This paper describes gradient methods based on a scaled metric on the Grassmann manifold for low-rank matrix completion. The proposed methods significantly improve canonical gradient methods, especially on ill-conditioned matrices, while maintaining established global convegence and exact recovery guarantees. A connection between a form of subspace iteration for matrix completion and the scaled...

1. Introduction. In this note we consider the solution of large, sparse linear systems of

and Applied Analysis 3 Additionally, we assume that there exist positive constants γ and γ such that 0 < γ ≤ 󵄩󵄩󵄩gk 󵄩󵄩󵄩 ≤ γ, ∀k ≥ 1, (21) then we have the following result. Theorem2. Consider the method (2), (8) and (12), where d k is a descent direction. If (21) holds, there exist positive constants ξ 1 , ξ 2 , and ξ 3 such that relations

and Applied Analysis 3 = ‖d k−1 ‖2 ‖g k−1 ‖4 + 1 ‖g k ‖2 − β2 k (gT k d k−1 ) 2 /‖g k ‖4

Because of its simplicity, low memory requirement, computational cost, and global convergence properties, the Conjugate Gradient (CG) method is most popular iterative mathematical technique for optimizing both linear nonlinear systems. Some classical CG methods, however, have drawbacks such as poor numerical performance in terms iterations function evaluations. To address these shortcomings, re...

Comparative study among Least Square Method: LSM, Steepest Descent Method: SDM, and Conjugate Gradient Method: CGM for atmospheric sounder data analysis (estimation of vertical profiles for water vapor) is conducted. Through simulation studies, it is found that CGM shows the best estimation accuracy followed by SDM and LSM. Method dependency on atmospheric models is also clarified. Keywords—non...

We compare the performance of several robust large-scale minimization algorithms for the unconstrained minimization of an ill-posed inverse problem. The parabolized Navier–Stokes equation model was used for adjoint parameter estimation. The methods compared consist of three versions of nonlinear conjugate-gradient (CG) method, quasiNewton Broyden–Fletcher–Goldfarb–Shanno (BFGS), the limited-mem...