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

作　　者：William Hager Cuong Ngo Maryam Yashtini Hong-Chao Zhang 

机构地区：[1]Department of Mathematics,University of Florida,PO Box 118105,Gainesville,FL 32611-8105,USA [2]School of Mathematics,Georgia Institute of Technology,686 Cherry Street,Atlanta [3]GA 30332-0160,USA 3 Department of Mathematics,Louisiana State University,Baton Rouge,LA 70803-4918,USA

出　　处：《Journal of the Operations Research Society of China》2015年第2期139-162,共24页中国运筹学会会刊（英文）

基　　金：This research was partly supported by National Science Foundation(Nos.1115568 and 1016204);by Office of Naval Research Grants(Nos.N00014-11-1-0068 and N00014-15-1-2048).

摘　　要：Analternating direction approximateNewton(ADAN)method is developed for solving inverse problems of the form min{φ(Bu)+(1/2)||Au−f||^(2)_(2)},whereφis convex and possibly nonsmooth,and A and B arematrices.Problems of this form arise in image reconstruction where A is the matrix describing the imaging device,f is the measured data,φis 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 aBarzilai–Borwein(BB)approximation.It is shown thatADAN 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.

关 键 词：Convex optimization Total variation regularization Nonsmooth optimization Global convergence Parallel MRI 

分 类 号：O17[理学—数学]