An inexact parallel splitting augmented Lagrangian method for large system of linear equations

2010 ◽  
Vol 216 (5) ◽  
pp. 1624-1636
Author(s):  
Zheng Peng ◽  
DongHua Wu
Keyword(s):  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Linear Equations ◽  
Lagrangian Method ◽  
Large System ◽  
System Of Linear Equations ◽  
Parallel Splitting

scholarly journals A Parallel Splitting Augmented Lagrangian Method for Two-Block Separable Convex Programming with Application in Image Processing

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Jing Liu ◽  
Yongrui Duan ◽  
Tonghui Wang
Keyword(s):  
Convex Programming ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Lagrangian Method ◽  
Step Size ◽  
Linear Equality Constraints ◽  
Separable Convex Programming ◽  
Parallel Splitting ◽  
Separable Convex Minimization ◽  
Alm Method

The augmented Lagrangian method (ALM) is one of the most successful first-order methods for convex programming with linear equality constraints. To solve the two-block separable convex minimization problem, we always use the parallel splitting ALM method. In this paper, we will show that no matter how small the step size and the penalty parameter are, the convergence of the parallel splitting ALM is not guaranteed. We propose a new convergent parallel splitting ALM (PSALM), which is the regularizing ALM’s minimization subproblem by some simple proximal terms. In application this new PSALM is used to solve video background extraction problems and our numerical results indicate that this new PSALM is efficient.

scholarly journals An augmented Lagrangian method for solving total variation (TV)-based image registration model

2020 ◽  
Vol 14 ◽  
pp. 174830262097353
Author(s):  
Noppadol Chumchob ◽  
Ke Chen
Keyword(s):  
Image Registration ◽  
Augmented Lagrangian ◽  
Numerical Solutions ◽  
Augmented Lagrangian Method ◽  
Closed Form Solution ◽  
Numerical Algorithms ◽  
Lagrangian Method ◽  
Form Solution ◽  
The Other ◽  
Other Hand

Variational methods for image registration basically involve a regularizer to ensure that the resulting well-posed problem admits a solution. Different choices of regularizers lead to different deformations. On one hand, the conventional regularizers, such as the elastic, diffusion and curvature regularizers, are able to generate globally smooth deformations and generally useful for many applications. On the other hand, these regularizers become poor in some applications where discontinuities or steep gradients in the deformations are required. As is well-known, the total (TV) variation regularizer is more appropriate to preserve discontinuities of the deformations. However, it is difficult in developing an efficient numerical method to ensure that numerical solutions satisfy this requirement because of the non-differentiability and non-linearity of the TV regularizer. In this work we focus on computational challenges arising in approximately solving TV-based image registration model. Motivated by many efficient numerical algorithms in image restoration, we propose to use augmented Lagrangian method (ALM). At each iteration, the computation of our ALM requires to solve two subproblems. On one hand for the first subproblem, it is impossible to obtain exact solution. On the other hand for the second subproblem, it has a closed-form solution. To this end, we propose an efficient nonlinear multigrid (NMG) method to obtain an approximate solution to the first subproblem. Numerical results on real medical images not only confirm that our proposed ALM is more computationally efficient than some existing methods, but also that the proposed ALM delivers the accurate registration results with the desired property of the constructed deformations in a reasonable number of iterations.

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