Spectral methods for Langevin dynamics and associated error estimates
Provides theoretical foundations for spectral methods in Langevin dynamics, relevant for researchers in computational statistical mechanics.
The paper proves consistency of Galerkin methods for solving Poisson equations involving the Langevin generator, leveraging hypocoercivity to establish invertibility and error bounds. Explicit convergence rates are derived for a 1D example with tensor basis, supported by numerical simulations.
We prove the consistency of Galerkin methods to solve Poisson equations where the differential operator under consideration is the generator of the Langevin dynamics. We show in particular how the hypocoercive nature of this operator can be used at the discrete level to first prove the invertibility of the rigidity matrix, and next provide error bounds on the approximation of the solution of the Poisson equation. We present general convergence results in an abstract setting, as well as explicit convergence rates for a simple one-dimensional example discretized using a tensor basis. Our theoretical findings are illustrated by numerical simulations.