The sound of the bees

in collaboration with B. Bourdin and D. Bucur

Here are some results obtained in collaboration with Dorin Bucur and Blaise Bourdin. More details are given in our paper Optimal partitions for eigenvalues. This work deals with the optimal partition problem for Dirichlet eigenvalues. Precisely, given a bounded open set $ D \subset {\mathbb R}^2 $ , we are looking for a family of subsets $\{\Omega_i\}_{i=1}^n $ such that

\[ \Omega_1 \cup \dots \cup \Omega_n \subseteq D, \,\Omega_i \cap \Omega_j = \emptyset \text{ for} \, i \neq j \]

and which minimizes

\[ \mathcal{J}_n(\Omega_1,\dots,\Omega_n) = \sum_{i=1}^n \lambda_k(\Omega_i) \]

among all possible such partitions. Above, $\lambda_k(\Omega) $ denotes the k-th eigenvalue of the Dirichlet-laplacian on $ \Omega $ (the eigenvalues are counted with multiplicity).

Existence of optimal partitions for this problem in the class of quasi-open sets was proved by Buttazzo and Bucur. For $k=1$ regularity and qualitative studies of the optimal partitions were obtained by Conti, Terracini, and Verzini and Caffarelli, and Lin. Caffarelli and Lin obtained regularity results for the optimal partition and estimates for the asymptotic behavior of the previous sum when $n\rightarrow +\infty $. In particular, they conjectured that for the optimal partition $ \{\Omega_i^*\}_{i=1}^n $

\[ \sum_{i=1}^n \lambda_1(\Omega_i^*) \simeq n^2 \lambda_1(H), \]

where $ H $ is the regular hexagon of area 1 in $ {\mathbb R}^2 $. Roughly speaking this estimate says that, far from $ \partial D $, a tiling by regular hexagons of area $ \frac{|D |}{n} $ is asymptotically close to the optimal partition.

In this work we propose a rigorous proof of the equivalence between that optimization problem and a relaxed formulation providing a complete justification of our numerical approach. Based on this method, we performed numerical simulations for $ k=1,2,3 $ and $ n\in{8,16,24,384,512} $.

As expected, and up to boundary effects, in our numerical experiments, we obtain partitions that are very close to a tiling by regular hexagons in the case $k=1$. Provided that the conjecture is true, it can be easily proved that the asymptotic optimal partition for $ k=2 $ is made of unions of pairs of regular hexagons (of measure $ \frac{|D |}{2n} $). Again our last numerical computations below illustrate this fact.

Optimal partitions with an important relaxation factor for 12 and 24 cells

Optimal partitions for 12, 16 and 24 cells

We were able to run a series of large computations on parallel supercomputers at the Texas Advanced Computing Center. The domain is again the unit square. Periodicity boundary conditions are not used, as the number of cell ( $ m=512 $) is large enough that we expect that the effect of boundary conditions vanishes in the center of the domain. The computations were run on four layers of recursively refined grid of respective dimension $(64\times64)$, $(127\times 127)$, and $(505\times 505)$. We used a simple projection operator, and the final objective functions. We observe that the solution corresponds to local patches of tiling by regular hexagons, as we would expect from a ``good'' local minimizer.

Move your mouse on the picture to zoom in

A recursive optimization algorithm to avoid local minima for 512 cells (that is 130000000 degrees of freedom...)

Our algorithm can easily be adapted to objective function involving higher order eigenvalues of linear combination of eigenvalues of different order. A classical numerical issue in this case comes from the potential non-differentiability of multiple eigenvalues with respect to changes of the density functions.

The figures below represent the optimal partitions obtained with $m=8$ for $k = 1$, $k = 2$ and $k=3$, respectively, using periodic boundary conditions. As explained previously, the optimal partition for $k=2$ is expected to be obtain by a partition made of pairs of regular hexagons. Again, modulo the flattening necessary to achieve periodicity on a unit cell, this is the configuration that we observe. For $k=3$, we obtain a periodic tiling by non-regular hexagons, which can be proven to be a sub-optimal solution, as a tiling by regular hexagons would lead to a lower energy. This lack of global optimality is most certainly due to the fact that our objective function admits a great deal of local minima, which are difficult to avoid in optimization problems of this size. Additionally difficulty when $k \geq 2$ is that the k-th eigenvalue of an optimal cell is expected to have multiplicity greater than 1 hence is not differentiable.

Move your mouse on a picture to see the evolution (slow connection be patient)

Evolution of the optimization process when optimizing the three third eigenvalues on a periodic square