scholarly journals Finite element algorithms for nonlocal minimal graphs

2021 ◽  
Vol 4 (2) ◽  
pp. 1-29
Author(s):  
Juan Pablo Borthagaray ◽  
◽  
Wenbo Li ◽  
Ricardo H. Nochetto ◽  
◽  
...  
Keyword(s):  
Mean Curvature ◽  
Numerical Experiments ◽  
Piecewise Linear ◽  
Gradient Flow ◽  
Minimal Graphs ◽  
Qualitative And Quantitative ◽  
Dirichlet Data ◽  
Data Truncation ◽  
Discrete Problems ◽  
Newton Scheme

<abstract><p>We discuss computational and qualitative aspects of the fractional Plateau and the prescribed fractional mean curvature problems on bounded domains subject to exterior data being a subgraph. We recast these problems in terms of energy minimization, and we discretize the latter with piecewise linear finite elements. For the computation of the discrete solutions, we propose and study a gradient flow and a Newton scheme, and we quantify the effect of Dirichlet data truncation. We also present a wide variety of numerical experiments that illustrate qualitative and quantitative features of fractional minimal graphs and the associated discrete problems.</p></abstract>

An Efficient Numerical Method for Mean Curvature-Based Image Registration Model

2017 ◽  
Vol 7 (1) ◽  
pp. 125-142 ◽  
Author(s):  
Jin Zhang ◽  
Ke Chen ◽  
Fang Chen ◽  
Bo Yu
Keyword(s):  
Fixed Point ◽  
Image Quality ◽  
Image Registration ◽  
Mean Curvature ◽  
Numerical Experiments ◽  
Energy Functional ◽  
Linear Search ◽  
Multilevel Method ◽  
Solution Approach ◽  
Newton Scheme

AbstractMean curvature-based image registration model firstly proposed by Chumchob-Chen-Brito (2011) offered a better regularizer technique for both smooth and nonsmooth deformation fields. However, it is extremely challenging to solve efficiently this model and the existing methods are slow or become efficient only with strong assumptions on the smoothing parameterβ. In this paper, we take a different solution approach. Firstly, we discretize the joint energy functional, following an idea of relaxed fixed point is implemented and combine with Gauss-Newton scheme with Armijo's Linear Search for solving the discretized mean curvature model and further to combine with a multilevel method to achieve fast convergence. Numerical experiments not only confirm that our proposed method is efficient and stable, but also it can give more satisfying registration results according to image quality.

scholarly journals Stable approximations for axisymmetric Willmore flow for closed and open surfaces

2021 ◽  
Author(s):  
John W. Barrett ◽  
Harald Garcke ◽  
Robert Nurnberg
Keyword(s):  
Mean Curvature ◽  
Numerical Experiments ◽  
Gradient Flow ◽  
Stability Result ◽  
Spontaneous Curvature ◽  
Line Energy ◽  
Geometric Evolution ◽  
Closed Surfaces ◽  
Willmore Flow ◽  
Willmore Energy

For a hypersurface in $\mathbb R^3$, Willmore flow is defined as the $L^2$--gradient flow of the classical Willmore energy: the integral of the squared mean curvature. This geometric evolution law is of interest in differential geometry, image reconstruction and mathematical biology. In this paper, we propose novel numerical approximations for the Willmore flow of axisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants we prove a stability result.  We consider both closed surfaces, and surfaces with a boundary. In the latter case, we carefully derive  weak formulations of suitable boundary conditions. Furthermore, we consider many generalizations of the classical Willmore energy, particularly those that play a role in the study of biomembranes. In the generalized models we include spontaneous curvature and area difference elasticity (ADE) effects,  Gaussian curvature and line energy contributions. Several numerical experiments demonstrate the efficiency and robustness of our developed numerical methods.

Some remarks on the existence of spacelike hypersurfaces of constant mean curvature

1977 ◽  
Vol 82 (3) ◽  
pp. 489-495 ◽  
Author(s):  
A. J. Goddard
Keyword(s):  
Minkowski Space ◽  
Mean Curvature ◽  
General Class ◽  
Constant Mean Curvature ◽  
Spacelike Hypersurface ◽  
De Sitter Space ◽  
De Sitter ◽  
Minimal Graphs ◽  
Complete Spacelike Hypersurface ◽  
Spacelike Hypersurfaces

AbstractBernstein's theorem states that the only complete minimal graphs in R3 are the hyperplanes. We shall produce evidence in favour of some conjectural generalizations of this theorem for the cases of spacelike hypersurfaces of constant mean curvature in Minkowski space and in de Sitter space. The results suggest that the class of asymptotically simple space-times admitting a complete spacelike hypersurface of constant mean curvature may well be considerably smaller than the general class of asymptotically simple space-times.

scholarly journals Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds

2021 ◽  
Author(s):  
Harald Garcke ◽  
Robert Nürnberg
Keyword(s):  
Boundary Value Problems ◽  
Riemannian Manifolds ◽  
Flow Curve ◽  
Mean Curvature Flow ◽  
Piecewise Linear ◽  
Boundary Value ◽  
Gradient Flow ◽  
Stability Result ◽  
Curvature Flow ◽  
Curve Shortening Flow

AbstractWe present variational approximations of boundary value problems for curvature flow (curve shortening flow) and elastic flow (curve straightening flow) in two-dimensional Riemannian manifolds that are conformally flat. For the evolving open curves we propose natural boundary conditions that respect the appropriate gradient flow structure. Based on suitable weak formulations we introduce finite element approximations using piecewise linear elements. For some of the schemes a stability result can be shown. The derived schemes can be employed in very different contexts. For example, we apply the schemes to the Angenent metric in order to numerically compute rotationally symmetric self-shrinkers for the mean curvature flow. Furthermore, we utilise the schemes to compute geodesics that are relevant for optimal interface profiles in multi-component phase field models.

