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Posts Tagged ‘gamma-convergence’

Relaxation of the Anisotropic Perimeter – Part 2

(Read this previous post to see the context and the statement of the problem)

While it is usually difficult to construct a recovery sequence for an arbitrary ${u \in BV(\Omega,\{0,1\})}$ (or all ${u \in L^1(\Omega)}$), the construction is often much simpler if the target function ${u}$ has some special structure.

It is useful to consider a subset ${\mathcal{D}\subset L^1(\Omega)}$ which is dense in ${BV(\Omega,\{0,1\})}$ such that for every ${u \in BV(\Omega,\{0,1\})}$ there exists ${(u_j) \subset \mathcal{D}}$ such that ${u_j \rightarrow u}$ in ${L^1(\Omega)}$ and ${P_\varphi(u)=\lim\limits_{j \rightarrow \infty} P_\varphi(u_j)}$.

A common choice for ${\mathcal{D}}$ is the class of characteristic functions of smooth sets, which is dense in ${BV(\Omega,\{0,1\})}$ in the sense that for every finite perimeter set ${E\subset \Omega}$ there exists a sequence ${E_n}$ of smooth sets such that ${|E \Delta E_n| \rightarrow 0}$ and ${|D\chi_{E_n}|(\Omega) \rightarrow |D\chi_E|(\Omega)}$ as ${n \rightarrow \infty}$. Note that this is different than the BV-norm convergence, but it is the notion of convergence very well suited for BV functions. In Approximation of Free-Discontinuity Problems by A. Braides a proof of the limsup estimate is given using the dense class of smooth functions.

When dealing with the anisotropic perimeter, a more natural way of working would be choosing sets ${E}$ whose boundary is polyhedral (piecewise affine). The main reason for doing that is the fact that polyhedrons have constant normals corresponding to one face. To be able to choose ${\mathcal{D}}$ as the class of characteristic functions of polyhedral subsets of ${\Omega}$ (with finite perimeter, of course), we must have a density result similar to the smooth case.

Relaxation of the Anisotropic Perimeter – Part 1

I have discussed in a previous post how Modica-Mortola theorem can provide a good framework for relaxing the perimeter functional in the single and multi-phase cases. The ideas can be extended further to a more generalized notion of perimeter, the anisotropic perimeter. (anisotropic = directionally dependent)

The main idea is that the anisotropic perimeter doesn’t count every part of the boundary in the same way; some directions are more favorized than others. The anisotropic perimeter associated to a norm ${\varphi}$ is defined by

$\displaystyle \text{Per}_\varphi(\Omega)=\int_{\partial \Omega} \varphi(\vec{n})d\mathcal{H}^{n-1}.$

There are variants of Modica-Mortola theorem for the anisotropic perimeter. Here is one of them:

Theorem – Relaxation of the Anisotropic Perimeter

Let ${\Omega}$ be a bounded open set with Lipschitz boundary. Let ${p>1}$, let ${W : \Bbb{R} \rightarrow [0,\infty)}$ be a continuous function such that ${W(z)=0}$ if and only if ${z \in \{0,1\}}$ and let ${\varphi : \Bbb{R}^n \rightarrow [0,\infty)}$ be a norm on ${\Bbb{R}^n}$. Let ${F_\varepsilon : L^1(\Omega) \rightarrow [0,\infty]}$ be defined by

$\displaystyle F_\varepsilon(u) = \begin{cases} \displaystyle \frac{1}{\varepsilon p'}\int_\Omega W(u)dx +\frac{1}{p}\varepsilon^p \int_\Omega \varphi^p(\nabla u)dx & \text{ if } u \in W^{1,p}(\Omega) \\ +\infty & \text{ otherwise} \end{cases}$

and let ${P_\varphi : L^1(\Omega) \rightarrow [0,\infty]}$ be defined by

$\displaystyle P_\varphi(u) =\begin{cases}\displaystyle c_p \int_{S(u)} \varphi(\nu_u)d\mathcal{H}^{n-1}& \text{ if }u \in SBV(\Omega) \text{ and } u \in \{0,1\} \text{ a.e.}\\ +\infty & \text{ otherwise} \end{cases}$

where ${c_p =\int_0^1 (W(s))^{1/p'}ds}$. Then ${\Gamma-\lim_{\varepsilon \rightarrow 0^+}F_\varepsilon(u)=P_\varphi(u)}$.

