Gaussian graphical models (GGMs) are well-established tools for probabilistic exploration of dependence structures using precision (inverse covariance) matrices. We propose a Bayesian method for estimating the precision matrix in GGMs. The method leads to a sparse and adaptively shrunk estimator of the precision matrix, and thus conduct model selection and estimation simultaneously. We extend this method in a regression setup with the presence of covariates. We consider both the linear as well as the nonlinear regressions in this GGM framework. Furthermore, to relax the assumption of the Gaussian distribution, we develop a quantile based approach for sparse estimation of graphs. We demonstrate that the resulting graph estimator is robust to outliers and applicable under general distributional assumptions. We discuss a few applications of the proposed models.

Excursions of random fields is an increasingly important topic within data analysis in medicine, cosmology, materials science, etc. In this talk, we will discuss some detailed results concerning their Betti numbers. Specifically, we shall consider a piecewise constant Gaussian field whose covariance function is positive and satisfies some local, boundedness, and decay rate conditions. We will discuss a way to model its excursion set using a Cech complex. For each Betti number of this complex, we shall then prove various limit theorems in different regimes based on how fast the window size and excursion level grow to infinity. These include asymptotic mean estimates, a vanishing to non-vanishing phase transition with a precise estimate of the transition threshold, and a weak law in the non-vanishing regime. We shall further see a Poisson and a central limit theorem close to the transition threshold. The expected vertex degree asymptotically vanishes in the regimes we shall deal with. This places all our above results in the so-called `sparse' regime. Our proofs combine tools from both extreme value theory and combinatorial topology. This is joint work with Sunder Ram Krishnan.

We consider here elliptical systems as Stokes and Navier-Stokes problems in a bounded domain, eventually multiply connected, whose boundary consists of multi- connected components. We investigate the solvability in L^p theory, with 1 < p < \infty, under non standard boundary conditions. We consider also the case of Navier boundary conditions. The main ingredients for this solvability are given by the Inf-Sup conditions, some Sobolev’s inequalities for vector fields and the theory of vector potentials. These inequalities play a fundamental key and are obtained thanks to Calderon-Zygmund inequalities and integral representations. In the study of ellpitical problems, we consider both generalized solutions and strong solutions that very weak solutions. In a second part, we will consider the nonstationary case for the Stokes equations.