Spectral methods for Langevin dynamics and associated error estimates

Abstract

We prove the consistency of Galerkin methods to solve Poisson equations where the differential operator under consideration is hypocoercive. We show in particular how the hypocoercive nature of the generator associated with Langevin dynamics 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 example discretized using a tensor basis. Our theoretical findings are illustrated by numerical simulations.1 Our results can be extended to the case of any symmetric positive definite mass matrix M but we focus on the case when M is proportional to the identity matrix for simplicity. AL ham Πp = [1 + (L ham Πp) * (L ham Πp)] −1 (L ham Πp) * (L ham Πp).Since (L ham Πp) * (L ham Πp) is self-adjoint and the function x → x/(1 + x) = 1 − 1/(1 + x) is increasing, the inequality (67) follows by spectral calculus.

Langevin Dynamics With General Kinetic Energies

“…Let us now turn to the proof of Proposition 2.4, written for real-valued functions. Its proof is very similar to the proof of [38,Proposition 1], with a few modifications except for the estimate given in Lemma A.3 below which requires a more involved treatment. First, note that…”

Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

“…Using the framework of [15], we conjecture that the assumption on the boundedness of the Hessian of V can be substantially weakened and more quantitative decay estimates (in particular with respect to μ and ν) can be obtained. This approach has recently been successfully applied to equilibrium and nonequilibirum Langevin dynamics, see [27,53]. We leave this work track for future study.…”

“…Due to the respresentation (53) and Theorem 4, the inverses of L and L − leave the Hermite spaces H m invariant. We will prove the claim from Proposition 2 under the assumption that P f = f which includes the case f = f (q).…”

Longtime convergence of the temperature-accelerated molecular dynamics method

“…The second operator on the right-hand side is bounded since ∇ q (∂ z U ) is uniformly bounded and Π p ∇ p is a bounded operator on L 2 (ν ref ), as can be seen by writing the action of this operator in a Hermite basis of the momenta variables (as made precise in [25] for instance). To conclude the proof, it remains to show that the first operator on the right-hand side of the last equality is bounded.…”

“…For the inertial dynamics, the first statement is proved as in [10,11], relying also on the above uniform in z Poincaré inequality. A careful inspection of the proof for the inertial dynamics shows that ρ is of order η (see for instance [25]).…”