Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising For an image denoising problem, the author has a functional $E$ defined 
$$E(u) = \iint_\Omega F \;\mathrm d\Omega$$
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:
$$\frac{\partial E}{\partial u} = \frac{\partial F}{\partial u} - 
\frac{\mathrm d}{\mathrm dx} \frac{\partial F}{\partial u_x} -
\frac{\mathrm d}{\mathrm dy} \frac{\partial F}{\partial u_y} = 0$$
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
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)
 A: You should note that a solution, $f$, to your differential equation, $\mathcal{L}[f] = 0$, is the steady state solution to the second equation, as $\partial_t f = 0$. By turning this into a parabolic equation, only the error term will depend on $t$, and it will decay with time.  This can be seen by letting 
$$h(x,y,t) = f(x,y) + \triangle f(x,y,t),$$ 
where $f$ is as before.  Then
$$\mathcal{L}[h] = \mathcal{L}[\triangle f] = \partial_t \triangle f$$ 
In general, this method makes the equations amenable to minimization routines like steepest descent.
Edit: Since you mentioned that you wanted a book to reference, when I was taking numerical analysis, we used v. 3 of Numerical Mathematics and Computing by Cheney and Kincaid, and I found it very useful.  Although, at points it lacked depth, however it provided a good jumping off point.  They also have a more mathematically in depth book Numerical analysis: mathematics of scientific computing that may be useful to you, which I have not read.
A: This is essentially a matter of definitions.  The steepest descent gradient flow of a functional $F$ in an inner product space $S(M,N)$ (for example) is a family $u:M\times [0,T)\rightarrow N$ which satisfies
$$
\partial_t F = - \lVert u_t \rVert^2.
$$
For example, suppose $\Sigma$ is a surface immersed in $\mathbb{R}^3$ (for simplicity) via an immersion $f:M\rightarrow\mathbb{R}^3$ and consider the Willmore functional
$$
\mathcal{W}(f) = \frac{1}{2}\int_M H^2 d\mu,
$$
where $H$ is the mean curvature of $M$.  We wish to compute from this functional, the Willmore flow, which is the steepest descent gradient flow in $L^2(M,\mathbb{R}^3)$.  To do this, one computes the first variation of $\mathcal{W}$ along normal variations of $f$ (the Willmore functional is invariant under tangential diffeomorphisms, (among other things) which are essentially reparametrisations).
Now, any critical point of the functional will have zero first variation.  This is a simple fact from basic calculus.  The equation "first-variation = 0" is the Euler-Lagrange equation.  It is a necessary condition that all minimal points of the functional must satisfy, although it is not in general sufficient.
The Euler-Lagrange equation is
$$
\Delta H + H|A^o|^2 = 0,
$$
where $A^o$ is the tracefree second fundamental form.  A detailed explanation of how one derives this equation can be found in the back of Riemannian Geometry by Willmore.  Any immersion satisfying this equation is a critical point of the Willmore functional and is called a Willmore surface.
Finally, suppose we have a one-parameter family of immersions $f:M\times[0,T)\rightarrow\mathbb{R}^3$ satisfying
$$
\partial_t^\perp f = \Delta H + H|A^o|^2.
$$
Along this family of immersions we have
$$
\partial_t\mathcal{W} = -\int_M |\Delta H + H|A^o|^2|^2 d\mu,
$$
and thus it is by definition the steepest descent gradient flow of $\mathcal{W}$ in $L^2$.  Usually one doesn't bother writing all that (or any of it) and just goes directly from the Euler-Lagrange operator in some function space to the gradient flow, since it is quiet straightforward.
