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

科学，技术和社会相关性的不同领域的问题，从新材料的开发到理解基本的生物机制，再到天气和气候预测，本质上是多尺度的。 从本质上讲，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，如分子动力学或蒙特卡罗方法，通常考虑原子和/或分子信息。 另一方面，理解材料的大尺度宏观性质需要在非常大的空间（例如10^23个原子）以及时间尺度上模拟微观系统。 一个这样的例子出现在设计具有预定属性的新材料中，在可行的情况下，数值模拟可以用作灵活且廉价的预测工具。 近年来发展起来的一类重要的计算工具，正是为了通过加速微观模拟方法来弥合这种尺度上的差距，这就是粗粒化方法。 这种方法的思想是通过将自由度集中到粗粒度变量中来降低分子系统的复杂性，从而产生加速的模拟方法。 这种粗粒度模型已经被开发用于微反应器（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。 现有的方法可以提供前所未有的速度，分子模拟，并可以在某些参数制度，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。 沿着这些路线，与许多应用相关的数学和统计目标，例如前面提到的那些，是开发系统诊断，用于确定粗粒度方法何时可以给出可靠的预测，以及如何进一步增强。 事实上，在我们提出的工作中，我们打算进行两个相关的主要任务：（a）了解现有的粗粒度方法的有效性制度，通过开发一个数学和统计误差量化分析;和（B）开发改进的算法，能够在更广泛的参数制度，自动调整的能力，一旦在模拟过程中检测到大量的偏差。 在我们提出的工作中，我们计划开发新的多尺度数学和计算方法，汇集来自概率论，统计力学，信息论，统计学和有限元的各种技术。 拟议的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，以互补的方式运作，可以提供一个数学系统的框架，为开发灵活和可靠的粗粒化算法的分子模拟。

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.