A structure-preserving numerical discretization of reversible diffusions

Abstract

Abstract. We propose a numerical discretization scheme for the infinitesimal generator of a diffusion process based on a finite volume approximation. The resulting discrete-space operator can be interpreted as a jump process on the mesh whose invariant distribution is precisely the cell approximation of the Boltzmann invariant measure and preserves the detailed balance property of the original stochastic process. Moreover this approximation is robust in the sense that these properties remain valid independently of the grid size.

Projected metastable Markov processes and their estimation with observable operator models

“…For convenience of simulation and analysis, here we utilize a reversibility preserving numerical discretization scheme proposed in Ref. 33 with spatial mesh size 0.2 × 0.2 to generate the simulation trajectories and calculate the eigenvalues and eigenfunctions of the system.…”

Overcoming the timescale barrier in molecular dynamics: Transfer operators, variational principles and machine learning

“…invariant measure, reversibility, stability, stochastic interpretation). One prominent example of a spatial discretization scheme meeting these requirements is the so-called square-root approximation (SQRA) (Donati, Heida, Keller and Weber 2018, Donati, Weber and Keller 2021, Latorre, Metzner, Hartmann and Schütte 2011. In order to see how it works, we consider diffusive molecular dynamics, i.e.…”

“…The master equation (4.2) inherits the main structural properties of the Fokker-Planck equation (4.1): is a rate matrix (non-negative off-diagonal entries and row-sums all equal to 0) with leading eigenvalues 0 and it satisfies the detailed balance condition, that is, ˆ = ˆ , where ˆ = Vol( ), which is thus also the invariant measure of the master equation (4.2). Donati et al (2021) and Latorre et al (2011) have given several alternative derivations of SQRA and its variants (leading to different ). SQRA is a method to calculate transition rates as a ratio of the invariant measure of neighbouring discretization sets times a flux.…”

On metastability and Markov state models for non-stationary molecular dynamics

“…The discretization can be made such that this matrix is a generator indeed. [12][13][14][15] The associated master equation, describing the propagation of probability distributions over the discrete state space, reads asμ…”

“…The fineness of the discretization is chosen such that further refinement essentially does not alter the results on the level of accuracy we consider. The discrete generators L(t) ∈ R N ×N are obtained by discretizating [13][14][15] the continuous Fokker-Planck equation (5). This yields the propagator P = P(0, 3) by solving…”