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I am solving elliptic PDE problem, for which, Euler scheme looks as following: $$ \nabla [\gamma ( |\nabla u|^2) \nabla u] = 0,$$ where $$\gamma(|\nabla u|^2) = (1 + |\nabla u|^2)^{-1/2}. $$

I am aiming to define minimization problem for problem above. Does anybody have any suggestion how it should look like? Or some recommended literature?

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For the particular equation you are solving (called the minimal surface equation), the functional you are trying to minimize is $$ J(u) = \int_\Omega \sqrt{1 + |\nabla u|^2} \; dx. $$

You can find a derivation of the equations, as well as a discussion of solution approaches in lectures 31.5 and following here: http://www.math.tamu.edu/~bangerth/videos.html . Of course there are many other sources for the same material as well. For example, nearly every book about the calculus of variations will have your example.

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  • $\begingroup$ is there any example, how to embedded framework into trust region framework ? $\endgroup$ Aug 29, 2015 at 12:54
  • $\begingroup$ No. But I also don't quite understand: you have an unconstrained optimization problem. Why not just use a line search? $\endgroup$ Aug 30, 2015 at 3:08
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Let's assume, that you have discretized problem and on every point you obtained evaluation $ g_{ij} $. Therefore $g(u) = 0$ is your system of nonlinear equations.

Now let $\gamma$ to be norm function defined as $\gamma(x) = || g(x)||^2$.

Then your nonlinear eq. problem is equivalent to the problem $$ \text{minimize} \ \ \gamma(x) .$$

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    $\begingroup$ This is not a useful approach because it yields degenerate optimization problems (that do not satisfy the second order sufficient conditions for optima) at places where $g(x)=0$ and $g'(x)=0$. You need to use a better approach. $\endgroup$ Aug 24, 2015 at 23:28

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