A Posteriori Error Estimation through Duality and Some Other Topics

通过对偶性和其他一些主题进行后验误差估计

• 批准号: 1522707
• 负责人: Zhiqiang Cai
• 金额: $ 26万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1522707&HistoricalAwards=false
• 关键词: Posteriori; Error; Estimation; through; Duality

Self-adaptive numerical methods provide a powerful and automatic approach in scientific computing. In particular, Adaptive Mesh Refinement (AMR) algorithms have been widely used in computational science and engineering and have become a common tool in computer simulations of complex natural science and engineering problems. As identified by the US National Research Council, AMR is one of two necessary tools (AMR and Parallel Computing) for computationally tackling Grand Challenge problems. The key ingredient for success of AMR algorithms is a posteriori error estimates that are able to accurately locate sources of global and local error in the current approximation. Another challenge in computer simulations of complex systems is the reliability of computer predictions. These considerations (efficiency in AMR algorithms and error control) demonstrate the need for an error estimator that can a posteriori be extracted from the computed numerical solution and the given data of the underlying problem. Such an a posteriori error estimate ideally should provide an underlying rigorous mathematical theory for estimating and quantifying discretization error in terms of the error's magnitude and its spatial distribution. Success in this project will allow AMR algorithms to automatically locate physical interfaces, detect layers and discontinuities, and resolve oscillations of various scales. The dual estimators to be developed in this project will resolve the most natural but extremely difficult question of discretization error control on coarse meshes for a class of problems and hence partially guarantee reliability of computer simulations.This research project focuses on the development, analysis, and test of a posteriori error estimators through the methodology of duality. The dual estimators to be developed in this project will have a guaranteed reliability bound with reliability constant being one. Hence, these estimators are perfect for discretization error control and may be used as an accurate stopping criterion for iterative solvers. The methodology of duality may be applied to a large class of problems arising from continuum mechanics including linear and nonlinear problems. Since these estimators will not use a priori knowledge on the locations and characteristics of interface singularities, discontinuities (in the form of shock-like fronts, and of interior and boundary layers), and/or oscillations of various scales (multi-scale phenomena), they may then be applied more readily to highly nonlinear problems and have the potential of being applied to complex systems arising in applications. The emphases and the difficulties of the proposed research are (1) explicit or local construction of an approximation to the dual variable such that the resulting indicator is efficient and robust, and (2) theoretical and numerical confirmation of the efficiency and robustness. Finally, a small portion of the proposed research addresses an open theoretical question on the robustness of estimators for interface problems.

自适应数值方法为科学计算提供了一种强大且自动的方法。特别是自适应网格细化（AMR）算法已广泛应用于计算科学和工程领域，并成为复杂自然科学和工程问题计算机模拟的常用工具。正如美国国家研究委员会所指出的，AMR 是计算解决重大挑战问题的两个必要工具（AMR 和并行计算）之一。 AMR 算法成功的关键因素是后验误差估计，能够准确定位当前近似值中的全局和局部误差源。复杂系统计算机模拟的另一个挑战是计算机预测的可靠性。这些考虑因素（AMR 算法和误差控制的效率）表明需要一个误差估计器，该误差估计器可以从计算的数值解和底层问题的给定数据中提取后验信息。理想情况下，这种后验误差估计应该提供一个基本的严格数学理论，用于根据误差的大小及其空间分布来估计和量化离散化误差。该项目的成功将使 AMR 算法能够自动定位物理接口、检测层和不连续性并解决各种规模的振荡。本项目要开发的对偶估计器将解决一类问题中最自然但又极其困难的粗网格离散误差控制问题，从而部分保证计算机模拟的可靠性。本研究项目的重点是通过对偶方法开发、分析和测试后验误差估计器。本项目中要开发的对偶估计器将具有保证的可靠性界限，其中可靠性常数为 1。因此，这些估计器非常适合离散化误差控制，并且可以用作迭代求解器的准确停止标准。对偶性方法可以应用于由连续介质力学引起的一大类问题，包括线性和非线性问题。由于这些估计器不会使用关于界面奇点、不连续性（以类激波前沿、内部和边界层的形式）和/或各种尺度的振荡（多尺度现象）的位置和特征的先验知识，因此它们可以更容易地应用于高度非线性问题，并且有可能应用于应用中出现的复杂系统。本研究的重点和难点是（1）显式或局部构造对偶变量的近似值，使得所得指标高效且稳健；（2）效率和稳健性的理论和数值确认。最后，所提出的研究的一小部分解决了关于界面问题估计量的鲁棒性的开放理论问题。