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)