Numerical minimization of dirichlet laplacian eigenvalues of four-dimensional geometries

We develop the first numerical study in four dimensions of optimal eigenmodes associated with the Dirichlet Laplacian. We describe an extension of the method of fundamental solutions adapted to the four-dimensional context. Based on our numerical simulation and a postprocessing adapted to the identification of relevant symmetries, we provide and discuss the numerical description of the eighth first optimal domains.The work of the first author was partially supported by FCT, Portugal, through the program “Investigador FCT” with reference IF/00177/2013 and the scientific project PTDC/MATCAL/4334/2014. The work of the second author was supported by the ANR, through the projects COMEDIC, PGMO, and OPTIFORMinfo:eu-repo/semantics/publishedVersio

Bounds and extremal domains for Robin eigenvalues with negative boundary parameter

We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking coordinate method, and a proof that in the fixed perimeter case the disk maximises the first eigenvalue for all values of the parameter. This is in contrast with what happens in the constant area problem, where the disk is the maximiser only for small values of the boundary parameter. We also present sharp upper and lower bounds for the first eigenvalue of the ball and spherical shells. These results are complemented by the numerical optimisation of the first four and two eigenvalues in 2 and 3 dimensions, respectively, and an evaluation of the quality of the upper bounds obtained. We also study the bifurcations from the ball as the boundary parameter becomes large (negative).Comment: 26 pages, 20 figure

Nodal and spectral minimal partitions -- The state of the art in 2015 --

In this article, we propose a state of the art concerning the nodal and spectral minimal partitions. First we focus on the nodal partitions and give some examples of Courant sharp cases. Then we are interested in minimal spectral partitions. Using the link with the Courant sharp situation, we can determine the minimal k-partitions for some particular domains. We also recall some results about the topology of regular partitions and Aharonov-Bohm approach. The last section deals with the asymptotic behavior of minimal k-partition

Multiclass Data Segmentation using Diffuse Interface Methods on Graphs

We present two graph-based algorithms for multiclass segmentation of high-dimensional data. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the functional's double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm is a uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, grayscale and color images, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current state-of-the-art multiclass segmentation algorithms.Comment: 14 page