A Posteriori Analysis of Multirate Numerical Methods

多速率数值方法的后验分析

• 批准号: 1016283
• 负责人: Victor Ginting
• 金额: $ 18.82万
• 依托单位: University of Wyoming
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016283&HistoricalAwards=false
• 关键词: Posteriori; Analysis; Multirate; Numerical; Methods

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.

本研究计画系关于一类多速率数值方法的后验分析，其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具有不同尺度的子系统的集合。每个子系统可以使用与其特征尺度有关的不同离散化参数。这是在对比使用一个单一的离散化参数决定的主导规模在原来的控制方程，如果标准的完全耦合的数值程序是使用。然而，多速率数值方法引入了一组误差，这些误差以明显和微妙的方式直接影响近似解的准确性和稳定性，这些方式难以精确量化。这些问题是通过进行后验分析的多速率数值方法的基础上变分伴随技术。这些技术是期望的，因为它们适合于在近似解的泛函方面表示的指定感兴趣的量的误差预测。能够直接关注基于应用的感兴趣的量对误差估计和自适应误差控制中的计算效率具有很强的影响。我们的目标是制定准确的估计技术，有能力区分和量化的误差分量，如多速率离散化，非线性解决方案中的不完全迭代，以及每个子系统的解决方案中的数值误差。然后，这可以用来更好地了解误差对精度，稳定性，多速率数值方法在许多应用中得到了广泛的应用。成功完成这一项目的主要指标将是更好地了解这些方法，并有能力量化其错误。这将对电力系统技术、核工程、石油生产和生物学等工程和科学领域产生广泛影响。在这些领域中的几个产生的多速率数值方法的应用程序被用来作为基准问题的误差估计技术的发展。该项目的设计可以解决采用多速率数值方法，如精度和稳定性，自适应性和效率的基本问题。这反过来将大大有助于开发高效和准确的多速率数值方法。预计该项目内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。该项目涉及培训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。