scholarly journals Bound Alternative Direction Optimization for Image Deblurring

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Xiangrong Zeng
Keyword(s):  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Experimental Studies ◽  
Image Deblurring ◽  
Minimization Method ◽  
Minimization Problems ◽  
Variable Splitting ◽  
Alternating Direction ◽  
Alternative Direction Method ◽  
Unconstrained Problem

This paper proposes a new method,bound alternative direction method(BADM), to address theℓp  (p∈0,1)minimization problems in image deblurring. The approach is to first obtain a bound unconstrained problem through bounding theℓpregularizer by a novel majorizer and then, based on a variable splitting, to reformulate the bound unconstrained problem into a constrained one, which is then addressed via an augmented Lagrangian method. The proposed algorithm actually combines the reweightedℓ1minimization method and thealternating direction method of multiples(ADMM) such that it succeeds in extending the application of ADMM toℓpminimization problems. The conducted experimental studies demonstrate the superiority of the proposed algorithm for the synthesisℓpminimization over the state-of-the-art algorithms for the synthesisℓ1minimization on image deblurring.

Alternating direction based method for optimal control problem constrained by Stokes equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Yu Gao ◽  
Jingzhi Li ◽  
Yongcun Song ◽  
Chao Wang ◽  
Kai Zhang
Keyword(s):  
Optimal Control ◽  
Discrete System ◽  
Augmented Lagrangian ◽  
Stokes Equations ◽  
Optimization Problems ◽  
Augmented Lagrangian Method ◽  
Lagrangian Method ◽  
The Finite Element Method ◽  
Alternating Direction ◽  
Different Types

Abstract We consider the optimal control problems constrained by Stokes equations. It has been shown in the literature, the problem can be discretized by the finite element method to generate a discrete system, and the error estimate has also been established. In this paper, we focus on solving the discrete system by the alternating splitting augmented Lagrangian method, which is a direct extension of alternating direction method of multipliers and possesses a global O ⁢ ( 1 / k ) \mathcal{O}({1}/{k}) convergence rate. In addition, we propose an acceleration scheme based on the alternating splitting augmented Lagrangian method to improve the efficiency of the algorithm. The error estimates and convergence analysis of our algorithms are presented for several different types of optimization problems. Finally, numerical experiments are performed to verify the efficiency of the algorithms.

On Variable Splitting and Augmented Lagrangian Method for Total Variation-Related Image Restoration Models

2021 ◽  
pp. 1-47
Author(s):  
Zhifang Liu ◽  
Yuping Duan ◽  
Chunlin Wu ◽  
Xue-Cheng Tai
Keyword(s):  
Image Restoration ◽  
Total Variation ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Lagrangian Method ◽  
Variable Splitting

scholarly journals Rotational self-friction problem of elastic rods

2021 ◽  
Author(s):  
Mohamed Ali Latrach ◽  
Mourad Chamekh
Keyword(s):  
Variational Formulation ◽  
Existence Result ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Contact Region ◽  
Large Displacements ◽  
Problem Type ◽  
Finite Element Discretization ◽  
Minimization Method ◽  
Element Discretization

Abstract The aim of this paper is to extend the modeling of a hyperelastic rod undergoing large displacements with tangential self-friction to their modeling with rotational self-friction. As well as the discontinuity of contact force into a contact region not known in advance, taking into account the effects of friction in this problem type underlies more serious modeling, mathematical and numerical analysis difficulties. In this paper, we present an accurate modeling of rotational and tangential self-friction with Coulomb's law and also describe an augmented Lagrangian method to present a weak variational formulation approach of this problem. We then use the minimization method of the total energy to present an existence result of solution for the nonlinear penalized formulation. Finally, we give the linearization and the finite-element discretization of the weak variational formulation that can be useful for a numerical implementation.

scholarly journals Smooth augmented lagrangian method for twin bounded support vector machine

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fatemeh Bazikar ◽  
Saeed Ketabchi ◽  
Hossein Moosaei
Keyword(s):  
Support Vector Machine ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Binary Classification ◽  
Unconstrained Minimization ◽  
Lagrangian Method ◽  
Superior Performance ◽  
Support Vector ◽  
Minimization Problems ◽  
Bounded Support

<p style='text-indent:20px;'>In this paper, we propose a method for solving the twin bounded support vector machine (TBSVM) for the binary classification. To do so, we use the augmented Lagrangian (AL) optimization method and smoothing technique, to obtain new unconstrained smooth minimization problems for TBSVM classifiers. At first, the augmented Lagrangian method is recruited to convert TBSVM into unconstrained minimization programming problems called as AL-TBSVM. We attempt to solve the primal programming problems of AL-TBSVM by converting them into smooth unconstrained minimization problems. Then, the smooth reformulations of AL-TBSVM, which we called AL-STBSVM, are solved by the well-known Newton's algorithm. Finally, experimental results on artificial and several University of California Irvine (UCI) benchmark data sets are provided along with the statistical analysis to show the superior performance of our method in terms of classification accuracy and learning speed.</p>

Micro-CT image reconstruction based on alternating direction augmented Lagrangian method and total variation

2013 ◽  
Vol 37 (7-8) ◽  
pp. 419-429 ◽  
Author(s):  
Varun P. Gopi ◽  
P. Palanisamy ◽  
Khan A. Wahid ◽  
Paul Babyn ◽  
David Cooper
Keyword(s):  
Image Reconstruction ◽  
Total Variation ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Lagrangian Method ◽  
Ct Image ◽  
Micro Ct ◽  
Alternating Direction ◽  
Ct Image Reconstruction

scholarly journals Augmented Lagrangian preconditioners for the Oseen–Frank model of nematic and cholesteric liquid crystals

2021 ◽  
Author(s):  
Jingmin Xia ◽  
Patrick E. Farrell ◽  
Florian Wechsung
Keyword(s):  
Liquid Crystals ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Mass Matrix ◽  
Unit Length ◽  
Mesh Size ◽  
Schur Complement ◽  
Cholesteric Liquid Crystals ◽  
Multigrid Algorithm ◽  
The Cost

AbstractWe propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen–Frank model arising in nematic and cholesteric liquid crystals. By applying the augmented Lagrangian method, the Schur complement of the director block can be better approximated by the weighted mass matrix of the Lagrange multiplier, at the cost of making the augmented director block harder to solve. In order to solve the augmented director block, we develop a robust multigrid algorithm which includes an additive Schwarz relaxation that captures a pointwise version of the kernel of the semi-definite term. Furthermore, we prove that the augmented Lagrangian term improves the discrete enforcement of the unit-length constraint. Numerical experiments verify the efficiency of the algorithm and its robustness with respect to problem-related parameters (Frank constants and cholesteric pitch) and the mesh size.

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.

Sign in / Sign up

Export Citation Format

Share Document