# Analysis of total variation flow and its finite element approximations

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- 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 37.3 (2010): 533-556. <http://eudml.org/doc/194177>.

@article{Feng2010,

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
ε, see (1.7)) 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 → 0, and to the
total variation gradient flow problem as h,k,ε → 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 ε. Optimal order error bounds are
derived for the numerical solution under the mesh
relation k = O(h2). In particular, it is shown that
all error bounds depend on $\frac\{1\}\{\varepsilon\}$ only
in some lower polynomial order for small ε.
},

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

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

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

language = {eng},

month = {3},

number = {3},

pages = {533-556},

publisher = {EDP Sciences},

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

url = {http://eudml.org/doc/194177},

volume = {37},

year = {2010},

}

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

DA - 2010/3//

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
ε, see (1.7)) 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 → 0, and to the
total variation gradient flow problem as h,k,ε → 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 ε. Optimal order error bounds are
derived for the numerical solution under the mesh
relation k = O(h2). In particular, it is shown that
all error bounds depend on $\frac{1}{\varepsilon}$ only
in some lower polynomial order for small ε.

LA - eng

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

UR - http://eudml.org/doc/194177

ER -

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