A convergence result for the Gradient Flow of ∫ |A|2 in Riemannian Manifolds

2015 ◽  
Vol 1 (1) ◽  
Author(s):  
Annibale Magni
Keyword(s):  
Riemannian Manifolds ◽  
Gradient Flow ◽  
Convergence Result

AbstractWe study the gradient flow of the L

ON THE SEMILOCAL CONVERGENCE OF NEWTON'S METHOD FOR SECTIONS ON RIEMANNIAN MANIFOLDS

2014 ◽  
Vol 07 (01) ◽  
pp. 1450007
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Convergence Analysis ◽  
Lipschitz Condition ◽  
Riemannian Manifolds ◽  
Newton’S Method ◽  
Newton's Method ◽  
Geometric Model ◽  
Semilocal Convergence ◽  
Convergence Result ◽  
Numer Anal ◽  
The One

We present a semilocal convergence analysis of Newton's method for sections on Riemannian manifolds. Using the notion of a 2-piece L-average Lipschitz condition introduced in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant α-theory, J. Complexity24 (2008) 423–451] in combination with the weaker center 2-piece L1-average Lipschitz condition given by us in this paper, we provide a tighter convergence analysis than the one given in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant α-theory, J. Complexity24 (2008) 423–451] which in turn has improved the works in earlier studies such as [R. L. Adler, J. P. Dedieu, J. Y. Margulies, M. Martens and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numer. Anal.22 (2002) 359–390; F. Alvarez, J. Bolte and J. Munier, A unifying local convergence result for Newton's method in Riemannian manifolds, Found. Comput. Math.8 (2008) 197–226; J. P. Dedieu, P. Priouret and G. Malajovich, Newton's method on Riemannian manifolds: Covariant α-theory, IMA J. Numer. Anal.23 (2003) 395–419].

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 On the total variation Wasserstein gradient flow and the TV-JKO scheme

2019 ◽  
Vol 25 ◽  
pp. 42
Author(s):  
Guillaume Carlier ◽  
Clarice Poon
Keyword(s):  
Total Variation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Minimum Principle ◽  
Gradient Flow ◽  
Convergence Result ◽  
Symmetric Case ◽  
Qualitative Properties ◽  
Time Step ◽  
Dimension One

We study the JKO scheme for the total variation, characterize the optimizers, prove some of their qualitative properties (in particular a form of maximum principle and in some cases, a minimum principle as well). Finally, we establish a convergence result as the time step goes to zero to a solution of a fourth-order nonlinear evolution equation, under the additional assumption that the density remains bounded away from zero, this lower bound is shown in dimension one and in the radially symmetric case.

Long time behavior of quasi-convex and pseudo-convex gradient systems on Riemannian manifolds

Filomat ◽  
2017 ◽  
Vol 31 (14) ◽  
pp. 4571-4578 ◽  
Author(s):  
P. Ahmadi ◽  
H. Khatibzadeh
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Gradient Flow ◽  
Complete Riemannian Manifold ◽  
Discrete Version ◽  
Gradient System ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Pseudo Convex

In this paper, we study the following gradient system on a complete Riemannian manifold M, {-x?(t) = grad'(x(t)) x(0) = x0, where ? : M ? R is a C1 function with Argmin ? ? ?. We prove that the gradient flow x(t) converges to a critical point of ? if ? is pseudo-convex, or if ? is quasi-convex and M is Hadamard. As an application to minimization, we consider a discrete version of the system to approximate a minimum point of a given pseudo-convex function ?.

scholarly journals Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

2021 ◽  
Author(s):  
Antonio Esposito ◽  
Francesco S. Patacchini ◽  
André Schlichting ◽  
Dejan Slepčev
Keyword(s):  
Gradient Flow ◽  
Wasserstein Distance ◽  
Convergence Result ◽  
Existence Theory ◽  
Nonlocal Interaction ◽  
Interaction Energies ◽  
Gradient Flows ◽  
Empirical Measures ◽  
Interaction Equation ◽  
The Continuum

AbstractWe consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$$^2$$ 2 IE). We develop the existence theory for the solutions of the NL$$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.

scholarly journals PASSING TO THE LIMIT IN MAXIMAL SLOPE CURVES: FROM A REGULARIZED PERONA–MALIK EQUATION TO THE TOTAL VARIATION FLOW

2012 ◽  
Vol 22 (08) ◽  
pp. 1250017 ◽  
Author(s):  
MARIA COLOMBO ◽  
MASSIMO GOBBINO
Keyword(s):  
Total Variation ◽  
General Result ◽  
General Principle ◽  
Time Scale ◽  
Space Dimension ◽  
Gradient Flow ◽  
Convergence Result ◽  
Gradient Flows ◽  
Slow Time ◽  
Total Variation Flow

We prove that solutions of a mildly regularized Perona–Malik equation converge, in a slow time scale, to solutions of the total variation flow. The convergence result is global-in-time, and holds true in any space dimension. The proof is based on the general principle that "the limit of gradient-flows is the gradient-flow of the limit". To this end, we exploit a general result relating the Gamma-limit of a sequence of functionals to the limit of the corresponding maximal slope curves.

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