, ill-conditioned matrix. The algorithm is based on the alternating direction method of multipliers (ADMM) and an approximation to Newton’s method in which a term in Newton’s Hessian is replaced by a Barzilai–Borwein (BB) approximation. It is shown that ADAN converges to a solution of the inverse problem. Numerical results are provided using test problems from parallel magnetic resonance imaging. ADAN was faster than a proximal ADMM scheme that does not employ a BB Hessian approximation, while it was more stable and much simpler than the related Bregman operator splitting algorithm with variable stepsize algorithm which also employs a BB-based Hessian approximation.