User-Friendly Solvers and Solver-Friendly Discretizations
用户友好的求解器和求解器友好的离散化
基本信息
- 批准号:0915153
- 负责人:
- 金额:$ 21.9万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2009
- 资助国家:美国
- 起止时间:2009-09-15 至 2013-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
In this proposal, various algorithms and theories will be developed to partially realize the following four-stage strategy for developing user-friendly solvers and solver-friendly discretizations: (1) develop user-friendly optimal solvers and relevant theories for a small number of basic solver-friendly systems, namely the discrete Poisson's equation and its variants; (2) extend the list of solver-friendly partial differential equations (such as the discrete Stokes and Maxwell equations) by reducing them to the solution of a handful of basic solver-friendly systems (for which optimal and user-friendly solvers can be applied); (3) develop solver-friendly discretization techniques for more complicated PDEs (systems) such that the discretized systems will join the list of solver-friendly systems (such as the Eulerian-Lagrangian method for the Navier-Stokes equation, the Johnson-Segalman equations, and the magnetohydrodynamics equations); and (4) solve the discretized system from a general discretization by using a solver-friendly discretization (if it is not a satisfactory discretization to obtain the numerical solution by itself) as an auxiliary discretization that can be used as a preconditioner or a means for obtaining a good initial guess for a linear or nonlinear iteration. These techniques will be developed with the purpose of making them effective for solving complicated problems such as non-Newtonian models and fuel cell model equations. Parallel implementations will be one major consideration in the design of these algorithms. Theoretical issues---such as the most fundamental open problem concerning the optimal convergence of algebraic multigrid methods---will be carefully investigated.Many problems in scientific and engineering computing can be reduced to the numerical solution of certain partial differential equations. Over the last few decades, researchers have expended significant effort on developing efficient iterative methods for solving discretized partial differential equations. Though these efforts have yielded many mathematically optimal solvers such as the multigrid method, the unfortunate reality is that multigrid methods have not been much used in practical applications. This marked gap between theory and practice is mainly due to the fragility of traditional multigrid methodology and the complexity of its implementation. This proposal aims to develop theories and techniques that will narrow this gap, specifically by developing mathematically optimal solvers that are robust and easy to use in practice. The proposed study will focus on an integrated application of user-friendly solvers and solver-friendly discretizations for various basic partial differential equations that arise in many applications; therefore, the results of this proposal are expected to be directly applicable in many areas of computational and applied mathematics. The solver and discretization techniques we produce, including mathematical algorithms, analyses, and software, will provide powerful tools for exploring multiscale models in physics, chemistry, and engineering. Through the accompanying Matrix-Solver Community Project (http://www.multigrid.org/solvers/), the results of this proposal will lead to timely and broad impacts. The proposed project will have a strong educational impact as well, as it focuses on training graduate students in theoretical and practical aspects of modern computational science and interdisciplinary applications.