scholarly journals DG approach to large bending plate deformations with isometry constraint

2021 ◽  
pp. 1-43
Author(s):  
Andrea Bonito ◽  
Ricardo H. Nochetto ◽  
Dimitrios Ntogkas
Keyword(s):  
Discontinuous Galerkin ◽  
Minimization Problem ◽  
Numerical Experiments ◽  
Gradient Flow ◽  
Kirchhoff Plate ◽  
Geometrically Nonlinear ◽  
Plate Model ◽  
Global Minimizers ◽  
Dg Method ◽  
Kirchhoff Plate Model

We propose a new discontinuous Galerkin (dG) method for a geometrically nonlinear Kirchhoff plate model for large isometric bending deformations. The minimization problem is nonconvex due to the isometry constraint. We present a practical discrete gradient flow that decreases the energy and computes discrete minimizers that satisfy a prescribed discrete isometry defect. We prove [Formula: see text]-convergence of the discrete energies and discrete global minimizers. We document the flexibility and accuracy of the dG method with several numerical experiments.

Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition

1990 ◽  
Vol 429 (1877) ◽  
pp. 505-532 ◽  
Keyword(s):  
Mean Curvature ◽  
Gradient Flow ◽  
Dirichlet Condition ◽  
Singularly Perturbed ◽  
Normal Velocity ◽  
Gradient Flows ◽  
Multiple Timescales ◽  
Angle Condition ◽  
Double Well Potential ◽  
Formal Asymptotic Solution

Minimizers and gradient flows are studied for the functional ∫ Ω W(u) + ϵ 2 ∣∇ u ∣ 2 d x , Ω ⊆ R n , ϵ > 0, where u satisfies a Dirichlet condition u = h ϵ on ∂ Ω . Here W is taken to be a double-well potential with minimum value zero attained at u = a and u = b . Questions of existence and structure of minimizers for small ϵ are resolved through the identification of a limiting variational problem, the so-called Γ-limit. A formal asymptotic solution is then constructed for the gradient flow ∂ t u ϵ = 2 ϵ ∆ u ϵ — ϵ -1 W' ( u ϵ ), u ϵ ( x , 0) = g ( x ), u ϵ ( x, t ) = h ϵ on ∂ Ω , valid when ϵ is small. Using multiple timescales we show that fronts rapidly develop and then propagate with normal velocity ϵk , where k is mean curvature. At the intersection of a front with ∂ Ω , the Dirichlet condition is shown to imply a contact angle condition for the front. This asymptotically correct evolution process represents gradient flow for the Γ-limit.

scholarly journals Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations

2020 ◽  
Author(s):  
Bart M. N. Smets ◽  
Jim Portegies ◽  
Etienne St-Onge ◽  
Remco Duits
Keyword(s):  
Total Variation ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Arbitrary Dimension ◽  
Gradient Flow ◽  
Image Data ◽  
Numerical Approach ◽  
Nonlinear Pdes ◽  
Total Variation Flow ◽  
Pde Framework

Abstract Two key ideas have greatly improved techniques for image enhancement and denoising: the lifting of image data to multi-orientation distributions and the application of nonlinear PDEs such as total variation flow (TVF) and mean curvature flow (MCF). These two ideas were recently combined by Chambolle and Pock (for TVF) and Citti et al. (for MCF) for two-dimensional images. In this work, we extend their approach to enhance and denoise images of arbitrary dimension, creating a unified geometric and algorithmic PDE framework, relying on (sub-)Riemannian geometry. In particular, we follow a different numerical approach, for which we prove convergence in the case of TVF by an application of Brezis–Komura gradient flow theory. Our framework also allows for additional data adaptation through the use of locally adaptive frames and coherence enhancement techniques. We apply TVF and MCF to the enhancement and denoising of elongated structures in 2D images via orientation scores and compare the results to Perona–Malik diffusion and BM3D. We also demonstrate our techniques in 3D in the denoising and enhancement of crossing fiber bundles in DW-MRI. In comparison with data-driven diffusions, we see a better preservation of bundle boundaries and angular sharpness in fiber orientation densities at crossings.

RANK ONE CHAOS IN SWITCH-CONTROLLED PIECEWISE LINEAR CHUA'S CIRCUIT

2007 ◽  
Vol 16 (05) ◽  
pp. 769-789 ◽  
Author(s):  
QIUDONG WANG ◽  
ALI OKSASOGLU
Keyword(s):  
Periodic Solution ◽  
Numerical Experiments ◽  
Piecewise Linear ◽  
Small Neighborhood ◽  
Hopf Bifurcations ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Pspice Simulations ◽  
Rank One ◽  
Stable Periodic

In this paper, we continue our study of rank one chaos in switch-controlled circuits. Periodically controlled switches are added to Chua's original piecewise linear circuit to generate rank one attractors in the vicinity of an asymptotically stable periodic solution that is relatively large in size. Our previous investigations relied heavily on the smooth nonlinearity of the unforced systems, and were, by large, restricted to a small neighborhood of supercritical Hopf bifurcations. Whereas the system studied in this paper is much more feasible for physical implementation, and thus the corresponding rank one chaos is much easier to detect in practice. The findings of our purely numerical experiments are further supported by the PSPICE simulations.

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