1,684 research outputs found
Nonlocal curvature flows
This paper aims at building a unified framework to deal with a wide class of
local and nonlocal translation-invariant geometric flows. First, we introduce a
class of generalized curvatures, and prove the existence and uniqueness for the
level set formulation of the corresponding geometric flows.
We then introduce a class of generalized perimeters, whose first variation is
an admissible generalized curvature. Within this class, we implement a
minimizing movements scheme and we prove that it approximates the viscosity
solution of the corresponding level set PDE.
We also describe several examples and applications. Besides recovering and
presenting in a unified way existence, uniqueness, and approximation results
for several geometric motions already studied and scattered in the literature,
the theory developed in this paper allows us to establish also new results
Multiclass Data Segmentation using Diffuse Interface Methods on Graphs
We present two graph-based algorithms for multiclass segmentation of
high-dimensional data. The algorithms use a diffuse interface model based on
the Ginzburg-Landau functional, related to total variation compressed sensing
and image processing. A multiclass extension is introduced using the Gibbs
simplex, with the functional's double-well potential modified to handle the
multiclass case. The first algorithm minimizes the functional using a convex
splitting numerical scheme. The second algorithm is a uses a graph adaptation
of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates
between diffusion and thresholding. We demonstrate the performance of both
algorithms experimentally on synthetic data, grayscale and color images, and
several benchmark data sets such as MNIST, COIL and WebKB. We also make use of
fast numerical solvers for finding the eigenvectors and eigenvalues of the
graph Laplacian, and take advantage of the sparsity of the matrix. Experiments
indicate that the results are competitive with or better than the current
state-of-the-art multiclass segmentation algorithms.Comment: 14 page
Periodic total variation flow of non-divergence type in Rn
We introduce a new notion of viscosity solutions for a class of very singular
nonlinear parabolic problems of non-divergence form in a periodic domain of
arbitrary dimension, whose diffusion on flat parts with zero slope is so strong
that it becomes a nonlocal quantity. The problems include the classical total
variation flow and a motion of a surface by a crystalline mean curvature. We
establish a comparison principle, the stability under approximation by
regularized parabolic problems, and an existence theorem for general continuous
initial data.Comment: 36 pages, 2 figure
A Non-Local Mean Curvature Flow and its semi-implicit time-discrete approximation
We address in this paper the study of a geometric evolution, corresponding to
a curvature which is non-local and singular at the origin. The curvature
represents the first variation of the energy recently proposed as a variant of
the standard perimeter penalization for the denoising of nonsmooth curves.
To deal with such degeneracies, we first give an abstract existence and
uniqueness result for viscosity solutions of non-local degenerate Hamiltonians,
satisfying suitable continuity assumption with respect to Kuratowsky
convergence of the level sets. This abstract setting applies to an approximated
flow. Then, by the method of minimizing movements, we also build an "exact"
curvature flow, and we illustrate some examples, comparing the results with the
standard mean curvature flow
Finite element algorithms for nonlocal minimal graphs
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
Neckpinch singularities in fractional mean curvature flows
In this paper we consider the evolution of sets by a fractional mean
curvature flow. Our main result states that for any dimension , there
exists an embedded surface in evolving by fractional mean
curvature flow, which developes a singularity before it can shrink to a point.
When this result generalizes the analogue result of Grayson for the
classical mean curvature flow. Interestingly, when , our result provides
instead a counterexample in the nonlocal framework to the well known Grayson
Theorem, which states that any smooth embedded curve in the plane evolving by
(classical) MCF shrinks to a point
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