Multiscale Computation of Highly Oscillatory Dynamical Systems

高振荡动力系统在计算上非常具有挑战性。传统的数值技术需要对每个波长进行多次函数评估。这通常是非常昂贵的计算。在早期的工作中，研究人员已经开发并分析了具有良好定义的尺度分离的有效解决此类系统的数值算法。在这个建议中，研究人员解决了更难的问题，其中尺度分离不太清楚，分解为慢速和快速变量是未知的。抽象地给出了一个状态变量为u的全尺寸模型。许多慢变量U，包括U(t)的局部平均值，将使用U(t)的短时间历史序列来计算，从U(t)一致定义的适当初始条件开始。因此，形成本提案的基本目标是：(1)多尺度方法在什么条件下和在什么意义上收敛？U的近似精度是多少，这些方法的稳定性又是什么？(2)当动力系统的右侧可以分解为刚性部分和非刚性部分时，如何系统地找到更高阶的精确算法？(3)与现有技术相比，设计计算复杂度大幅降低的方法。所提出的研究将从数学上描述具有慢模式的高度振荡的动力系统意味着什么。有了这些信息，新的有效的数值算法可以设计和严格测试。所提出的研究将在各种平均理论和新的有效的数值积分器和滤波技术之间建立数学联系。从天体物理学到量子力学，拟议的研究是科学和工程中许多重要应用的数学和计算核心。在这些应用中存在快速振荡，例如，原子的振动会在更慢的时间尺度上影响整个系统。这在传统上很难用合理的计算成本来模拟。在这个提议中发展的理论和算法使这样的模拟具有实用性，并直接与用于代替实际物理实验的重要原子模拟相关。它们将产生直接的影响，例如，在分子生物学中理解细胞水平上的药物功能，以及在材料特性和不稳定事件（如裂纹的形成和传播）的研究中，这可能会影响电子元件以及飞机的结构安全。

Highly oscillatory dynamical systems are computationally very challenging. Traditional numerical techniques require several function evaluations per wavelength. This is typically prohibitingly computationally costly. In earlier work the investigators have developed and analyzed numerical algorithms for efficient solution of such systems with well-defined scale separation. In this proposal the investigators tackle the harder problems where the scale separation is less clear and where the decomposition into slow and fast variables is not known. Abstractly, a full-scale model with state variables u is given. A number of slow variables, U, which include the local averages of u(t), is to be computed using a sequence of short time histories of u(t), starting from appropriate initial conditions consistently defined by U(t). Hence, the essential objectives that form this proposal are: (1) Under what conditions and in what sense do the multiscale approaches converge? What is the accuracy in the approximation of U, and what are the stability properties of the methods? (2) How can we find higher order accurate algorithms in a systematic way, when the dynamical system's right hand side can be decomposed into stiff and non-stiff parts? (3) Design such methods with a substantial reduction of computational complexity compared with existing techniques. The proposed research will delineate mathematically what it means for a dynamical system that appear to be highly oscillatory to possess slow modes. With this information, new efficient numerical algorithms can be devised and tested rigorously. The proposed research will establish a mathematical link between a variety of averaging theories and new effective numerical integrators and filtering techniques.The proposed research is at the very mathematical and computational heart of many important applications in science and engineering, from astrophysics to quantum mechanics. In these applications there are rapid oscillations as, for example, vibrations of atoms that affects the overall system on a much slower time scale. This is traditionally difficult to simulate with reasonable computational cost. The theories and algorithms developed in this proposal make such simulations practical and directly relate to important atomistic simulations that are used in place of actual physical experiments. They will have direct consequences, for example, in molecular biology for understanding drug functions at the cellular level, and also in the study of material properties and unstable events such as the formation and propagation of cracks, which may affect electronic components as well as the structural safety of airplanes.