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 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.

scholarly journals A time dependent Stokes interface problem: well-posedness and space-time finite element discretization

2018 ◽  
Vol 52 (6) ◽  
pp. 2187-2213 ◽  
Author(s):  
Igor Voulis ◽  
Arnold Reusken
Keyword(s):  
Finite Element ◽  
Variational Formulation ◽  
Time Dependent ◽  
Interface Problem ◽  
Phase Flow ◽  
Finite Element Discretization ◽  
Two Phase ◽  
Element Discretization ◽  
Divergence Free ◽  
Well Posed

In this paper a time dependent Stokes problem that is motivated by a standard sharp interface model for the fluid dynamics of two-phase flows is studied. This Stokes interface problem has discontinuous density and viscosity coefficients and a pressure solution that is discontinuous across an evolving interface. This strongly simplified two-phase Stokes equation is considered to be a good model problem for the development and analysis of finite element discretization methods for two-phase flow problems. In view of theunfitted finite element methods that are often used for two-phase flow simulations, we are particularly interested in a well-posed variational formulation of this Stokes interface problem in a Euclidean setting. Such well-posed weak formulations, which are not known in the literature, are the main results of this paper. Different variants are considered, namely one with suitable spaces of divergence free functions, a discrete-in-time version of it, and variants in which the divergence free constraint in the solution space is treated by a pressure Lagrange multiplier. The discrete-in-time variational formulation involving the pressure variable for the divergence free constraint is a natural starting point for a space-time finite element discretization. Such a method is introduced and results of numerical experiments with this method are presented.

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.

scholarly journals An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

2021 ◽  
Author(s):  
Christian Kanzow ◽  
Andreas B. Raharja ◽  
Alexandra Schwartz
Keyword(s):  
Constrained Optimization ◽  
Numerical Experiments ◽  
Penalty Method ◽  
Augmented Lagrangian ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Augmented Lagrangian Method ◽  
Constrained Optimization Problems ◽  
Strongly Stationary ◽  
Type Constraint

AbstractA reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided.

scholarly journals Augmented Lagrangian method for second-order cone programs under second-order sufficiency

2021 ◽  
Author(s):  
Nguyen T. V. Hang ◽  
Boris S. Mordukhovich ◽  
M. Ebrahim Sarabi
Keyword(s):  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Lagrangian Method ◽  
Second Order ◽  
Second Order Cone

The Augmented Lagrangian Method for Parameter Estimation in Elliptic Systems

1990 ◽  
Vol 28 (1) ◽  
pp. 113-136 ◽  
Author(s):  
Kazufumi Ito ◽  
Karl Kunisch
Keyword(s):  
Parameter Estimation ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Elliptic Systems ◽  
Lagrangian Method
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