We propose a method to quantify the complexity of conditional probability measures by a Hilbert space seminorm of the logarithm of its density. The concept of Reproducing Kernel Hilbert Spaces (RKHS) is a flexible tool to define such a seminorm by choosing an appropriate kernel. We present several examples with artificial datasets where our kernel-based complexity measure is consistent with our...

We describe a time-dependent functional involving the relative entropy and $\dot{H}^1$ seminorm, which decreases along solutions to spatially homogeneous Landau equation with Coulomb potential. The study of this monotone sheds light on competition between dissipation nonlinearity for equation. It enables us obtain new results concerning regularity/blowup issues

In this paper we present one approach to build optimal meshes for P1 interpolation. Considering classical geometric error estimates based on the Hessian matrix of a solution, we show it is possible to generate optimal meshes in H 1 semi-norm via a simple minimization procedure.

The normal cycle TK associated with a convex body K ⊂ Rn is a current which in principle contains complete information about K. It is known that if a sequence of convex bodies Ki, i ∈ N, converges to a convex body K in the Hausdorff metric, then the associated normal cycles TKi converge to TK in the dual flat seminorm. We give a quantitative improvement of this convergence result by providing a...

In the original Virtual Element space with degree of accuracy k, projector operators in theH-seminorm onto polynomials of degree ≤ k can be easily computed. On the other hand, projections in the L norm are available only on polynomials of degree ≤ k − 2 (directly from the degrees of freedom). Here we present a variant of VEM that allows the exact computations of the L projections on all polynom...