Algorithms and Numerical Analysis for Nested Approximations of Stochastic Particle Dynamics

随着计算能力的不断提高，大型随机相互作用粒子系统的计算技术得到了快速发展。 不断增长的计算能力有助于获得前所未有的洞察力，从材料中的物理现象，化学反应和生物过程到图像处理等众多问题。 然而，正如在模拟方法的最初发展中所常见的那样，新的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的竞争目标仍然是大型多尺度系统模拟的中心问题之一。 所提出的研究概述了一个框架的数值分析和实现的模拟算法的基础上粗粒化的微观系统。 在拟议的工作中，我们认为粗粒化的粗观测值，这将不得不从计算昂贵的微观模拟估计的数值近似。 建议的工作的目标是开发数值工具，用于评估近似的质量，并使用误差指标，以实现可靠的仿真算法。 理解近似及其局限性在随机模拟的背景下特别重要，因为即使是收敛模拟也可能无法提供对模拟现象的任何可靠见解（例如，相变、临界现象）。 此外，计算复杂度可以大大降低，如果准确性和效率之间的权衡是适当的调整。 在对具有大量相互作用实体的系统进行模拟时，我们通常最终会估计依赖于系统随机分布的微观配置的某些可观测量的平均值。 在所提出的方法中，我们采用分层微观-宏观计算范式，并探索其潜力，以提高效率，可靠性和算法的复杂性的方法用于采样概率的措施，在高维空间。 我们的目标是开发工具，也适合近似瞬态行为可能无法正确捕获，实验相关的时间尺度上，通过采样的平衡分布。 所提出的框架将得到嵌套近似的基础多尺度系统与不同尺度之间的最佳相互作用。 这种规模分解策略将被应用于模拟随机系统的并行算法的发展，由大量的自由度。 建议的工作是在数值分析，随机过程和统计力学的交叉。 所提出的数学主题的潜在影响包括物理，化学，材料科学和其他使用大型多尺度模拟的领域的广泛应用。 因此，目标之一是实施灵活的工具，这些工具可以在较短的学习曲线内针对具体问题和现有软件包进行调整。

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.