A Posteriori Analysis of Multirate Numerical Methods

本研究项目涉及一类多速率数值方法的后验分析，这些方法的自然应用出现在多尺度微分系统中。多速率数值方法的一个共同特点是将原控制方程分解为不同尺度的子系统集合。每个子系统可以根据其特征尺度使用不同的离散化参数。如果要使用标准的完全耦合数值过程，这与在原始控制方程中使用由主导尺度决定的单个离散化参数形成对比。然而，多速率数值方法引入了一组误差，这些误差以明显或微妙的方式直接影响近似解的精度和稳定性，难以精确量化。这些问题是通过进行基于变分伴随技术的多速率数值方法的后验分析来解决的。这些技术是需要的，因为它们适合于以近似解的泛函形式表示的特定量的误差预测。能够直接关注基于应用程序的兴趣量对误差估计和自适应误差控制的计算效率有很大的影响。目标是制定准确的估计技术，该技术具有区分和量化误差成分的能力，例如多速率离散化，非线性解决方案中的不完全迭代，以及每个子系统解决方案中的数值误差。这可以用来更好地了解误差对方法的准确性，稳定性和适应性等问题的影响。多速率数值方法在许多领域都有广泛的应用。成功完成这个项目的主要指标将是更好地理解这些方法和量化其误差的能力。这将对诸如电力系统技术、核工程、石油生产和生物学等工程和科学领域产生广泛的影响。在这些领域中出现的多速率数值方法的应用被用作开发误差估计技术的基准问题。该项目设计允许解决采用多速率数值方法的基本问题，如准确性和稳定性、适应性和效率。这反过来又将大大有助于发展有效和准确的多速率数值方法。预计该项目的活动将加强正在进行的合作，特别是与落基山脉地区的调查人员的合作。该项目包括对一名研究生进行后验分析及其在多速率数值方法中的应用方面的培训。

This research project is concerned with a posteriori analysis of a class of multirate numerical methods whose natural application appears in multiscale differential systems. A common trait in the multirate numerical methods is the decomposition of the original governing equations into collection of subsystems with different scales. Each subsystem can use a different discretization parameter pertaining to its characteristic scale. This is in contrast to using a single discretization parameter dictated by the dominating scale in the original governing equations if standard fully coupling numerical procedure is to be used. The multirate numerical methods, however, introduces a set of errors which directly affect the accuracy and stability of the approximate solutions in both obvious and subtle ways that are difficult to quantify accurately. These issues are addressed by conducting a posteriori analysis of the multirate numerical methods based on variational adjoint techniques. These techniques are desired because of their suitability for error prediction in the specified quantities of interest expressed in terms of functional of the approximate solutions. Being able to focus directly on application-based quantities of interest has strong consequences for computational efficiency in error estimation and adaptive error control. The goal is to formulate accurate estimation techniques that have the capability to distinguish and quantify the error components, such as the multirate discretization, incomplete iteration in the nonlinear solution, and numerical errors in the solution of each subsystem This can then be used to gain better insights of the effects of the errors on issues such as accuracy, stability, and adaptivity of the methods.Multirate numerical methods are widely used in many applications. The main indicator of the successful completion of this project will be a better understanding of these methods and a capability to quantify their errors. This will have a broad impact on areas of engineering and science such as power system technology, nuclear engineering, petroleum production, and biology. Applications of multirate numerical methods arising in several of these areas are used as benchmark problems for developing the error estimation techniques. The project design allows for addressing fundamental issues in employing multirate numerical methods, such as accuracy and stability, adaptivity, and efficiency. This in turn will significantly contribute to developing efficient and accurate multirate numerical methods. Activities within this project are expected to strengthen ongoing collaborations, especially with investigators in the Rocky Mountain region. The project involves training of a graduate student in the area of a posteriori analysis and their application to multirate numerical methods.