Numerical Approximation using Relaxed Formulation

April 23, 2013 1 comment

Sometimes it is easier to replace an optimization problem with a sequence of relaxed problems whose solutions approximate the solution to the initial problem.

This kind of procedure can be useful when we need to approximate numerically discontinuous functions (in particular the characteristic function). Modica Mortola theorem states that the functionals

$\displaystyle F_\varepsilon (u) = \begin{cases} \varepsilon \int_D |\nabla u|^2+\frac{1}{\varepsilon} \int_D W(u) & u \in H^1(D) \\ \infty & \text{ otherwise} \end{cases}$

${\Gamma}$-converges to the functional

$\displaystyle F(u) = \begin{cases} \text{Per}(\{u=1\}) & \text{ if }\{u=1\} \text{ has finite perimeter} \\ \infty & \text{ otherwise} \end{cases}.$

(Recall that ${W}$ is a real function which is positive except for ${0}$ and ${1}$ where it is zero.)

The Basic properties of Gamma Convergence

Let ${X}$ be a metric space, and for ${\varepsilon >0}$ let be given ${F_\varepsilon : X \rightarrow [0,\infty]}$. We say that ${F_\varepsilon}$ ${\Gamma}$-converges to ${F}$ on ${X}$ as ${\varepsilon \rightarrow 0}$, and we write ${\Gamma-\lim F_\varepsilon =F}$, if the following conditions hold:

(LI) For every ${u \in X}$ and every sequence ${(u_\varepsilon)}$ such that ${u_\varepsilon \rightarrow u}$ in ${X}$ we have

$\displaystyle \liminf_{\varepsilon \rightarrow 0}F_\varepsilon(u_\varepsilon)\geq F(u)$

(LS) For every ${u \in X}$ there exists a sequence ${(u_\varepsilon)}$ such that ${u_\varepsilon \rightarrow u}$ in ${X}$ and

$\displaystyle \limsup_{\varepsilon \rightarrow 0}F_\varepsilon(u_\varepsilon)\leq F(u).$

The ${\Gamma}$-convergence has the following properties:

1. The ${\Gamma}$-limit ${F}$ is always lower semicontinuous on ${X}$;

2. ${\Gamma}$-convergence is stable under continuous perturbations: if ${F_\varepsilon \stackrel{\Gamma}{\longrightarrow} F}$ and ${G}$ is continuous, then

$\displaystyle F_\varepsilon + G \stackrel{\Gamma}{\longrightarrow} F+G$

3. If ${F_\varepsilon \stackrel{\Gamma}{\longrightarrow} F}$ and ${v_\varepsilon}$ minimizes ${F_\varepsilon}$ over ${X}$, then every limit point of ${(v_\varepsilon)}$ minimizes ${F}$ over ${X}$.

I have seen these properties stated in many places, but the proofs are usually left to the reader. I will try and give the proofs below.

Master 7

(For the context and pervious posts look on the Shape Optimization page for the links)

As a result of the theorem proved in the previous post and of the fact that every ${\Gamma}$-limit is lower semicontinuous we can see that the functional

$\displaystyle F_0(u)=\sum_{i,j=1}^k d(\alpha_i,\alpha_j) \mathcal{H}^{N-1}(\partial^* S_i \cap \partial^* S_j \cap \Omega)$

is lower semicontinuous on the space ${X=\{ u \in L^1(\Omega;\Bbb{R}^N) : u=\sum_{i=1}^k \alpha_i \chi_{S_i},S_i \subset \Omega,\text{Per}_\Omega(S_i)<\infty,|\Omega\setminus (S_1 \cup ... \cup S_k)|=0, \text{ and } \sum_{i=1}^k |S_i|\alpha_i=m \}}$.

Definition 1 We say that a set of ${\sigma_{ij},\ 1\leq i,j \leq k}$, ${i \neq j}$ is an admissible configuration if there exist linearly independent vectors ${\alpha_1,..,\alpha_k \in \Bbb{R}_+^n}$ and a continuous function ${W : \Bbb{R}^n_+ \rightarrow [0,\infty)}$ with zeros precisely at points ${\alpha_i,\ i=1..k}$ such that we have ${d(\alpha_i,\alpha_j)=\sigma_{ij},\ i,j=1..n}$.

