# American Institute of Mathematical Sciences

December  2017, 10(6): 1207-1232. doi: 10.3934/dcdss.2017066

## Iterative finite element solution of a constrained total variation regularized model problem

 Department of Applied Mathematics, Mathematical Institute, University of Freiburg, Hermann-Herder-Str. 9,79104 Freiburg i. Br., Germany

* Corresponding author: Sören Bartels

Dedicated to Professor T. Roubíčcek on the occasion of his 60th birthday.

Received  May 2016 Revised  November 2016 Published  June 2017

The discretization of a bilaterally constrained total variation minimization problem with conforming low order finite elements is analyzed and three iterative schemes are proposed which differ in the treatment of the non-differentiable terms. Unconditional stability and convergence of the algorithms is addressed, an application to piecewise constant image segmentation is presented and numerical experiments are shown.

Citation: Sören Bartels, Marijo Milicevic. Iterative finite element solution of a constrained total variation regularized model problem. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1207-1232. doi: 10.3934/dcdss.2017066
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##### References:
$L^2$-error between approximate minimizer $\tilde{u}_h$ of $E$ (generated by the split-split method) and iterates $u_h^j$, $s_h^j$ of the Heron-penalty method, $h=\sqrt{2}2^{-6}$, $\varepsilon=h$, for Example 1. Note that the iterates $u_h^j$ serve as good approximations of $\tilde{u}_h$ when $\delta=h/\alpha$
$L^2$-error between approximate minimizer $\tilde{u}_h$ of $E$ (generated by the split-split method) and iterates $u_h^j$, $s_h^j$ of the Heron-penalty method, $h=\sqrt{2}2^{-6}$, $\varepsilon=h$, for Example 2. Again, the iterates $u_h^j$ approximate $\tilde{u}_h$ properly when $\delta=h/\alpha$
Original image and outputs of Algorithm 5 using the split-split, Heron-penalty and Heron-split method in step (2), respectively (horizontal white lines are due to image conversion)
Iteration numbers with (5), (6), (7) for Example 1 and $\sigma_1=h^{-3/2}$ for the split-split method and $\tau=1$ for the Heron-penalty and the Heron-split method
 Split-split Heron-penalty Heron-split $\sigma_2$ $\delta$ $\sigma$ $1$ $\alpha$ $1/h$ $h/\alpha$ $h$ $1$ $\alpha$ $1/h$ $\sqrt{2}/2^5$ $344$ $39$ $37$ $89$ $42$ $892$ $52$ $57$ $\sqrt{2}/2^6$ $289$ $79$ $79$ $143$ $55$ $576$ $77$ $77$ $\sqrt{2}/2^7$ $283$ $77$ $78$ $211$ $69$ $473$ $94$ $94$ $\sqrt{2}/2^8$ $287$ $115$ $120$ $426$ $101$ $373$ $148$ $151$
 Split-split Heron-penalty Heron-split $\sigma_2$ $\delta$ $\sigma$ $1$ $\alpha$ $1/h$ $h/\alpha$ $h$ $1$ $\alpha$ $1/h$ $\sqrt{2}/2^5$ $344$ $39$ $37$ $89$ $42$ $892$ $52$ $57$ $\sqrt{2}/2^6$ $289$ $79$ $79$ $143$ $55$ $576$ $77$ $77$ $\sqrt{2}/2^7$ $283$ $77$ $78$ $211$ $69$ $473$ $94$ $94$ $\sqrt{2}/2^8$ $287$ $115$ $120$ $426$ $101$ $373$ $148$ $151$
Iteration numbers with (5), (6), (7) for Example 2 and $\sigma_1=h^{-3/2}$ for the split-split method and $\tau=1$ for the Heron-penalty and the Heron-split method
 Split-split Heron-penalty Heron-split $\sigma_2$ $\delta$ $\sigma$ $1$ $\alpha$ $1/h$ $h/\alpha$ $h$ $1$ $\alpha$ $1/h$ $\sqrt{2}/2^5$ $2745$ $10$ $124$ $74$ $24$ $4592$ $15$ $189$ $\sqrt{2}/2^6$ $2808$ $15$ $65$ $156$ $27$ $3586$ $21$ $72$ $\sqrt{2}/2^7$ $2847$ $23$ $35$ $298$ $35$ $3133$ $25$ $42$ $\sqrt{2}/2^8$ $-$ $29$ $30$ $572$ $43$ $-$ $33$ $37$
 Split-split Heron-penalty Heron-split $\sigma_2$ $\delta$ $\sigma$ $1$ $\alpha$ $1/h$ $h/\alpha$ $h$ $1$ $\alpha$ $1/h$ $\sqrt{2}/2^5$ $2745$ $10$ $124$ $74$ $24$ $4592$ $15$ $189$ $\sqrt{2}/2^6$ $2808$ $15$ $65$ $156$ $27$ $3586$ $21$ $72$ $\sqrt{2}/2^7$ $2847$ $23$ $35$ $298$ $35$ $3133$ $25$ $42$ $\sqrt{2}/2^8$ $-$ $29$ $30$ $572$ $43$ $-$ $33$ $37$
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