Error Analysis of Modified Langevin Dynamics

Abstract

We consider Langevin dynamics associated with a modified kinetic energy vanishing for small momenta. This allows us to freeze slow particles, and hence avoid the re-computation of inter-particle forces, which leads to computational gains. On the other hand, the statistical error may increase since there are a priori more correlations in time. The aim of this work is first to prove the ergodicity of the modified Langevin dynamics (which fails to be hypoelliptic), and next to analyze how the asymptotic variance on ergodic averages depends on the parameters of the modified kinetic energy. Numerical results illustrate the approach, both for low-dimensional systems where we resort to a Galerkin approximation of the generator, and for more realistic systems using Monte Carlo simulations.

Convergence rates for nonequilibrium Langevin dynamics

“…where Λ− is defined in (24). We next follow a discussion similar to the one performed at the end of Section 5.1.…”

Spectral methods for Langevin dynamics and associated error estimates

“…Moreover, it may be preferable in practice to create bases adapted to the operators ∇q, ∇ * q , ∇p and ∇ * q in order to simplify the algebra involved in the computation of the elements of the rigidity matrix. For these two reasons basis functions are rarely of mean 0 with respect to µ in the literature, see for instance [20,28,1] for recent examples. We therefore need to extend the results of Section 3 to the non-conformal case VM ⊂ L 2 0 (µ).…”

Langevin Dynamics With General Kinetic Energies

“…respectively scale as ∆t 3 and ∆t 3/2 (see Lemmas 3.1 and 3.5). We consider three kinds of ARkinetic energies: the original function interpolation (33), and two interpolation functions (34) (a) Comparison of the AR-kinetic energy functions (33) and (34).…”