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有力的自动化方法。特别是自适应网格细化（AMR）算法在计算科学和工程中得到了广泛的应用，并已成为复杂自然科学和工程问题的计算机模拟中的常用工具。正如美国国家研究理事会所确定的那样，AMR是计算解决大挑战问题的两个必要工具之一（AMR和并行计算）。AMR算法成功的关键因素是后验误差估计，它能够准确地定位当前近似中的全局和局部误差源。计算机模拟复杂系统的另一个挑战是计算机预测的可靠性。这些考虑（AMR算法和误差控制的效率）表明需要一个误差估计，可以从计算的数值解和给定的数据的基本问题的后验提取。理想情况下，这种后验误差估计应该提供一个基本的严格的数学理论，用于估计和量化离散化误差的误差大小和空间分布。该项目的成功将使AMR算法能够自动定位物理界面，检测层和不连续性，并解决各种尺度的振荡。本计画所发展之对偶估计量将解决一类问题在粗网格上离散化误差控制这一最自然但又极为困难的问题，并因此部分地保证计算机模拟的可靠性。本研究计画的重点是透过对偶方法发展、分析及检验后验误差估计量。在这个项目中开发的对偶估计将有一个保证的可靠性界限，可靠性常数为1。因此，这些估计器是完美的离散误差控制，并可用作迭代求解器的精确停止准则。对偶方法可应用于连续介质力学中的一大类问题，包括线性和非线性问题。由于这些估计将不使用先验知识的位置和特性的界面奇点，不连续性（在形式上的冲击波一样的前线，和内部和边界层），和/或振荡的各种尺度（多尺度现象），他们可以更容易地应用到高度非线性问题，并有可能被应用到复杂的系统中出现的应用。研究的重点和难点在于：（1）显式或局部构造对偶变量的近似值，使得所得到的指标是有效的和稳健的;（2）理论和数值验证所得到的指标的有效性和稳健性。最后，一小部分拟议的研究解决了一个开放的理论问题的鲁棒性估计接口问题。