Phase diversity phase retrieval (PDPR) has been a popular technique for quantitatively measuring wavefront errors of optical imaging systems by extracting the information from several designated intensity measurements. As problem is inverse and non-convex in general, accuracy robustness most such algorithms rely greatly on initial conditions. In this work, we propose new strategy that combines ...

A major weakness of the limited memory BFGS (LBFGS) method is that it may converge very slowly on ill-conditioned problems when the identity matrix is used for initialization. Very often, the LBFGS method can adopt a preconditioner on the identity matrix to speed up the convergence. For this purpose, we propose a class of diagonal preconditioners to boost the performance of the LBFGS method. In...

The standard formulations of the Kalman filter (KF) and extended Kalman filter (EKF) require the storage and multiplication of matrices of size n × n, where n is the size of the state space, and the inversion of matrices of size m × m, where m is the size of the observation space. Thus when both m and n are large, implementation issues arise. In this paper, we advocate the use of the limited me...

Using Taylor's series we propose a modified secant relation to get a more accurate approximation of the second curvature of the objective function. Then, based on this modified secant relation we present a new BFGS method for solving unconstrained optimization problems. The proposed method make use of both gradient and function values while the usual secant relation uses only gradient values. U...

This report is devoted to preconditioning techniques for the matrix-free truncated Newton method. After a review of basic known approaches, we propose new results concerning tridiagonal and pentadiagonal preconditioners based on the standard BFGS updates and on numerical differentiation. Furthermore, we present results of extensive numerical experiments serving for the careful comparison of sui...