在本建议中,将开发各种算法和理论,以部分实现开发用户友好的求解器和求解器友好的离散化的以下四个阶段的策略:(1)为少数基本求解器友好的系统(即离散泊松方程及其变体)开发用户友好的最佳求解器和相关理论;(2)扩展了求解器友好的偏微分方程列表(如离散的斯托克斯和麦克斯韦方程),通过减少他们的解决方案的一小部分基本的解决方案友好的系统(可应用最佳和用户友好的求解器);(3)为更复杂的偏微分方程(系统)开发求解友好的离散化技术,使得离散化系统将加入求解友好的系统列表(例如用于Navier-Stokes方程、Johnson-Segalman方程和磁流体动力学方程的欧拉-拉格朗日方法);以及(4)通过使用求解器友好的离散化来从一般离散化求解离散化系统(如果它本身不是一个令人满意的离散化来获得数值解)作为辅助离散化,可以用作预处理器或用于获得线性或非线性迭代的良好初始猜测的手段。这些技术将被开发,目的是使它们有效地解决复杂的问题,如非牛顿模型和燃料电池模型方程。并行实现将是这些算法设计中的一个主要考虑因素。理论问题---在过去的几十年里,研究人员花费了大量的精力来开发有效的迭代方法来求解离散偏微分方程。虽然这些努力已经产生了许多数学上的最优解,如多重网格方法,不幸的是,多重网格方法在实际应用中并没有得到太多的应用。这种理论与实践之间的显著差距主要是由于传统多重网格方法的脆弱性及其实现的复杂性。该提案旨在开发缩小这一差距的理论和技术,特别是通过开发在实践中强大且易于使用的数学最优解算器。拟议的研究将集中在一个综合的应用程序的用户友好的求解器和求解器友好的离散化的各种基本偏微分方程,在许多应用中出现的,因此,这个建议的结果预计将直接适用于计算和应用数学的许多领域。我们生产的求解器和离散化技术,包括数学算法,分析和软件,将为探索物理,化学和工程中的多尺度模型提供强大的工具。 通过附带的矩阵求解社区项目(http://www.multigrid.org/solvers/),该提案的结果将产生及时和广泛的影响。拟议的项目也将产生强大的教育影响,因为它侧重于培养现代计算科学和跨学科应用的理论和实践方面的研究生。
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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Jinchao Xu其他文献
span style=line-height:150%;font-family:Times New Roman;font-size:12pt;A discontinuous Galerkin method for the fourth order Curl problem/span
求解四阶Curl问题的间断伽辽金法
- DOI:
- 发表时间:
2012 - 期刊:
- 影响因子:0
- 作者:
Qingguo Hong;Jun Hu;Shi Shu;Jinchao Xu - 通讯作者:
Jinchao Xu
Extended Regularized Dual Averaging Methods for Stochastic Optimization
用于随机优化的扩展正则化双平均方法
- DOI:
- 发表时间:
2019 - 期刊:
- 影响因子:0
- 作者:
Jonathan W. Siegel;Jinchao Xu - 通讯作者:
Jinchao Xu
<span style="line-height:150%;font-family:'Times New Roman';font-size:12pt;">Two-grid Methods for Time-harmonic Maxwell Equations</span>
- DOI:
- 发表时间:
2013 - 期刊:
- 影响因子:
- 作者:
Liuqiang Zhong;Shi Shu;Junxian Wang;Jinchao Xu; - 通讯作者:
Surges generated by water export from an impounded channel
从蓄水渠道排水所产生的涌浪
- DOI:
10.1016/j.oceaneng.2025.121160 - 发表时间:
2025-06-15 - 期刊:
- 影响因子:5.500
- 作者:
Feidong Zheng;Qiang Liu;Xueming Wu;Xiaofen Liu;Shuai Zhang;Jinchao Xu;Xueyi Li - 通讯作者:
Xueyi Li
Efficient degradation of methylene blue at near neutral pH based on heterogeneous Fenton-like system catalyzed by Fe<sub>2</sub>O<sub>3</sub>/MnO<sub>2</sub>
- DOI:
10.1016/j.rechem.2024.101795 - 发表时间:
2024-10-01 - 期刊:
- 影响因子:
- 作者:
Tie Geng;Jiaguo Yan;Bin Li;Haiyuan Yan;Lei Guo;Qiang Sun;Zengfu Guan;Chunning Zhao;Jinchao Xu;Weichao Wang - 通讯作者:
Weichao Wang
Jinchao Xu的其他文献
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{{ truncateString('Jinchao Xu', 18)}}的其他基金
Workshop on Mathematical Machine Learning and Application
数学机器学习与应用研讨会
- 批准号:
2020623 - 财政年份:2020
- 资助金额:
$ 21.9万 - 项目类别:
Standard Grant
US Participation at the Twenty-sixth Internaltional Domain Decomposition Conference
美国参加第二十六届国际域分解会议
- 批准号:
1930036 - 财政年份:2019
- 资助金额:
$ 21.9万 - 项目类别:
Standard Grant
Multigrid Methods and Machine Learning
多重网格方法和机器学习
- 批准号:
1819157 - 财政年份:2018
- 资助金额:
$ 21.9万 - 项目类别:
Continuing Grant
Integrated Geometric and Algebraic Multigrid Methods
综合几何和代数多重网格方法
- 批准号:
1522615 - 财政年份:2015
- 资助金额:
$ 21.9万 - 项目类别:
Continuing Grant
Single-grid Multi-level Solvers for Coupled PDE Systems
耦合偏微分方程系统的单网格多级求解器
- 批准号:
1217142 - 财政年份:2012
- 资助金额:
$ 21.9万 - 项目类别:
Continuing Grant
SCREMS: Scientific Computing Environments for Mathematical Sciences
SCEMS:数学科学的科学计算环境
- 批准号:
0619587 - 财政年份:2006
- 资助金额:
$ 21.9万 - 项目类别:
Standard Grant
Adaptive Multigrid Methods for a Multiphase Fuel Cell Model
多相燃料电池模型的自适应多重网格方法
- 批准号:
0609727 - 财政年份:2006
- 资助金额:
$ 21.9万 - 项目类别:
Continuing Grant
Mathematical and Computational Studies of Fuel Cell Dynamics
燃料电池动力学的数学和计算研究
- 批准号:
0308946 - 财政年份:2005
- 资助金额:
$ 21.9万 - 项目类别:
Standard Grant
Multiscale Methods for Partial Differential Equations
偏微分方程的多尺度方法
- 批准号:
0209497 - 财政年份:2002
- 资助金额:
$ 21.9万 - 项目类别:
Standard Grant
Scientific Computing Research Environments for the Mathematical Sciences
数学科学的科学计算研究环境
- 批准号:
0215392 - 财政年份:2002
- 资助金额:
$ 21.9万 - 项目类别:
Standard Grant
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