Seminários

Morse homology of the area minus volume functional

Resumo: We use Morse homology to determine a lower bound for the number of prescribed mean curvature Alexandrov embedded hyperspheres in a $(d+1)$-dimensional torus. This is the content of arXiv:1601.03437.

Numerics of miscible displacement in porous media: heterogeneous permeability and intermediate concentration

Resumo: We study the motion of viscous, miscible liquids in porous media. Injection of a less viscous fluid to a more viscous one leads to the growth of an instability often refereed as viscous fingering. The main goal is finding sharp estimates for the size of the mixing zone containing the instabilities.We perform two series of numerical experiments.

The first one devoted to study size of mixing zone in heterogeneous environments. We observe the non-monotonic nature of the dependence of the front end of the mixing zone on the correlation length of the permeability of the reservoir. Interestigly, while for small values of correlation length the pattern of viscous fingers looks stochastic, for intermediate values of correlation lenght it is almost periodic.

In the second we showed an important role of intermediate concentrations (different from minimum and maximum values) on the speed of the front and back end of mixing zones. As a result we demonstrate a potential for improvement of existing estimates on the size of the mixing zone and suggest a theoretical approach (work in progress) for this.

Non-uniformly hyperbolic endomorphisms of the torus

Resumo: I will present a recent result obtained in collaboration with Pablo Carrasco and Radu Saghin about Lyapunov exponents of conservative non-invertible maps of the torus. Consider a non-invertible linear endomorphism of the two torus (e.g. an expanding map). Is it possible to deform it through a homotopy to obtain a map which preserves the Haar measure and has a negative Lyapunov exponent almost everywhere? If so, can this exponent be made as small as we want? We give a positive answer to both questions, provided that the topological degree is not too small.

Lorenz Knots and their Geometric Classification

Resumo: Lorenz knots are periodic orbits of the Lorenz system, which is a system of three ordinary differential equations in $\mathbb{R}^3$. This family of knots piqued the interest of mathematicians as it appears in this unexpected setting. Thurston proved that a non-trivial knot in $S^{3}$ is either a torus, a satellite, or a hyperbolic knot, which we call the geometric type of a knot. The geometric type of a knot is the information that helps to classify it. In this talk we will summarize what we have done and what we should do to complete the geometric classification of Lorenz links.

On projective stochastic-gradient type methods for solving large scale systems of nonlinear ill-posed equations: Applications to machine learning

Resumo: A distinctive feature of our method resides in the a posteriori choice of the stepsize, which promotes a relaxed orthogonal projection of the current iterate onto a conveniently chosen convex set. This characteristic distinguish our method from other SGD type methods in the literature (where the stepsize is typically chosen a priori) and accounts for the faster convergence observed in the numerical experiments conducted in this manuscript.

The convergence analysis discussed here includes: monotonicity and mean square convergence of the iteration error (exact data case), stability and semi-convergence (noisy data case). In the later case, our method is coupled with an a priori stopping rule.

Numerical experiments are presented for two large scale nonlinear inverse problems in machine learning (both with real data): (i) we address, using neural networks, the big data problem of CO-concentration prediction considered in the above cited article; (ii) we tackle the classification problem for the MNIST database (http://yann.lecun.com/exdb/mnist/). The obtained numerical results demonstrate the efficiency of the proposed method.