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

Analternating direction approximateNewton (ADAN)method is developed for solving inverse problems of the form min{φ(Bu)+ (1/2)‖Au− f ‖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 This research was partly supported by National Science Foundation (Nos. 1115568 and 1016204) and by Office of Naval Research Grants (Nos. N00014-11-1-0068 and N00014-15-1-2048). B Hong-Chao Zhang [email protected] https://www.math.lsu.edu/∼hozhang/ William Hager [email protected] http://people.clas.ufl.edu/hager/ Cuong Ngo [email protected] http://people.clas.ufl.edu/ngocuong/ Maryam Yashtini [email protected] http://people.math.gatech.edu/∼myashtini3/ 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, GA 30332-0160, USA 3 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA