AMC-SS: Multiscale Methods for Many-Particle Stochastic Systems: Coarse-Graining and Microscopic Reconstruction

与科学、技术和社会相关的不同领域的问题，从新材料的开发到对基本生物机制的理解，再到天气和气候预测，本质上是多尺度的。本质上，这意味着在非常小的空间和时间尺度上的信息，例如原子或分子的性质和运动，可以深刻地影响中尺度和大尺度的行为，例如膜的渗透性或流体的粘度。从建模和计算的角度来看，小的时空尺度可以用微观模型来详细描述，例如分子动力学或蒙特卡罗方法，它们通常解释原子和/或分子信息。另一方面，要理解材料的大尺度、宏观性质，需要在大得令人望而却步的空间(例如10^23个原子)以及时间尺度上用微观系统进行模拟。一个这样的例子出现在具有预先指定的性质的新型材料的设计中，在这种情况下，数值模拟--如果可行--可以用作一种灵活而廉价的预测工具。近年来开发的一类重要的计算工具，正是为了通过加快微观模拟方法来弥合这种规模差距，就是粗粒化方法。这种方法的想法是通过将自由度集中到粗粒度变量中来降低S分子系统的复杂性，从而产生一种加速的模拟方法。这种粗粒度模型已被开发用于研究和模拟微反应器(例如，便携式能源)、聚合物、蛋白质和生物流体(例如，小血管中的红细胞流动)等。现有的方法可以给分子模拟带来前所未有的速度，并且可以在某些参数范围内很好地工作，例如高温。另一方面，它们也可能对扩散、结晶和相变等重要特征给出错误的预测。沿着这些思路，一个与许多应用相关的数学和统计学目标，如前面提到的那些，是开发系统的诊断学，以确定粗粒化方法何时可以给出可靠的预测，以及如何进一步加强这些预测。事实上，在我们拟议的工作中，我们打算执行两项相关的主要任务：(A)通过开发数学和统计误差量化分析来了解现有粗粒度方法的有效性范围；以及(B)开发能够在更广泛的参数范围内运行的改进算法，一旦在模拟过程中检测到明显的偏差，就能够自动调整。在我们提议的工作中，我们计划开发新的多尺度数学和计算方法，将概率论、统计力学、信息论、统计学和有限元的不同技术结合在一起。拟议工作的一个关键组成部分依赖于应用数学和统计方法之间的协同作用，这些方法以互补的方式运作，可以为开发用于分子模拟的灵活和可靠的粗粒化算法提供数学系统的框架。

Problems in diverse areas of scientific, technological, and societal relevance, ranging from the development of novel materials to understanding fundamental biological mechanisms to weather and climate prediction, are intrinsically multi-scale. In essence this means that information at very small spatial and temporal scales, for instance properties and motions of atoms or molecules, can profoundly impact intermediate- and large-scale behavior, such as the permeability of a membrane or the viscosity of a fluid. From a modeling and computational perspective, small spatiotemporal scales can be described in detail with microscopic models such as Molecular Dynamics or Monte Carlo methods that typically account for atomistic and/or molecular information. On the other hand, understanding large-scale, macroscopic properties of materials requires simulations with microscopic systems at prohibitively large spatial (e.g. 10^23 atoms), as well as temporal scales. One such example arises in the design of novel materials with pre-specified properties, where numerical simulations--when feasible--could be used as a flexible and inexpensive predictive tool. An important class of computational tools that has been developed in recent years, precisely to bridge such scales gaps by speeding up microscopic simulation methods, is the method of coarse-graining. The idea of this approach is to reduce the molecular system''s complexity by lumping together degrees of freedom into coarse-grained variables, thus yielding an accelerated simulation methodology. Such coarse-grained models have been developed for the study and simulation of micro-reactors (e.g., portable energy sources), polymers, proteins, and bio-fluids (e.g., red-cell flow in small blood vessels), among others. Existing approaches can give unprecedented speed-ups to molecular simulations and can work well in certain parameter regimes, such as high temperatures. On the other hand, they can also give wrong predictions on important features such as diffusion, crystallization, and phase transitions. Along these lines, a relevant mathematical and statistical goal to numerous applications, such as the ones mentioned earlier, is to develop systematic diagnostics for determining when coarse-graining methods can give reliable predictions, and how they can be further enhanced. Indeed, in our proposed work we intend to carry out two related main tasks: (a) understand the validity regimes of existing coarse-graining methods by developing a mathematical and statistical error quantification analysis; and (b) develop improved algorithms capable of operating in much wider parameter regimes, with the capacity to automatically adjust once substantial deviations are detected during simulation. In our proposed work, we plan to develop novel multi-scale mathematical and computational methods, bringing together diverse techniques from probability theory, statistical mechanics, information theory, statistics, and finite elements. A critical component of the proposed work relies on the synergy between applied mathematics and statistics methods, operating in a complementary fashion, that can provide a mathematically systematic framework for developing flexible and reliable coarse-graining algorithms for molecular simulations.