Algorithms and Numerical Analysis for Nested Approximations of Stochastic Particle Dynamics

不断增长的计算能力推动了模拟大型随机相互作用粒子系统的计算技术的快速发展。不断增长的计算能力帮助人们对从材料中的物理现象、化学反应、生物过程到图像处理等众多问题获得了前所未有的洞察力。然而，正如在模拟方法的初始开发中常见的那样，新计算技术的迅速出现远远超过了对算法的理论理解。从计算的角度来看，平衡数值精度和计算效率这两个相互竞争的目标仍然是大型多尺度系统仿真的中心问题之一。提出的研究概述了基于微观系统粗粒化的模拟算法的数值分析和实现的框架。在拟议的工作中，我们将粗粒化视为粗略可观测值的数值近似，这将不得不通过计算代价高昂的微观模拟来估计。拟议工作的目标是开发用于评估近似质量的数值工具，并使用误差指标来实施可靠的模拟算法。在随机模拟的背景下，理解近似及其局限性尤其重要，因为即使是收敛的模拟也不能提供对模拟现象(例如，相变、临界现象)的任何可靠的洞察。此外，如果适当地调整精度和效率之间的权衡，可以显著降低计算复杂性。在具有大量相互作用实体的系统的模拟中，我们通常最终估计依赖于系统随机分布的微观构型的某些可观测值的平均值。在提出的方法中，我们应用了分层的微观-宏观计算范例，并探索了它们在提高高维空间上抽样概率度量方法的效率、可靠性和算法复杂性方面的潜力。我们的目标是开发适合于通过对平衡分布进行采样，在实验相关的时间尺度上近似可能无法正确捕获的瞬时行为的工具。所提出的框架将得到底层多尺度系统的嵌套近似，并在不同尺度之间进行最佳相互作用。这种尺度分解策略将被应用于模拟由大量自由度组成的随机系统的并行算法的开发。这项拟议的工作是数值分析、随机过程和统计力学的交集。拟议的数学主题的潜在影响包括在物理、化学、材料科学和其他使用大型多尺度模拟的领域中的广泛应用。因此，目标之一是实现灵活的工具，这些工具可以在较短的学习曲线内针对特定问题和现有包进行定制。

The ever-increasing computational power has instigated rapid development of computational techniques for simulating large stochastic interacting particle systems. Growing computing capabilities have helped to obtain unprecedented insight into numerous problems ranging from physical phenomena in materials, chemical reactions, and biological processes to image processing. However, as is common in the initial development of simulation methodologies the rapid emergence of new computational techniques far outstrips theoretical understanding of the algorithms. From the computational point of view, balancing the competing aims of numerical accuracy and computational efficiency still remains one of the central problems in simulations of large multi-scale systems. The proposed research outlines a framework for the numerical analysis and implementation of simulation algorithms based on coarse-graining of the microscopic system. In the proposed work we view coarse-graining as numerical approximation of coarse observables which would have to be estimated from computationally expensive microscopic simulations. The goal of the proposed work is to develop numerical tools for assessing the quality of the approximation and to use error indicators in order to implement reliable simulation algorithms. Understanding approximations and their limitations is particularly important in the context of stochastic simulations, since even a convergent simulation may not provide any reliable insight into simulated phenomena (e.g., phase transitions, critical phenomena). Furthermore, the computational complexity can be substantially decreased if a trade-off between accuracy and efficiency is properly adjusted. In simulations of systems with a large number of interacting entities we often end up estimating average values of certain observables that depend on randomly distributed microscopic configurations of the system. In the proposed approach we apply hierarchical microscopic-macroscopic computational paradigms and explore their potential for improving efficiency, reliability, and algorithmic complexity of methods used for sampling probability measures on high-dimensional spaces. We aim at developing tools suitable also for approximating transient behavior which may not be properly captured, on experimentally relevant time-scales, by sampling the equilibrium distribution. The proposed framework will derive nested approximations of the underlying multi-scale system with optimal interactions between different scales. This scale decomposition strategy will be applied to the development of parallel algorithms for simulating stochastic systems composed of a large number of degrees of freedom. The proposed work is in the intersection of numerical analysis, stochastic processes, and statistical mechanics. The potential impact of the proposed mathematical topics encompasses a wide range of applications in physics, chemistry, materials science, and other fields where large multi-scale simulations are used. Therefore, one of the objectives is to implement flexible tools that can be tailored to specific problems and existing packages within a short learning curve.