Résumé :
We present some regularity estimates for solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to the zero set of a modulating coefficient $a=a(\cdot,\cdot)$. The model case is driven by the following nonlocal double phase operator,
$$
\int \!\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\,{\rm d}y
+ \int \!a(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+tq}}\,{\rm d}y,
$$
where $q\geq p$ and $a(\cdot,\cdot)\geqq 0$.
Our results do also apply for inhomogeneous equations, for very general classes of measurable kernels.
By simply assuming the boundedness of the modulating coefficient, we are able to prove that the solutions are Hölder continuous, whereas similar sharp results for the classical local case
do require $a$ to be Hölder continuous.
To our knowledge, this is the first result for nonlocal double phase problems.
Lieu : Salle MIP
Résumé : Many application domains such as ecology or genomics have to deal with multivariate count data. A typical example is the joint observation of the respective abundances of a set of species in a series of sites, aiming to understand the co-variations between these species. The Gaussian setting provides a canonical way to model such dependencies, but does not apply in general. We adopt here the Poisson lognormal (PLN) model, which is attractive since it allows one to describe multivariate count data with a Poisson distribution as the emission law, while all the dependencies is kept in an hidden friendly multivariate Gaussian layer. While usual maximum likelihood based inference raises some issues in PLN, we show how to circumvent this issue by means of a variational algorithm for which gradient descent easily applies. We then derive several variants of our algorithm to apply PLN to PCA, LDA and sparse covariance inference on multivariate count data. We illustrate our method on microbial ecology datasets, and show the importance of accounting for covariate effects to better understand interactions between species.
Lieu : Salle 106 (bâtiment 1R1)