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I am reading this pdf which is about an image denoise model. Essentially, we want to find a function $u$ such that $u$ minimize the following functional: $$F(u) = \lambda \int_\Omega |f-u|^2 \,dx\,dy + \int_\Omega \sqrt{\epsilon^2 + |\nabla u|^2} \,dx\,dy$$.

From there, we can get two Euler-Lagrange equations, which are formula (1) and (2) in the paper. $$u = f+\frac{1}{2\lambda} \nabla\cdot\left(\frac{|\nabla u|}{\sqrt{\epsilon^2 + |\nabla u|^2}}\right)$$ $$\frac{\partial u}{\partial \text{n}} = 0 \text{ on the boundary of } \Omega$$

However, I am having trouble understanding how do we go from the Euler-Lagrange equation to the following time-dependent problem: $$\frac{\partial u}{\partial t} = f-u+\frac{1}{2\lambda}\nabla\cdot\left(\frac{|\nabla u|}{\sqrt{\epsilon^2 + |\nabla u|^2}}\right) $$

The idea is that the functional $F$ should get smaller and smaller as $t$ increases, so I am thinking of something like $\frac{dF}{du} < 0$ but I am not sure.

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In general, when examining the following type models. Suppose that we can define the Euler-Lagrange Equation in a Partial Differential Operator Form. We define the operator $\Phi$ such that: $$\Phi u=0.$$ As such, from your obtained model, we have: $$f-u+\frac{1}{2\lambda} \nabla\cdot\left(\frac{|\nabla u|}{\sqrt{\epsilon^2 + |\nabla u|^2}}\right)=0$$ For such equations, natural time flows are defined as equations of the form: $$\frac{\partial}{\partial t}u=\Phi u$$ Such that steady state, time-independent solutions, are extrema of $\Phi$ (Since steady-state simplifications would result in the equation $\Phi u=0$). Thus applying the following back to your equation, we have: $$\frac{\partial}{\partial t}u=f-u+\frac{1}{2\lambda} \nabla\cdot\left(\frac{|\nabla u|}{\sqrt{\epsilon^2 + |\nabla u|^2}}\right)$$ There is more information on the Wikipedia page entitled "Geometric Flow"

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