无方程多尺度计算：使微观模拟器能够执行系统级任务
Equation-Free Multiscale Computation: enabling microscopic simulators to perform system-level tasks
摘要：
我们提出并讨论了一个框架，计算机辅助多尺度分析，使模型在“精细”（微观/随机）的描述水平，以执行建模任务在“粗糙”（宏观，系统）的水平。这些宏观建模任务，产生信息在很长的时间和大的空间尺度，通过适当的初始化调用微观模拟器，只有很短的时间和小的空间域完成。我们的无方程（EF）的方法，当成功时，可以绕过推导的宏观演化方程时，这些方程概念上存在，但不能在封闭的形式。我们将讨论如何辅助发展的计算超结构的几何学可能使替代描述的问题物理（例如，在相对短的时间和空间尺度上执行的晶格玻尔兹曼（LB）、动力学蒙特卡罗（KMC）或分子动力学（MD）微观模拟器（在相对大的时间和空间尺度上的积分、“粗略”分叉分析、优化和控制）。实际上，该程序构成了一个系统识别的基础上，“按需关闭”的计算工具包，桥梁微观/随机模拟与传统的连续科学计算和数值分析。我们通过化学动力学（LB，KMC），流变学（布朗动力学），均匀化和计算的“粗糙自相似”的解决方案的例子来说明这些想法，并讨论各种功能，局限性和潜在的扩展的方法。
英文摘要：
We present and discuss a framework for computer-aided multiscale analysis, which enables models at a "fine" (microscopic/stochastic) level of description to perform modeling tasks at a "coarse" (macroscopic, systems) level. These macroscopic modeling tasks, yielding information over long time and large space scales, are accomplished through appropriately initialized calls to the microscopic simulator for only short times and small spatial domains. Our equation-free (EF) approach, when successful, can bypass the derivation of the macroscopic evolution equations when these equations conceptually exist but are not available in closed form. We discuss how the mathematics-assisted development of a computational superstructure may enable alternative descriptions of the problem physics (e.g. Lattice Boltzmann (LB), kinetic Monte Carlo (KMC) or Molecular Dynamics (MD) microscopic simulators, executed over relatively short time and space scales) to perform systems level tasks (integration over relatively large time and space scales,"coarse" bifurcation analysis, optimization, and control) directly. In effect, the procedure constitutes a systems identification based, "closure on demand" computational toolkit, bridging microscopic/stochastic simulation with traditional continuum scientific computation and numerical analysis. We illustrate these ideas through examples from chemical kinetics (LB, KMC), rheology (Brownian Dynamics), homogenization and the computation of "coarsely self-similar" solutions, and discuss various features, limitations and potential extensions of the approach.
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