1,721,017 research outputs found
Backtracking strategies for accelerated descent methods with smooth composite objectives
No full textWe present and analyze a backtracking strategy for a general fast iterative shrinkage/thresholding algorithm proposed by Chambolle and Pock [Acta Numer., 25 (2016), pp. 161–319] for strongly convex composite objective functions. Unlike classical Armijo-type line searching, our backtracking rule allows for local increasing and decreasing of the descent step size (i.e., proximal parameter) along the iterations. We prove accelerated convergence rates and show numerical results for some exemplar problems
Convergence of an algorithm for the anisotropic and crystalline mean curvature flow
Full text linkWe give a simple proof of convergence of the anisotropic variant of a well-known algorithm for mean curvature motion, introduced in 1992 by Merriman, Bence, and Osher. The algorithm consists in alternating the resolution of the (anisotropic) heat equation, with initial datum the characteristic function of the evolving set, and a thresholding at level 1/2
Approximation of the anisotropic mean curvature flow
No full textIn this paper, we provide simple proofs of consistency for two well-known algorithms for mean curvature motion, Almgren-Taylor-Wang's(1) variational approach, and Merriman-Bence-Osher's algorithm. Our techniques, based on the same notion of strict sub- and superflows, also work in the (smooth) anisotropic case
Mumford–Shah functionals on graphs and their asymptotics
Full text linkWe consider adaptations of the Mumford–Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford–Shah functional. Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the graph constructions under which the minimizers of graph Mumford–Shah functionals converge to a minimizer of a continuum Mumford–Shah functional. Furthermore we explicitly identify the limiting functional. Moreover we describe an efficient algorithm for computing the approximate minimizers of the graph Mumford–Shah functional
Existence and qualitative properties of isoperimetric sets in periodic media
No full textWe review and extend here some recent results on the existence of minimal surfaces and isoperimetric sets in non homogeneous and anisotropic periodic media. We also describe the qualitative properties of the homogenized surface tension, also known as stable norm (or minimal action) in Weak KAM theory. In particular we investigate its strict convexity and differentiability properties
Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy
No full textWe prove that special functions of bounded deformation with small jump set are close in energy to functions which are smooth in a slightly smaller domain. This permits to generalize the decay estimate by De Giorgi, Carriero, and Leaci to the linearized context in dimension n and to establish the closedness (up to negligible sets for the (n - 1)-dimensional Hausdorff measure) of the jump set for local minimizers of the Griffith energy. (C) 2019 Elsevier Masson SAS. All rights reserved
A posteriori error estimates for the effective Hamiltonian of dislocation dynamics
No full textWe study an implicit and discontinuous scheme for a non-local Hamilton-Jacobi equation modelling dislocation dynamics. For the evolution problem, we prove an a posteriori estimate of Crandall-Lions type for the error between continuous and discrete solutions. We deduce an a posteriori error estimate for the effective Hamiltonian associated to a stationary cell problem. In dimension one and under suitable assumptions, we also give improved a posteriori estimates. Numerical simulations are provided. © 2011 Springer-Verlag
Regularity for solutions of the total variation denoising problem
No full textThe main purpose of this paper is to prove a local Holder regularity result for the solutions of the total variation based denoising problem assuming that the datum is locally Holder continuous. We also prove a global estimate on the modulus of continuity of the solution in convex domains of R(N) and some extensions of this result for the total variation minimization flow
Some remarks on uniqueness and regularity of Cheeger sets
No full textWe show that generically the subsets of R^N with finite volume have a unique Cheeger set, in the sense that there always exists a nearby set which has a unique Cheeger set. We also prove that Cheeger sets are C^(1,1), when the ambient set is C^(1,1)
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