Project C8: Numerical approximation of high-dimensional Fokker-Planck equations

Stochastic processes driven by Brownian motion, which play a fundamental role in soft matter physics, can also be described by associated deterministic Fokker-Planck equations for probability distributions, where the dimensionality of the space on which this equation is posed increases linearly with respect to the number of particles. The aim of this project is to develop numerical solution methods for such high-dimensional problems that allow for the efficient extraction of quantities of interest, which typically take the form of certain integrals with respect to the computed distributions. In the high-dimensional case, beyond the basic numerical feasibility, a central issue is to ensure the accuracy of the computed solutions by suitable a posteriori error control.

The initial focus of the project, which started during the second funding period, was on the development of numerical methods. On the one hand, we considered adaptive low-rank methods using hierarchical tensor decompositions, where we could start from a detailed understanding of such methods for stationary high-dimensional diffusion problems. We constructed a novel space-time adaptive low-rank scheme that yields rigorous computable error bounds and is scalable to high dimensions. On the other hand, we consider approximations by anisotropic Gaussian functions, which have many features that make them particularly attractive for the problems under consideration. However, compared to low-rank methods, more fundamental mathematical questions remain open in the case of anisotropic Gaussians. Adapting some of the basic strategies that have proved successful for low-rank tensor approximations, we have arrived at a first numerical scheme of splitting type for Gaussian approximations that treats diffusion and advection steps separately.

In the third funding period, we intend to further develop these new methods and to apply them to models of practical interest from soft matter physics. Our focus will be in particular on methods suitable for the treatment of problems involving pair potentials, where Gaussian-type approximations offer significant advantages. In addition to the splitting approach, we will consider gradient-flow type methods for approximation by a fixed number of Gaussian terms as well as hybrid deterministic-stochastic methods that provide a natural connection to importance sampling strategies. We will apply these methods to two one-dimensional test cases for validation: single-file diffusion of colloidal particles in a channel and water molecules confined to a pore. Finally, we will explore the full potential of this numerical approach studying pathways and kinetics of a many-body system undergoing a phase transition, specifically a supersaturated Lennard-Jones fluid as it nucleates the liquid
phase. We will combine particle-based Brownian dynamics simulations with the methods developed here to construct approximations of the probability distribution and explore how these approximations can inform importance sampling schemes.