CDS&E: Matrix-Free Algorithms for Large-Scale Hydrodynamic Brownian Simulations

布朗动力学(BD)是一种用于大分子模拟的计算方法，在生物和化学工程等多个领域有着广泛的应用。目前，许多BD模拟忽略了流体中颗粒之间的远程流体动力相互作用(HI)的影响。这种选择是为了减少这些模拟的计算成本，尽管现在人们意识到HI是我们在现实中观察到的大分子动态行为的主要原因。该项目将开发无矩阵算法和软件，以便能够在大规模BD模拟中对水动力相互作用进行建模。此外，尽管BD已经成为一种主要技术，但它不能准确地模拟高体积分数系统中的HI，例如生物细胞内拥挤的环境。一种解决方案是使用斯托克斯动力学(SD)。SD的一个缺点是它的计算成本很高，但对于大型系统，使用无矩阵方法也可以降低这一成本。将开发新的无基质的SD方法，这些方法比目前可用的方法更有效，并且也适用于BD。HI的高计算成本是由于构造和操作了一个大的流体动力迁移率或扩散矩阵。这个项目的方法是用一个基于FFT的粒子网格Ewald求和算子来代替这个迁移率矩阵，从而与常用的方法相比逐渐降低了计算复杂性和存储要求。由于迁移率矩阵不再可用，这种方法需要使用无矩阵算法来计算布朗位移。这些算法将在用于多核节点和GPU的高性能软件库中实现。该库可以被其他研究人员用来为他们现有的BD代码添加快速HI功能和SD功能。还将开发处理较长和较大轨迹输出文件的数据分析工具。这些软件工具将在大规模生物模拟中进行演示，以研究大肠杆菌类核细胞扩散背后的机制。BD模拟是生物学中用于研究细胞内运输和几乎所有细胞过程背后的扩散机制的主要工具。这项工作将为其他研究人员在大范围内进行更真实的BD模拟铺平道路。此外，这里开发的算法还可以广泛应用于机械、化学工程和材料科学中的其他粒子模拟方法，在这些方法中，布朗粒子通过长程力相互作用。这项工作的结果将向应用科学界和计算科学界传播。实施新算法的软件将在开放源码许可下以库的形式发布，计算机科学和数学专业的学生可以在网上免费获得关于粒子模拟的教育材料。

Brownian dynamics (BD) is a computational method for macromolecular simulation with a myriad of applications in multiple areas including biology and chemical engineering. Currently, many BD simulations ignore the effect of the long-range hydrodynamic interactions (HI) between particles in a fluid. This choice is made to reduce the computational cost of these simulations, although it is now appreciated that HI is responsible for much of the dynamic behavior of macromolecules that we observe in reality. This project will develop matrix-free algorithms and software to enable the modeling of hydrodynamic interactions in large-scale BD simulations. Further, although BD has been a staple technique, it is not able to accurately model HI in high volume fraction systems, such as the crowded environment inside biological cells. A solution is to use Stokesian dynamics (SD). A drawback of SD is its high computational cost, but this cost can also be reduced for large systems using a matrix-free approach. New matrix-free free approaches for SD will be developed that are more efficient than those currently available, and that are also applicable to BD. The high computational cost of HI is due to the construction and manipulation of a large hydrodynamic mobility or diffusion matrix. This project's approach is to replace this mobility matrix by an operator applied with a FFT-based particle-mesh Ewald summation, thereby asymptotically reducing the computational complexity and storage requirements compared to commonly used methods. Since the mobility matrix is no longer available, this approach requires the use of matrix-free algorithms for computing Brownian displacements. The algorithms will be implemented in a high-performance software library for multicore nodes and GPUs. The library can be used by other researchers to add fast HI capabilities and also SD functionality to their existing BD codes. Data analysis tools to handle long and large trajectory output files will also be developed. The software tools will be demonstrated on large-scale biological simulations to study the mechanisms behind diffusion in the E. coli nucleoid. BD simulation is the main tool used in biology for studying intra-cellular transport and the diffusive mechanisms that lie behind almost all cellular processes. This work will pave the way for other researchers to perform much more realistic BD simulations at large scales. Furthermore, the algorithms developed here may be applied to a wide range of other particle simulation methods in mechanical and chemical engineering and materials science where Brownian particles interact through long-range forces. The results of this work will be disseminated to both the application and computational science communities. Software implementing the new algorithms will be released in library form under an open-source license, and educational materials on particle simulations, accessible to computer science and mathematics students, will be made freely available on the Web.