# Analysis of total variation flow and its finite element approximations

- Volume: 37, Issue: 3, page 533-556
- ISSN: 0764-583X

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topFeng, Xiaobing, and Prohl, Andreas. "Analysis of total variation flow and its finite element approximations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.3 (2003): 533-556. <http://eudml.org/doc/245487>.

@article{Feng2003,

abstract = {We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter $\{\varepsilon \}$, and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence to the regularized gradient flow problem as $h,k\rightarrow 0$, and to the total variation gradient flow problem as $h,k,\{\varepsilon \}\rightarrow 0$ in general cases. Provided that the regularized gradient flow problem possesses strong solutions, which is proved possible if the datum functions are regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter $\varepsilon $. Optimal order error bounds are derived for the numerical solution under the mesh relation $k=O(h^2)$. In particular, it is shown that all error bounds depend on $\frac\{1\}\{\varepsilon \}$ only in some lower polynomial order for small $\varepsilon $.},

author = {Feng, Xiaobing, Prohl, Andreas},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {bounded variation; gradient flow; variational inequality; equations of prescribed mean curvature and minimal surface; fully discrete scheme; finite element method},

language = {eng},

number = {3},

pages = {533-556},

publisher = {EDP-Sciences},

title = {Analysis of total variation flow and its finite element approximations},

url = {http://eudml.org/doc/245487},

volume = {37},

year = {2003},

}

TY - JOUR

AU - Feng, Xiaobing

AU - Prohl, Andreas

TI - Analysis of total variation flow and its finite element approximations

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2003

PB - EDP-Sciences

VL - 37

IS - 3

SP - 533

EP - 556

AB - We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow, and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship between the regularized gradient flow (characterized by a small positive parameter ${\varepsilon }$, and the minimal surface flow [21] and the prescribed mean curvature flow [16]. Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence to the regularized gradient flow problem as $h,k\rightarrow 0$, and to the total variation gradient flow problem as $h,k,{\varepsilon }\rightarrow 0$ in general cases. Provided that the regularized gradient flow problem possesses strong solutions, which is proved possible if the datum functions are regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter $\varepsilon $. Optimal order error bounds are derived for the numerical solution under the mesh relation $k=O(h^2)$. In particular, it is shown that all error bounds depend on $\frac{1}{\varepsilon }$ only in some lower polynomial order for small $\varepsilon $.

LA - eng

KW - bounded variation; gradient flow; variational inequality; equations of prescribed mean curvature and minimal surface; fully discrete scheme; finite element method

UR - http://eudml.org/doc/245487

ER -

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