Collaborative Research: Numerical methods for high-dimensional sticky diffusions

数值模拟可以提供对科学或工程中许多问题的洞察，例如通过提供对以其他方式难以或不可能通过实验观察的变量的访问，或者通过允许用户比通过实验更便宜地优化变量。然而，数值模拟的实现可能是一个挑战，因为计算机无法再现从量子力学到宏观尺度的所有尺度。一个特别具有挑战性的模拟系统是相互作用粒子的集合，这些粒子在各种应用中被研究，从设计新材料（如抗冲击或节能材料）到了解细胞内部如何工作，再到生物医学应用（如设计携带mRNA疫苗的脂质纳米颗粒）。该项目将开发方法来模拟相互作用的粒子，目前需要计算机采取的时间步长比感兴趣的时间尺度小很多倍。我们将建立在最近的数学描述的有效相互作用，这样的粒子，使模拟需要显着更大的时间步长。这将允许模拟更长的时间和更大的粒子集合，从而使科学家能够使用计算来理解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（SDEs）建模系统的颗粒，可以重复形成，打破和重新形成键，由于刚性，短程力。这些颗粒是胶体、交联聚合物（凝胶）、DNA纳米技术、肌动蛋白丝或其他细胞骨架组分的网络、细胞中的染色质等系统的模型。由于粒子力的刚性，当前的模拟方法需要极小的时间步长，因此需要过长的模拟时间。该项目将开发允许更大时间步长的方法，从而可以适用于数百到数千个粒子的系统，该方法基于两个关键的发展。第一个是一个分析结果，消除了刚性力，并取代他们与刚性债券时，颗粒接触，这可以实现与粘性边界条件的帮助。由此产生的粘性扩散允许粒子随机地受到刚性距离约束，但至关重要的是，允许这些约束发生变化。第二个是空间离散化的SDES和数值PDE理论离散化的无穷小生成的粘性扩散，稍后用于模拟马尔可夫跳跃过程。这种方法允许处理粘性边界条件，因为可以直接在边界上选择离散点。该方法将被应用于研究系统，如DNA涂层胶体和网络的肌动蛋白filaments.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的知识价值和更广泛的影响审查标准。

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.