Complejidad en Diseño Óptimo
An introduction to mean curvature flow
G. Belletini
Definition of classical mean curvature flow.
First properties.
The reaction-diffusion approximation to classical mean curvature flow.
Weak solutions: the minimal barriers approach.
A short course on asymptotics in media containing inhomogeneities of low volume fraction
E. Bonnetier
We give a review of recent work on composite media containing inhomogeneities of
low volume fraction. In such a medium, we consider an elliptic PDE, the coefficients
of which are perturbed by the presence of the inhomogeneities. One can then derive
asymptotics of the solution to the PDE. The first order error term scales like the volume
of the inhomogeneity and writes in terms of a polarization tensor and of the gradient
of the Green's function of a reference ideal medium.
Such asymptotics can be obtained using PDE techniques, integral equations, for a
large range of differential operators: the conduction equation, elasticity, the Helmholtz
equation and the Maxwell system. We also present a general structure result, with
a flavor of compensated compactness, that was obtained by Y. Capdeboscq and M.
Vogelius.
We relate polarization tensors and low volume fraction homogenization, and we discuss
bounds on the polarization tensors for a mixture of two isotropic phases.
In the last part of the course, we show how the structure of these asymptotics can be
applied to design efficient algorithms for inverse impedance tomography, and to recent
experiments of "super-resolution" in inhomogeneous media.
Mathematical and computational methods for coarse-graining of stochastic systems
P. Plechác