Multiscale Stochastic Modeling, Analysis and Computation

多尺度随机建模、分析和计算

• 批准号: 0413864
• 负责人: Markos Katsoulakis
• 金额: --
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-15 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0413864&HistoricalAwards=false
• 关键词: Multiscale; Stochastic; Modeling; Analysis; Computation

Problems in diverse scientific disciplines ranging from materials science to macromolecular dynamics, to atmosphere/ocean science and climate modeling involve the nonlinear interaction of physical processes across many length and time scales ranging from the microscopic to the macroscopic. From a modeling and computational perspective, microscopic simulation methods such as Molecular Dynamics (MD) and Monte Carlo (MC) algorithms can describe complex, out of equilibrium interactions at small scales (e.g. between atoms or molecules). Although there is substantial progress in improving aspects of these computational methods, they are still limited to short length and time scales, while on the other hand device sizes and morphological features observed in experiments often involve much larger scales; at the same time stochastic fluctuations--inherited from the microscopics--can be important, for instance in self-organization problems characterized by small coherent structures such as pattern formation in nanotechnology applications. In addition to such challenges posed by the disparity in scales within the same model, in many instances we are faced with an additional disparity in models: for example in phenomena with detailed fluid/surface or boundary layer interactions it is necessary to couple microscopic, possibly stochastic models describing the dynamics of atoms or molecules on a surface, along with continuum PDE for species, fluid and thermodynamic variables on the overlying to the surface gas phase. It is therefore inevitable that features of the microscopic model will essentially enter as a subgrid effect in the coupling with the coarse computational grid of the macroscopic PDE models. In this case, the proper incorporation and simulation of stochastic effects from the subgrid microscale is a critical element in the modeling and simulations. The proposed projects focus on aspects of the aforementioned issues by putting forward a combination of interconnected modeling, computational and analysis questions, roughly divided in two categories: (i) Mathematical strategies for the coarse-graining of microscopic models and the corresponding simulators, addressing problems which are currently intractable with conventional MD/MC due to scale limitations. Here, it is not directly attempted to speed up microscopic simulation algorithms; instead, a hierarchy of new coarse-grained stochastic models (referred to as Coarse Grained Monte Carlo methods) is derived, ordered by the magnitude of space/time scales. This new set of models involves a reduced set of observables over the original microscopic models incorporating microscopic details and noise, as well asthe interaction of the unresolved degrees of freedom. (ii) Hybrid stochastic/deterministic systems describing detailed fluid-surface interactions arise in applications that range from deposition process and catalysis to fuel cell design and microreactors, to biology and atmosphere and ocean science. Here two such applications are addressed, namely catalytic reactors and stochastic parametrizations of tropical convection.Due to their inherent complexity it is also necessary to develop simpler, test bed problems that capture significant features of the multi-scale nature of the physical models, but are still amenable to asymptotics, mathematical analysis and tractable computations.The modeling and simulation of problems with multiple interrelating length and time scales is one of the preeminent issues in essentially all timely scientific and engineering challenges, ranging from the design of nanodevices, to biomolecular dynamics, to the spread of epidemics and climate modeling. In spite of a continuously increasing computing power many of these problems remain intractable, at least in realistic conditions, and new modeling and simulation strategies need to be developed. As several paradigmshave recently demonstrated, the critical step in this process is the use of (the limited in number and flexibility) existing multiscale mathematical and statistical tools as well as the development of new ones, that enable the creation of new algorithms for complex systems. In the proposed work an array of such novel multiscale mathematics and computing methods is developed. The research is motivated by and targeted to anumber of the aforementioned applications. Two such particular examples are, (i) the surface processes in catalytic reactors and fuel cells, and (ii) the tropical and open ocean convection.

从材料科学到大分子动力学等不同科学学科中的问题， 大气/海洋科学和气候模拟涉及从微观到宏观的许多长度和时间尺度上的物理过程的非线性相互作用。从建模和计算的角度来看，微观模拟方法，如分子动力学（MD）和蒙特卡罗（MC）算法， 可以描述复杂的，在小尺度上的平衡相互作用（例如原子或分子之间）。尽管在改进这些计算方法方面有实质性的进展，但它们仍然限于短的长度和时间尺度，而另一方面，在实验中观察到的器件尺寸和形态特征通常涉及更大的尺度;与此同时，随机波动--从微观上继承下来的--可能是重要的，例如在以小相干结构为特征的自组织问题中，例如在纳米技术应用中的图案形成。除了同一模型中尺度的差异所带来的挑战之外，在许多情况下，我们还面临着模型中的额外差异：例如，在具有详细的流体/表面或边界层相互作用的现象中，有必要将描述表面上的原子或分子的动力学的微观的、可能随机的模型，沿着与物质的连续PDE，流体和热力学变量对上覆到表面的气相的影响。因此，不可避免的是，微观模型的功能将基本上进入与宏观偏微分方程模型的粗计算网格耦合的亚网格效应。在这种情况下，适当的纳入和模拟的随机效应，从亚电网微尺度是一个关键因素的建模和模拟。拟议的项目侧重于上述方面 通过提出相互关联的建模，计算和分析问题的组合，大致分为两类：（i）微观模型和相应模拟器的粗粒度的数学策略，解决目前由于规模限制而难以解决的问题。在这里，它不是 直接试图加快微观模拟算法，而是一个新的粗粒度随机模型（称为粗粒度蒙特卡罗方法）的层次结构推导，空间/时间尺度的大小排序。这组新的模型涉及到一组减少的观测量超过原来的微观模型，将微观细节和噪音，以及未解决的自由度的相互作用。（二） 混合 描述详细的流体-表面相互作用的随机/确定性系统出现在从沉积过程和催化到燃料电池设计和微反应器到生物学和大气和海洋科学的应用范围内。这里讨论两个这样的应用，即催化反应器和热带对流的随机参数化。由于它们固有的复杂性，也有必要开发更简单的测试床问题，这些问题捕获物理模型的多尺度性质的重要特征，但仍然适合于渐近，数学分析和易于处理的计算。具有多个相互关联的长度和时间尺度的问题的建模和仿真是基本上所有领域中的突出问题之一。及时的科学和工程挑战，从纳米器件的设计，生物分子动力学，流行病的传播和气候建模。尽管不断增加的计算能力，许多这些问题仍然难以解决，至少在现实条件下，和新的建模和仿真策略需要开发。正如最近几个范例所证明的那样，这一过程的关键步骤是使用（数量和灵活性有限）现有的多尺度数学和统计工具以及开发新的工具，从而为复杂系统创建新的算法。在拟议的工作中，开发了一系列这样的新的多尺度数学和计算方法。该研究的动机和针对上述应用的数量。两个这样的具体例子是，（一）在催化反应器和燃料电池的表面过程，和（二）热带和开放的海洋对流。