Home > shape optimization > Shape Optimization course – Day 5

Shape Optimization course – Day 5

Speaker: Edouard Oudet

Consider the problem

\displaystyle \min_{\Omega \in \mathcal{C}} F(\Omega)

where {F} is a given functional and {\mathcal{C}} is a class of admissible sets.

One obvious way to represent a set is by using its characteristic function. In general, from a numerical point of view, it is harder to work with characteristic function than to work with smooth functions. For this we could think to associate to the initial problem a series of problems

\displaystyle \min_{u \in \mathcal{F}_\varepsilon} F_\varepsilon(u)

where {u} denotes here a smooth function which approximates the characteristic function in the sense that |\{ \chi_\Omega \neq u \}|<C\varepsilon and therefore for \varepsilon small we have

\displaystyle |\Omega|= \int \chi_\Omega \simeq \int u.

If we try, for example, to approximate {F(\Omega)=\text{Per}(\Omega)} we may think of a formulation of the type

\displaystyle |\partial \Omega| = \int_{\partial \Omega} d\sigma \simeq \varepsilon \int |\nabla u|^2 +\frac{1}{\varepsilon} \int W(u),

for {\varepsilon >0}, {u} a smooth approximation of {\chi_\Omega} and {W} a continuous, positive function with only two zeros, at {0} and {1}. Why could this be a valid choice? The integral of the gradient forces the minimizer to have a relatively small variation, and thus try and minimize the interface region, which in fact, gives the perimeter. The second term, the integral of {W(u)}, forces (when {\varepsilon} is small) the minimizer {u} to be close to a characteristic function. Therefore we would expect that as {\varepsilon \rightarrow 0} the minimizers of {F_\varepsilon} should converge to minimizers of the perimeter.

This fact is made rigorous by using the {\Gamma}-convergence. See more details in this post, where you can find a rigorous proof of the above approximation of the perimeter, called the Modica-Mortola theorem.

For the last two points of the course, I will point you to the webpage of Edouard Oudet, where more accurate details and pictures of the numerical results can be found.

Irrigation Problems.

Regularization of Images.

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