Multiscale Computation of Highly Oscillatory Dynamical Systems

高度振荡的动力系统在计算上是非常具有挑战性的。传统的数值方法需要对每个波长进行多个函数求值。这通常在计算上代价高昂。在早期的工作中，研究人员已经开发和分析了数值算法，用于有效解决具有明确尺度分离的此类系统。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，分解成慢变量和快变量是未知的。抽象地，给出了一个具有状态变量u的全尺寸模型。从U（t）一致定义的适当初始条件开始，使用u（t）的一系列短时间历史来计算一些慢变量U，其中包括u（t）的局部平均值。因此，形成这一建议的基本目标是：（1）在什么条件下，在什么意义上的多尺度方法收敛？近似U的精度是多少，方法的稳定性是什么？(2)当动力系统的右侧可以分解为刚性和非刚性部分时，我们如何以系统的方式找到高阶精确算法？(3)设计这样的方法，与现有技术相比，大大降低了计算复杂性。拟议的研究将从数学上描述一个似乎是高度振荡的动力系统拥有慢模式意味着什么。有了这些信息，新的有效的数值算法可以设计和严格测试。这项研究将在各种平均理论与新的有效数值积分器和滤波技术之间建立数学联系，是从天体物理学到量子力学等科学和工程领域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，原子的振动在慢得多的时间尺度上影响整个系统。这在传统上难以以合理的计算成本来模拟。 在这个提议中开发的理论和算法使这种模拟变得实用，并直接关系到用于代替实际物理实验的重要原子模拟。它们将产生直接的后果，例如，在分子生物学中理解药物在细胞水平上的功能，以及在材料特性和不稳定事件的研究中，例如裂纹的形成和传播，这可能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.