Using the above result we can see that if ${\sigma_{ij},\ i,j=1..k}$ is an admissible configuration then the functional ${\mathcal{F} :\mathcal{K} \rightarrow [0,\infty)}$ defined by

$\displaystyle \mathcal{F}(S_1,..,S_k)=\sum_{1 \leq i

is lower semicontinuous on

$\displaystyle \mathcal{K}=\{ (S_1,..,S_k) : S_i \subset \Omega, \text{Per}_\Omega(S_i)<\infty, |S_i|=c_i>0,\ c_1+...+c_k=|\Omega|\},$

where ${(c_i)}$ is the unique solution of the system

$\displaystyle \sum_{i=1}^k c_i\alpha_i=m,$

where ${m}$ is chosen such that all ${c_i}$ are positive.

Master 6

March 3, 2013 1 comment

(For the context see the Shape Optimization page where you can find links to the first 5 parts)

A particular consequence of the Modica-Mortola Theorem is that the functional

$\displaystyle \mathcal{F}(E_1,E_2)=\sigma \text{Per}_\Omega(\partial^* E_1 \cap \partial^* E_2)$

is lower semicontinuous with respect to the ${L^1(\Omega)}$ convergence for ${\sigma>0}$ on the set

$\displaystyle \mathcal{K}=\{ (E_1,E_2) : E_1\cup E_2=\Omega,\ E_1\cap E_2=\emptyset, |E_i|=c_i>0\}$

where the equalities are, as usual, up to a set of measure zero. It would be nice if a similar result would be true for multi-phase systems, where a functional of the form

$\displaystyle \mathcal{F}(E_1,E_2,...,E_k)=\sum_{1\leq i

is a ${\Gamma}$-limit and therefore semicontinuous, for ${E=(E_i) \in \mathcal{K}}$ where

$\displaystyle \mathcal{K}=\{ (E_1,...,E_k) : \bigcup_{i=1}^k E_i=\Omega,\ E_i\cap E_j=\emptyset, \text{ for }i\neq j, |E_i|=c_i>0\}.$

Let’s first remark that allowing the function ${W}$ in the Modica-Mortola theorem to have more than two zeros does not suffice. Indeed, if we allow ${W}$ to have zeros ${\alpha<\beta<\gamma}$, then the limiting phase will take only two values ${\alpha}$ and ${\beta}$ or ${\beta}$ and ${\gamma}$, depending on the constraint ${\int_\Omega u=c}$. This means that functionals of the form we presented above cannot be represented as a ${\Gamma}$-limit when the function ${W}$ is scalar, but with more than two zeros. This obstacle can be overcome by passing to the multidimensional case. This approach is presented by Sisto Baldo in [1] and we will present the ideas of this approach below.

Modica-Mortola Theorem

The notion of ${\Gamma}$-convergence was introduced by E. De Giorgi and T. Franzioni in the article Su un tipo di convergenza variazionale 1975.

Let ${X}$ be a metric space, and for ${\varepsilon >0}$ let be given ${F_\varepsilon : X \rightarrow [0,\infty]}$. We say that ${F_\varepsilon}$ ${\Gamma}$-converges to ${F}$ on ${X}$ as ${\varepsilon \rightarrow 0}$, and we write ${\Gamma-\lim F_\varepsilon =F}$, if the following conditions hold:

• (LI) For every ${u \in X}$ and every sequence ${(u_\varepsilon)}$ such that ${u_\varepsilon \rightarrow u}$ in ${X}$ we have

$\displaystyle \liminf_{\varepsilon \rightarrow 0}F_\varepsilon(u_\varepsilon)\geq F(u)$

• (LS) For every ${u \in X}$ there exists a sequence ${(u_\varepsilon)}$ such that ${u_\varepsilon \rightarrow u}$ in ${X}$ and

$\displaystyle \limsup_{\varepsilon \rightarrow 0}F_\varepsilon(u_\varepsilon)\leq F(u).$

Categories: shape optimization