Convergences of the squareroot approximation scheme to the Fokker–Planck operator

Abstract

We study the qualitative convergence behavior of a novel FV-discretization scheme of the Fokker–Planck equation, the squareroot approximation scheme (SQRA), that recently was proposed by Lie, Fackeldey and Weber [A square root approximation of transition rates for a markov state model, SIAM J. Matrix Anal. Appl. 34 (2013) 738–756] in the context of conformation dynamics. We show that SQRA has a natural gradient structure and that solutions to the SQRA equation converge to solutions of the Fokker–Planck equation using a discrete notion of G-convergence for the underlying discrete elliptic operator. The SQRA does not need to account for the volumes of cells and interfaces and is tailored for high-dimensional spaces. However, based on FV-discretizations of the Laplacian it can also be used in lower dimensions taking into account the volumes of the cells. As an example, in the special case of stationary Voronoi tessellations, we use stochastic two-scale convergence to prove that this setting satisfies the G-convergence property.

A variational finite volume scheme for Wasserstein gradient flows

“…Theorem 2.3 For all n ≥ 1, there exists a unique solution ρ n ∈ P T to (32). Moreover, energy is dissipated along the time steps.…”

“…To bypass this difficulty, we adopt a formalism based on dissipation potentials inspired from the one of generalized gradient flows introduced by Mielke in [48]. This framework was used for instance to study the convergence of the semi-discrete in space squareroot Finite Volume approximation of the Fokker-Planck equation, see [32].…”

“…Therefore, it admits a unique minimum on P T . The energy / energy dissipation estimate (33) is obtained by choosing ρ = ρ n−1 as a competitor in (32).…”

“…The scheme (32) is based on a "first discretize then optimize" approach. We have built a discrete counterpart of 1 2 W 2 2 and a discrete energy E T , then the discrete dynamics is chosen in an optimal way by (32). In opposition, the continuous equation ( 1) can be thought as the Euler-Lagrange optimality condition for the steepest descent of the energy.…”

Extending Transition Path Theory: Periodically Driven and Finite-Time Dynamics

“…As an alternative for estimating the transition matrix for an overdamped Langevin process (such as (40)), one can use the square root approximation to get a cheap estimate of the rate matrix(Lie et al 2013;Heida 2018) and thereof get the transition matrix.…”