We prove sharp rate of convergence to stationarity for a natural random walk on a compact Riemannian manifold (M, g). The proof includes a detailed study of the spectral theory of the associated operator.
A. Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates… (More)
This paper gives geometric tools: comparison, Nash and Sobolev inequalities for pieces of the relevent Markov operators, that give useful bounds on rates of convergence for the Metropolis algorithm. As an example, we treat the random placement of N hard discs in the unit square, the original application of the Metropolis algorithm.
We consider the problem of the numerical approximation of the linear controllability of waves. All our experiments are done in a bounded domain Ω of the plane, with Dirichlet boundary conditions and internal control. We use a Galerkin approximation of the optimal control operator of the continuous model, based on the spectral theory of the Laplace operator… (More)
We prove sharp rates of convergence to stationarity for a simple case of the Metropolis algorithm: the placement of a single disc of radius h randomly into the interval [−1 − h, 1 + h], with h > 0 small. We find good approximations for the top eigenvalues and eigenvectors. The analysis gives rigorous proof for the careful numerical work in ([DN04]). The… (More)
1 Introduction Consider in the plane the flow of two ideal incompressible fluids of constant densities ρ ± > 0 in the gravity field, with ρ + = ρ −. Velocity field satisfies the Euler equation ∂ρu ∂t + u · ∇ρu = −∇p + ρg (1) div u = 0 (2) ρ t + div (ρu) = 0 (3) with initial data u(x, 0) = u 0 (x). (4) We suppose, that u 0 (x) satisfies the continuity… (More)
A new Carleman inequality for parabolic systems with a single observation and applications Une nouvelle inégalité de Carleman pour des systèmes paraboliques avec une seule observation et applications a r t i c l e i n f o a b s t r a c t In this Note, we present Carleman estimates for linear reaction–diffusion–convection systems of two equations and linear… (More)