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 \}| 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.