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# Euler-LagranceEuler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

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For a image denoising problem (below):

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)