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

Abstract

An alternating direction approximate Newton (ADAN) method is developed for solving inverse problems of the form min{φ(Bu) + (1/2) Au − f 2 2 }, where 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, 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.

Block-wise Alternating Direction Method of Multipliers for Multiple-block Convex Programming and Beyond

“…But it should be mentioned that it is significantly different from the so-called linearized ADMM in the literature (e.g., [8,35,36]), which aims at linearizing the quadratic terms in ADMM's subproblems with a sufficiently large proximal parameter (e.g., it should be greater than β · A T i A i if the x i -subproblem in (3.1) is considered) and thus alleviating the linearized subproblems. In other words, the proximal parameters in current linearized ADMM literature are dependent on the involved matrices of the corresponding quadratic terms and they may need to be sufficiently large to ensure the convergence if the matrices happen to be ill-conditioned, see, e.g., [16]. In (3.1), however, the proximal parameters τ 1 and τ 2 are just constants independent of the matrices.…”

Iterative Methods for Computing the Resolvent of Composed Operators in Hilbert Spaces

“…In recent years, Newton-type methods have been combined with the forward-backward splitting (FBS) algorithm to accelerate the speed of the original FBS algorithm. See, for example, [24][25][26]. Argyriou et al [27] considered the following convex optimization problem:…”

An Alternating Direction Nonmonotone Approximate Newton Algorithm for Inverse Problems

“…In Hong-Chao Zhang's paper [6], he uses the Alternating Direction Approximate Newton method (ADAN) based on Alternating Direction Method (ADMM) which originaly in [7] to solve (1). He employs the BB approximation to increase the iterations.…”

“…In many applications, the optimization problems in ADMM are either easily resolvable, since ADMM iterations can be performed at a low computational cost. Besides, com-bine different Newton-based methods with ADMM have become a trend, see [6] [8] [9], since those methods may achieve the high convergent speed.…”