权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Optimal Convergence Rates for Adaptive Finite Element Techniques

自适应有限元技术的最佳收敛率

基本信息

批准号：
1720297
负责人：
Peter Binev
金额：
$ 27.5万
依托单位：
University of South Carolina at Columbia
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-06-15 至 2022-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1720297&HistoricalAwards=false
关键词：
Optimal Convergence Rates Adaptive Finite

项目摘要

Numerical simulation is an indispensable tool for acquiring deeper and more quantitative insight into increasingly complex scientific and technological processes. Despite the ever-increasing power of digital computing facilities, numerical simulation technology is somewhat of a weak link. This research project aims to develop improved adaptive numerical algorithms, which have the ability to optimally allocate computational resources -- viz. degrees of freedom -- in the course of the solution process based on information gathered so far. Economizing as much as possible the number of degrees of freedom with the aid of adaptive solution techniques while still accurately capturing the structures of interest remains central to large-scale simulation and a fundamental prerequisite for ultimately further advancing the frontiers of computability. While large-scale scientific computation usually takes place in a highly interdisciplinary arena, the design of adaptive algorithms with rigorously-founded certifiable performance guarantees is an inherently mathematical task that is pursued in this project. The many conceptual facets of this research project additionally offer unique opportunities for talented young researchers to develop their potential.This project aims at developing and analyzing hp-adaptive approximations through a process of locally distributing the degrees of freedom through a coarse-to-fine procedure based on local error estimators. The challenges in accomplishing the goals of the project have two major sources. On the one hand, the type of the partial differential equation to which such methods are to be applied, of course, matters very much. On the other hand, there are several fundamental problem aspects that are independent of the particular application and are primarily of approximation-theoretic nature. Even when restricting the problem to a fixed mesh refinement depth and a largest allowable polynomial degree, finding the optimal degree distribution in conjunction with an adequately locally-refined partition is an NP-hard problem. In particular, when progressing from coarse to successively refined meshes seemingly good degree assignments could turn out at a much later stage to prevent near-optimal results. It is therefore of crucial importance to address such core approximation theoretic issues and understand to what extent they are affected by the particular type of partial differential equation. For instance, when using conforming methods for the important class of elliptic boundary value problems, trial functions need to be globally continuous, which severely impedes the analysis of local refinements due to "smoothness pollution," particularly in the multivariate case. To address these aspects and build a solid footing for future specifications to different application areas is the primary goal of this project. Some of the envisaged theoretical results are expected to be of asymptotic nature. Therefore, the theoretical investigations will be accompanied by implementing the strategies for model problems that shed light on the quantitative behavior of the methods. A high level of adaptivity interferes with parallelization, opening yet another direction of research, especially regarding modern processor technologies.

数值模拟是获取更深入和更定量的洞察日益复杂的科学和技术过程中不可或缺的工具。尽管数字计算设备的能力不断增强，但数值模拟技术在某种程度上是一个薄弱环节。本研究项目旨在开发改进的自适应数值算法，该算法能够根据迄今为止收集的信息，在求解过程中最佳分配计算资源-即自由度。在自适应求解技术的帮助下尽可能地节省自由度的数量，同时仍然准确地捕获感兴趣的结构，这仍然是大规模模拟的核心，也是最终进一步推进可计算性前沿的基本先决条件。虽然大规模的科学计算通常发生在一个高度跨学科的竞技场，自适应算法的设计与严格成立的可认证的性能保证是一个固有的数学任务，在这个项目中追求。这个研究项目的许多概念方面也为有才华的年轻研究人员提供了独特的机会，以发展他们的potential.This项目的目的是开发和分析hp-自适应近似通过一个过程中的局部分布的自由度，通过一个由粗到细的程序的基础上局部误差estimators。实现项目目标的挑战主要来自两个方面。一方面，这种方法所适用的偏微分方程的类型当然非常重要。另一方面，有几个基本的问题方面是独立的特定应用程序，主要是近似理论的性质。即使将问题限制为固定的网格细化深度和最大允许多项式次数，找到最佳度分布与充分局部细化分区相结合也是NP难问题。特别是，当从粗网格到连续细化网格的过程中，看似良好的度分配可能会在更晚的阶段出现，以防止接近最优的结果。因此，解决这些核心近似理论问题并了解它们在多大程度上受特定类型的偏微分方程的影响是至关重要的。例如，当使用协调方法处理一类重要的椭圆边值问题时，试函数需要是全局连续的，这严重阻碍了由于“光滑污染”而导致的局部精化分析，特别是在多变量情况下。为了解决这些方面，并建立一个坚实的基础，为未来的规范，以不同的应用领域是本项目的主要目标。一些设想的理论结果预计是渐近性质的。因此，理论研究将伴随着实现模型问题的策略，这些策略揭示了方法的定量行为。高水平的自适应性干扰了并行化，开辟了另一个研究方向，特别是关于现代处理器技术。