Some Facts About Operator-Splitting and Alternating Direction Methods

Glowinski, Roland; Pan, Tsorng‐Whay; Tai, Xue–Cheng

doi:10.1007/978-3-319-41589-5_2

Cited by 38 publications

(22 citation statements)

References 147 publications

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“…To fully discretize system (13)- (19), we reduce it first to a finite dimensional initial value problem using the above finite element spaces (after dropping most of the sub-scripts h's). Next, we combine the Lozinski-Owens factorization approach (see, e.g., [21], [20]) with the Lie scheme (e.g., see [34], [35], and [36]) to decouple the above finite element analogue of system (13)- (19) into a sequence of subproblems and apply the backward Euler schemes to time-discretize some of these subproblems. Finally we obtain thus the following sequence of sub-problems (where △t(> 0) is a time-discretization step and t n = n△t):…”

Section: Methodsmentioning

confidence: 99%

A 3D DLM/FD method for simulating the motion of spheres in a bounded shear flow of Oldroyd-B fluids

2018

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We present a novel distributed Lagrange multiplier/fictitious domain (DLM/FD) method for simulating fluid-particle interaction in Oldroyd-B fluids under creeping conditions. The results concerning two ball interaction in a three dimensional (3D) bounded shear flow are obtained for Weissenberg numbers up to 1 . The pass and return trajectories of the two ball mass centers are similar to those in a Newtonian fluid; but they lose the symmetry due to the effect of elastic force arising from viscoelastic fluids. A tumbling chain of two balls (a dipole) may occur, depending on the value of the Weissenberg number and the initial vertical displacement of the ball mass center to the middle plane between two walls.

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Section: Methodsmentioning

confidence: 99%

A 3D DLM/FD method for simulating the motion of spheres in a bounded shear flow of Oldroyd-B fluids

2018

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“…Most of the authors introduce auxiliary variables [9,26,10], for instance representing the orientation, which is already close to the idea of the Gauss map or Roto-translational representation. Recent techniques based on Augmented Lagrangian methods (for coupling the auxiliary variables) [74,80,78,79,43,42,8] have shown to be quite efficient, despite the lack of convexity and hence convergence guaranties.…”

Section: + DIVmentioning

confidence: 99%

Total roto-translational variation

Chambolle

Pock

2019

Numer. Math.

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We consider curvature depending variational models for image regularization, such as Euler's elastica. These models are known to provide strong priors for the continuity of edges and hence have important applications in shape-and image processing. We consider a lifted convex representation of these models in the roto-translation space: In this space, curvature depending variational energies are represented by means of a convex functional defined on divergence free vector fields. The line energies are then easily extended to any scalar function. It yields a natural generalization of the total variation to the roto-translation space. As our main result, we show that the proposed convex representation is tight for characteristic functions of smooth shapes. We also discuss cases where this representation fails. For numerical solution, we propose a staggered grid discretization based on an averaged Raviart-Thomas finite elements approximation. This discretization is consistent, up to minor details, with the underlying continuous model. The resulting non-smooth convex optimization problem is solved using a first-order primal-dual algorithm. We illustrate the results of our numerical algorithm on various problems from shape-and image processing. vision) since at least the 90s [12,13,16,14,17,15,30]. It is shown already in [12] that the boundary of many sets with cusps can be approximated by sets with smooth boundaries of bounded energy, showing that the relaxation of the Elastica for boundaries of sets is already far from trivial. The study of this lower semi-continuity of course enters the long history of the study of general curvature dependent energies of manifolds and in particular the Willmore energy [77].Quite early, it has been suggested to lift the manifold in a larger space where a variable represents its direction or orientation, by means in particular (for co-dimension one manifolds) of the Gauss map (x, ν(x)), ν(x) being the normal to the manifold at x [5,6]. Such approach allows to study very general curvature energies and has been successfully used for establishing lower-semicontinuity and existence results [6,7,32,31], and in particular to lines energies such as ours in higher codimension [2,1] or in dimension 2 [31]. We must mention also in this class an older approach based on "curvature varifolds" (which is very natural since varifolds are defined on the cross product of spatial and directional variables) which has allowed to show existence results since the 80s [45], see also [50].Interestingly, it was understood much earlier [44] that (cats') vision was functioning in a similar way (eg., using a sort of "Gauss map"), thanks to neurons sensitive to particular directions which were found to be stacked inside the visual cortex into ordered columns, making us sensitive to changes of orientations (and thus curvature intensity). These findings (which might explain some of Kanizsa's experiments) inspired some mathematical models quite early [48], however they were formalized into a consistent geometric inte...

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“…However, the convergence speed will be affected. In order to guarantee the convergence speed, we use the rule introduced in [35] to tune θ 1 , θ 2 and θ 3 . Namely, the gives a good convergence speed as each pair of curves in Fig.…”

Section: Parametersmentioning

confidence: 99%

Introducing diffusion tensor to high order variational model for image reconstruction

Duan

Ward

Sibbett

et al. 2017

Digital Signal Processing

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The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.uk JID:YDSPR AID:2153 /FLA [m5G; v1.219; Prn:17/07/2017; 11:02] Second order total variation (SOTV) models have advantages for image restoration over their first order counterparts including their ability to remove the staircase artefact in the restored image. However, such models tend to blur the reconstructed image when discretised for numerical solution [1][2][3][4][5]. To overcome this drawback, we introduce a new tensor weighted second order (TWSO) model for image restoration. Specifically, we develop a novel regulariser for the SOTV model that uses the Frobenius norm of the product of the isotropic SOTV Hessian matrix and an anisotropic tensor. We then adapt the alternating direction method of multipliers (ADMM) to solve the proposed model by breaking down the original problem into several subproblems. All the subproblems have closed-forms and can be solved efficiently. The proposed method is compared with state-of-the-art approaches such as tensor-based anisotropic diffusion, total generalised variation, and Euler's elastica. We validate the proposed TWSO model using extensive experimental results on a large number of images from the Berkeley BSDS500. We also demonstrate that our method effectively reduces both the staircase and blurring effects and outperforms existing approaches for image inpainting and denoising applications.

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Some Facts About Operator-Splitting and Alternating Direction Methods

Cited by 38 publications

References 147 publications

A 3D DLM/FD method for simulating the motion of spheres in a bounded shear flow of Oldroyd-B fluids

A 3D DLM/FD method for simulating the motion of spheres in a bounded shear flow of Oldroyd-B fluids

Total roto-translational variation

Introducing diffusion tensor to high order variational model for image reconstruction

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