Investigation of Auxiliary Subspace Techniques as a General Tool for A Posteriori Error Estimation
辅助子空间技术作为后验误差估计通用工具的研究
基本信息
- 批准号:1414365
- 负责人:
- 金额:$ 14.39万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Continuing Grant
- 财政年份:2013
- 资助国家:美国
- 起止时间:2013-10-01 至 2016-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
A posteriori error estimation is an essential component of high-performance finite element computations. Such estimates are used in practice not only to reliably determine when an approximate solution is accurate enough, but also to (efficiently) adaptively improve the approximation. This proposal considers auxiliary subspace error estimates, which are derived from computing an approximate error function in an auxiliary space. Such an approximate error function provides great flexibility, in principle, in how it may be used for adaptive finite, and we are here concerned estimation of and adaptivity with respect to: error in a variety of norms, higher-order derivatives in a variety norms, functional error for general classes of linear functionals, and error in eigenvalue and invariant subspace computations. The robustness of hierarchical error estimates of energy-norm error is well-established both in theory and in practice for low-order finite elements and second-order linear elliptic boundary value problems in two dimensions. This proposal aims to significantly extend both theory and practice not only to include the various error measures mentioned above, but also higher-order elements in two and three dimensions (p- and hp-adaptivity), as well as to different types of operators and finite elements, including systems of partial differential equations. Additionally, an adaptive convergence theory would also be developed, where possible. A key component of the proposed research is the development of a basic framework in which clear guidance concerning an appropriate choice of auxiliary space for computing the approximate error function is provided by considering a few basic properties of the underlying problem and the space which was used for the approximate solution.The ability to automatically detect and adapt to relevant fine and coarse-scale features in the modeling of composite materials is often indispensable as an aid for design of such materials, as well as for remote sensing in the presence of complex media (e.g. non-destructive exploration for natural resources). This proposal concerns the development of a general and very flexible approach to error estimation and adaptive improvement of approximations in a variety of contexts, providing careful development of specific cases of interest, such as those mentioned above. A clear theoretical framework for error estimation and adaptivity, together with several important practical realizations, will not only aid practitioners in making appropriate choices in their particular contexts, but will also make it easier to train students to be able to develop such tools for problem where little (if any) are available. The various projects in the proposal include both national and international collaboration,as well as the education and involvement of graduate students. Additionally, much of the software developed in conjunction with this proposal will be made freely available by the proposer from his website.
后验误差估计是高性能有限元计算的重要组成部分。 这种估计在实践中不仅用于可靠地确定近似解何时足够准确,而且用于(有效地)自适应地改进近似。该建议考虑辅助子空间误差估计,这是来自计算一个近似的误差函数在辅助空间。这样一个近似的误差函数提供了很大的灵活性,在原则上,它可以用于自适应有限,我们在这里关注的估计和自适应性方面:误差在各种规范,高阶导数在各种规范,功能错误的一般类的线性泛函,和错误的特征值和不变子空间计算。对于低阶有限元和二维二阶线性椭圆边值问题,能量模误差的分层误差估计的稳健性在理论和实践中都得到了很好的证明.该建议旨在显著扩展理论和实践,不仅包括上述各种误差测量,还包括二维和三维中的高阶元素(p-和hp-自适应),以及不同类型的算子和有限元,包括偏微分方程系统。 此外,在可能的情况下,还将开发自适应收敛理论。所提出的研究的一个关键组成部分是发展一个基本框架,在该框架中,通过考虑基本问题的一些基本性质和用于近似解的空间,提供关于用于计算近似误差函数的辅助空间的适当选择的明确指导。复合材料建模中的比例特征通常是不可或缺的,作为设计此类材料的辅助手段,以及在复杂介质存在下的遥感(例如,自然资源的非破坏性勘探)。 该建议涉及开发一种通用且非常灵活的方法,以在各种情况下进行误差估计和近似值的自适应改进,并仔细开发特定的感兴趣案例,例如上文提到的案例。 一个清晰的误差估计和自适应性的理论框架,加上几个重要的实际实现,不仅可以帮助从业者在他们的特定环境中做出适当的选择,而且还可以更容易地训练学生能够开发这样的工具来解决几乎没有(如果有的话)的问题。 提案中的各种项目包括国家和国际合作,以及研究生的教育和参与。 此外,与本提案一起开发的许多软件将由提案人在其网站上免费提供。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Jeffrey Ovall其他文献
Jeffrey Ovall的其他文献
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{{ truncateString('Jeffrey Ovall', 18)}}的其他基金
Computational Tools for Exploring Eigenvector Localization
用于探索特征向量定位的计算工具
- 批准号:
2208056 - 财政年份:2022
- 资助金额:
$ 14.39万 - 项目类别:
Standard Grant
A Fitted Finite Element Method for the Modeling of Complex Materials
复杂材料建模的拟合有限元方法
- 批准号:
2012285 - 财政年份:2020
- 资助金额:
$ 14.39万 - 项目类别:
Continuing Grant
Cluster-Robust Estimates for Galerkin and Petrov-Galerkin Discretizations of Elliptic Eigenvalue Problems
椭圆特征值问题的 Galerkin 和 Petrov-Galerkin 离散化的聚类鲁棒估计
- 批准号:
1522471 - 财政年份:2015
- 资助金额:
$ 14.39万 - 项目类别:
Standard Grant
Investigation of Auxiliary Subspace Techniques as a General Tool for A Posteriori Error Estimation
辅助子空间技术作为后验误差估计通用工具的研究
- 批准号:
1216672 - 财政年份:2012
- 资助金额:
$ 14.39万 - 项目类别:
Continuing Grant
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