Convex approximations of functionals with curvature

Bellettini, Giovanni, Paolini, Maurizio, and Verdi, Claudio. "Convex approximations of functionals with curvature." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 2.4 (1991): 297-306. <http://eudml.org/doc/244157>.

@article{Bellettini1991,
abstract = {We address the numerical minimization of the functional \( \mathcal\{F\} (v) = \int\_\{\Omega\} |Dv| + \int\_\{\partial \Omega\} \mu v \, d\mathcal\{H\}^\{n-1\} - \int\_\{\Omega\} x v \, dx \), for \( v \in BV(\Omega; \\{-1,1\\}) \). We note that \( \mathcal\{F\} \) can be equivalently minimized on the larger, convex, set \( BV(\Omega; \left[-1,1\right]) \) and that, on that space, \( \mathcal\{F\} \) may be regularized with a sequence \( \\{ \mathcal\{F\}\_\{\epsilon\}(v) = \int\_\{\Omega\} \sqrt\{ \epsilon^\{2\} + |Dv|^\{2\}\} + \int\_\{\partial \Omega\} \mu v \, d\mathcal\{H\}^\{n-1\} - \int\_\{\Omega\} xv \, dx \\}\_\{\epsilon\} \)of regular functionals. Then both \( \mathcal\{F\} \) and \( \mathcal\{F\}\_\{\epsilon\} \) can be discretized by continuous linear finite elements. The convexity of the functionals in \( BV(\Omega; \left[-1,1\right]) \) is useful for the numerical minimization of \( \mathcal\{F\} \). We prove the \( \Gamma - L^\{1\} (\Omega) \)-convergence of the discrete functionals to \( \mathcal\{F\} \) and present a few numerical examples.},
author = {Bellettini, Giovanni, Paolini, Maurizio, Verdi, Claudio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {calculus of variations; surfaces with prescribed mean curvature; convergence; computer vision; convex approximations; finite elements},
language = {eng},
month = {12},
number = {4},
pages = {297-306},
publisher = {Accademia Nazionale dei Lincei},
title = {Convex approximations of functionals with curvature},
url = {http://eudml.org/doc/244157},
volume = {2},
year = {1991},
}

TY - JOUR
AU - Bellettini, Giovanni
AU - Paolini, Maurizio
AU - Verdi, Claudio
TI - Convex approximations of functionals with curvature
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1991/12//
PB - Accademia Nazionale dei Lincei
VL - 2
IS - 4
SP - 297
EP - 306
AB - We address the numerical minimization of the functional \( \mathcal{F} (v) = \int_{\Omega} |Dv| + \int_{\partial \Omega} \mu v \, d\mathcal{H}^{n-1} - \int_{\Omega} x v \, dx \), for \( v \in BV(\Omega; \{-1,1\}) \). We note that \( \mathcal{F} \) can be equivalently minimized on the larger, convex, set \( BV(\Omega; \left[-1,1\right]) \) and that, on that space, \( \mathcal{F} \) may be regularized with a sequence \( \{ \mathcal{F}_{\epsilon}(v) = \int_{\Omega} \sqrt{ \epsilon^{2} + |Dv|^{2}} + \int_{\partial \Omega} \mu v \, d\mathcal{H}^{n-1} - \int_{\Omega} xv \, dx \}_{\epsilon} \)of regular functionals. Then both \( \mathcal{F} \) and \( \mathcal{F}_{\epsilon} \) can be discretized by continuous linear finite elements. The convexity of the functionals in \( BV(\Omega; \left[-1,1\right]) \) is useful for the numerical minimization of \( \mathcal{F} \). We prove the \( \Gamma - L^{1} (\Omega) \)-convergence of the discrete functionals to \( \mathcal{F} \) and present a few numerical examples.
LA - eng
KW - calculus of variations; surfaces with prescribed mean curvature; convergence; computer vision; convex approximations; finite elements
UR - http://eudml.org/doc/244157
ER -