ALEX 2018 Workshop: Abstracts

On convergences of the squareroot approximation scheme to the Fokker–Planck operator

Luca Donati${}^{\mathrm{(}\mathrm{1}\mathrm{)}}$, Bettina Keller${}^{\mathrm{(}\mathrm{1}\mathrm{)}}$, Martin Heida${}^{\mathrm{(}\mathrm{1}\mathrm{)}}$, and Marcus Weber${}^{\mathrm{(}\mathrm{3}\mathrm{)}}$

(1) Freie Universität Berlin, Physical and Theoretical Chemistry

(2) Weierstrass Institute for Applied Analysis and Stochastics, Berlin

(3) Zuse Institute Berlin

We study the qualitative convergence behavior [3] of a novel FV-discretization scheme of the Fokker-Planck equation, the squareroot approximation scheme (SQRA), that recently was proposed by [4] in the context of conformation dynamics. We show that SQRA has a natural gradient structure related to the Wasserstein gradient flow structure of the Fokker-Planck equation and that solutions to the SQRA converge to solutions of the Fokker-Planck equation. This is done using a discrete notion of G-convergence for the underlying discrete elliptic operator. The gradient structure of the FV-scheme guaranties positivity of solutions and preserves asymptotic behavior of the Fokker–Planck equation for large times. Furthermore, the SQRA does not need to account for the volumes of cells and interfaces and is taylored 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. The long term goal of the method is to deal with high dimensional state spaces of large molecules such as in [1]. As a first test, we apply the method to Alanine Dipeptide[2].

Acknowledgments: The work was financed by DFG through SFB1114 ”Scaling Cascades in Complex Systems” project C05.