Collaborative Research: Numerical Methods for High-Dimensional Sticky Diffusions

数值模拟可以提供对科学或工程中的许多问题的洞察，例如，通过提供对本来难以或不可能通过实验观察的变量的访问，或者通过允许用户以比通过实验更低的成本来优化变量。然而，数值模拟的实现可能是一个挑战，因为计算机不能复制从量子力学到感兴趣的宏观尺度的所有尺度。模拟一个特别具有挑战性的系统是相互作用的粒子的集合，这些粒子在广泛的应用中进行研究，从设计抗冲击或节能材料等新材料，到了解细胞内部的工作原理，再到生物医学应用，如设计携带RNA疫苗的脂质纳米粒子。该项目将开发模拟相互作用的粒子的方法，目前需要计算机采取的时间步长比感兴趣的时间尺度小许多倍。我们将建立在最近对这些粒子之间有效相互作用的数学描述的基础上，以允许模拟花费更大的时间步长。这将允许在更长的时间和更大的粒子集合上进行模拟，从而使科学家能够使用计算来了解在各种重要应用中出现的更丰富的系统集合。学生将参与并接受跨学科应用的培训。该项目旨在开发数值方法来模拟高维随机微分方程(SDE)粒子建模系统，这些粒子可以由于刚性的短程力而重复形成、断裂和重新形成键。这些颗粒是胶体、交联聚合物(凝胶)、DNA纳米技术、肌动蛋白细丝网络或其他细胞骨架组件、细胞中染色质等系统的模型。由于质点力的刚性，目前的模拟方法需要极小的时间步长，因此模拟时间长得令人望而却步。该项目将开发允许更大的时间步长的方法，从而可以对数百到数千个粒子的系统起作用，该方法基于两个关键发展。第一种是一个解析结果，它消除了颗粒接触时的刚性作用力，代之以刚性键，这可以通过粘性边界条件来实现。由此产生的粘性扩散允许粒子在严格的距离约束下随机进化，但关键是，允许这些约束发生变化。第二种是离散空间中的SDE和数值PDE理论，将粘性扩散的无穷小生成元离散，然后用来模拟马尔可夫跳跃过程。这种方法使人们能够处理粘性边界条件，因为人们可以直接在边界上选择离散点。这些方法将被应用于研究DNA包被的胶体和肌动蛋白细丝网络等系统。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.