Collaborative Research: Numerical Methods for High-Dimensional Sticky Diffusions

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Numerical simulations can provide insight into many problems in science or engineering, for example by providing access to variables that are otherwise difficult or impossible to observe experimentally, or by allowing a user to optimize over variables more cheaply than though experiments. Yet, numerical simulations can be a challenge to implement, because computers cannot reproduce all scales from the quantum mechanical to the macroscopic scales of interest. A particularly challenging system to simulate are collections of interacting particles, which are studied in a wide variety of applications, from designing new materials such as impact-resistant or energy-efficient materials, to understanding how the interior of a cell works, to biomedical applications such as designing the lipid nanoparticles that carry the mRNA vaccines. This project will develop methods to simulate interacting particles which currently require the computer to take timesteps that are many times smaller than the timescales of interest. We will build upon a recent mathematical description of the effective interactions between such particles to allow a simulation to take significantly larger timesteps. This will allow for simulations over significantly longer times and of larger collections of particles, and hence will enable scientists to use computations to understand a richer collection of systems that arise in a variety of important applications. Students will be involved and trained in interdisciplinary applications. This project aims to develop numerical methods to simulate high-dimensional stochastic differential equations (SDEs) modeling systems of particles that can repeatedly form, break and re-form bonds due to stiff, short-ranged forces. Such particles are models for systems such as colloids, cross-linked polymers (gels), DNA nanotechnology, networks of actin filaments or other cytoskeletal components, chromatin in the cell, among many others. Because of the stiffness of the particle forces, current simulation methods require extremely small time steps and thus prohibitively long simulation times. The project will develop methods that allow significantly larger timesteps and thus can work for systems of hundreds to thousands of particles, and the approach is based on two key developments. The first is an analytic result which eliminates the stiff forces and replaces them with rigid bonds when particles are in contact, which can be achieved with the help of sticky boundary conditions. The resulting sticky diffusion allows particles to evolve stochastically subject to rigid distance constraints, but crucially, allows these constraints to change. The second is a discretization of SDEs in space and numerical PDE theory to discretize the infinitesimal generator of the sticky diffusion to be later used to simulate a Markov Jump Process. This approach allows one to handle sticky boundary conditions because one can choose discretization points directly on the boundary. The methods will be applied to study systems such as DNA-coated colloids and networks of actin filaments.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.