46,488 research outputs found
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
Mean curvature flow for generating discrete surfaces with piecewise constant mean curvatures
Piecewise constant mean curvature (P-CMC) surfaces are generated using the mean curvature flow (MCF). As an extension of the known fact that a CMC surface is the stationary point of an energy functional, a P-CMC surface can be obtained as the stationary point of an energy functional of multiple patch surfaces and auxiliary surfaces between them. A new formulation is presented for the MCF as the negative gradient flow of the energy functional for multiple patch continuous surfaces, which are further discretized so as to determine the change in the vertex positions of triangular meshes on the surface as well as along the internal boundaries between patches. Numerical examples show that multiple patch surfaces approximately reach the specified mean curvatures through the proposed method, which can diversify the options for the shape design using CMC surfaces
Volume-preserving mean curvature flow of revolution hypersurfaces in a Rotationally Symmetric Space
In an ambient space with rotational symmetry around an axis (which include
the Hyperbolic and Euclidean spaces), we study the evolution under the
volume-preserving mean curvature flow of a revolution hypersurface M generated
by a graph over the axis of revolution and with boundary in two totally
geodesic hypersurfaces (tgh for short). Requiring that, for each time t, the
evolving hypersurface M_t meets such tgh ortogonally, we prove that: a) the
flow exists while M_t does not touch the axis of rotation; b) throughout the
time interval of existence, b1) the generating curve of M_t remains a graph,
and b2) the averaged mean curvature is double side bounded by positive
constants; c) the singularity set (if non-empty) is finite and discrete along
the axis; d) under a suitable hypothesis relating the enclosed volume to the
n-volume of M, we achieve long time existence and convergence to a revolution
hypersurface of constant mean curvature.Comment: 24 pages. We have added some lines at the beginning explaining the
notation, and clarified a little bit more the proofs of Proposition 1 and
Theorems 5 and 10, the statements of Proposition 2 and Corollary 3 and an
argument in Remark 1. We have also completed reference 18. A version of this
paper will appear in Mathematische Zeitschrif
Mean curvature, threshold dynamics, and phase field theory on finite graphs
In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study.
We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow.
We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as ``freezing'' or ``pinning'') and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation
A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs
We consider the probabilistic numerical scheme for fully nonlinear PDEs
suggested in \cite{cstv}, and show that it can be introduced naturally as a
combination of Monte Carlo and finite differences scheme without appealing to
the theory of backward stochastic differential equations. Our first main result
provides the convergence of the discrete-time approximation and derives a bound
on the discretization error in terms of the time step. An explicit
implementable scheme requires to approximate the conditional expectation
operators involved in the discretization. This induces a further Monte Carlo
error. Our second main result is to prove the convergence of the latter
approximation scheme, and to derive an upper bound on the approximation error.
Numerical experiments are performed for the approximation of the solution of
the mean curvature flow equation in dimensions two and three, and for two and
five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations
arising in the theory of portfolio optimization in financial mathematics
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