Variational principle for probabilistic learning Yet another justification More simplification of updates for mean-field family Examples Dirichlet Process Mixture On minimization of divergence measures Energy minimization justifications Variational learning with exponential family Mean parametrization and marginal polytopes Convex dualities The log-partition function and conjugate duality Belie...

In this paper we study a non-homogeneous eigenvalue problem involving variable growth conditions and a sign-changing potential. We prove that any λ > 0 sufficiently small is an eigenvalue of the nonhomogeneous eigenvalue problem { −div(a(|∇u|)∇u) = λV(x)|u|q(x)−2u, in Ω, u = 0, on ∂Ω. The proofs of the main results are based on Ekeland’s variational principle.

We construct finite element subspaces of the space of symmetric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger–Reissner mixed formulation of the elasticty equations, when standard discontinuous finite element spaces are used to approximate the displacement field. These finite element sp...

Abstract We study how three-dimensional linearized elasticity for thin plates can be approximated by a two-dimensional projection. The classical approach using formal asymptotic expansions in powers of the thickness in the Hellinger-Reissner formulation, only provides error estimates in the H1 norm for the displacements, assuming at least L2 regularity for the applied forces, plus additional re...