For a image denoising problem (below):


<http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf>


the author has a functional E defined 


$E(u) = \int\int\_\Omega F \\ d\Omega$


which he wants to minimize. F is defined as 


$F = ||\nabla u ||^2 = u\_x^2 + u\_y^2$


Then, the E-L equations are derived:


$\frac{\partial E}{\partial u} = \frac{\partial F}{\partial u} - 
\frac{d}{dx} \frac{\partial F}{\partial u\_x} -
\frac{d}{dy} \frac{\partial F}{\partial u\_y} = 0$


Then it is mentioned that gradient descent method is used to minimize the functional E by using 


$\frac{\partial u}{\partial t} = u\_{xx} + u\_{yy}$ 


which is the heat equation. I understand both equations, and have solved the heat equation numerically before. I also worked with functionals. I do not understand however how the author jumps from the E-L equations to the gradient descent method. How is the time variable t included? Any detailed derivation, proof on this relation would be welcome. I found some papers on the Net, the one by Colding et al looked promising. 


References:


<http://arxiv.org/pdf/1102.1411> (Colding et al)


<http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.117.1675&rep=rep1&type=pdf> (ROF)


<http://dl.dropbox.com/u/1570604/tmp/functional-grad-descent.pdf>


<http://dl.dropbox.com/u/1570604/tmp/gelfand_var_time.ps> (Gelfand and Romin)



Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising
If the solution to 


$
u\_t=u\_{xx}+u\_{yy}
$


reaches an equilibrium solution, then $u\_{t}=0$ at that equilibrium, so $u\_{xx}+u\_{yy}=0$. The author has shown that a necessary condition for $u$ to be minimizer of $E(u)$ is $u\_{xx}+u\_{yy}=0$. 


This isn't steepest descent in the way that it is normally presented as an optimization algorithm for minimizing a function $f(x)$, but it is conceptually the same.


Of course you don't want to simply minimize $E(u)$ without respecting the original image- you want to somehow balance the minimization of $E(u)$ with keeping the original image. By starting the time dependent PDE with the original image and then stopping after a finite time, you can achieve this balance. 


