Collaborative Research: Numerical Methods for High-Dimensional Sticky Diffusions

合作研究：高维粘性扩散的数值方法

• 批准号: 2111224
• 负责人: Nawaf Bou-Rabee
• 金额: $ 8.29万
• 依托单位: Rutgers University Camden
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2111224&HistoricalAwards=false
• 关键词: Collaborative; Research; Numerical; Methods; Dimensional

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.

数值模拟可以深入了解科学或工程中的许多问题，例如，通过提供对难以或不可能通过实验观察的变量的访问，或者允许用户比通过实验更便宜地优化变量。然而，数值模拟的实施可能是一个挑战，因为计算机无法再现从量子力学到感兴趣的宏观尺度的所有尺度。模拟的一个特别具有挑战性的系统是相互作用的粒子的集合，这些粒子在各种应用中进行研究，从设计新材料（例如抗冲击或节能材料）到了解细胞内部的工作原理，再到生物医学应用（例如设计携带 mRNA 疫苗的脂质纳米颗粒）。该项目将开发模拟相互作用粒子的方法，目前要求计算机采取比感兴趣的时间尺度小很多倍的时间步长。我们将基于这些粒子之间有效相互作用的最新数学描述，以允许模拟采用更大的时间步长。这将允许在更长的时间内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。学生将参与跨学科应用并接受培训。该项目旨在开发数值方法来模拟高维随机微分方程 (SDE) 建模系统，这些颗粒可以由于刚性、短程力而重复形成、破坏和重新形成键。这些颗粒是胶体、交联聚合物（凝胶）、DNA 纳米技术、肌动蛋白丝或其他细胞骨架成分的网络、细胞中的染色质等系统的模型。由于粒子力的刚度，当前的模拟方法需要极小的时间步长，因此模拟时间过长。该项目将开发允许更大时间步长的方法，因此可以适用于数百到数千个粒子的系统，并且该方法基于两项关键进展。第一个是当颗粒接触时消除刚性力并用刚性键代替的分析结果，这可以借助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演化，但最重要的是，允许这些约束发生变化。第二个是 SDE 在空间和数值偏微分方程理论中的离散化，以离散粘性扩散的无穷小生成器，以便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化点。这些方法将应用于研究系统，例如 DNA 涂层胶体和肌动蛋白丝网络。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。