Collaborative Research: Advancing Theoretical Understanding of Accelerated Nonlinear Solvers, with Applications to Fluids

合作研究:推进对加速非线性求解器的理论理解及其在流体中的应用

基本信息

  • 批准号:
    2011490
  • 负责人:
  • 金额:
    $ 14.53万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2020
  • 资助国家:
    美国
  • 起止时间:
    2020-08-01 至 2024-07-31
  • 项目状态:
    已结题

项目摘要

Many mathematical models used to describe and predict behavior of physical, biological, chemical, and financial systems lead to systems of equations for which the problem coefficients depend on an unknown solution. These are known as nonlinear problems, and they are solved iteratively, by generating a sequence of successive approximations. For many such problems, even state of-the-art solution methods can be slow, can fail, and may not be robust with respect to changes in the underlying problem data. This project aims to develop faster and more reliable iterative solution techniques using methods that recombine information from previous approximations to create a more accurate next approximation. Theory will be developed to mathematically show how these methods improve current solution techniques, and the improved methods will be demonstrated on a wide range of systems that arise from important practical problems in optics and fluid mechanics. This project provides research training for graduate students.The efficient solution of systems of nonlinear equations is essential to the high-fidelity simulation technology necessary for predictive physical modeling throughout engineering and the life sciences. An extrapolation technique commonly referred to as Anderson acceleration (AA) has been known since 1965 to often improve the efficiency and robustness of iterative solvers for nonlinear problems. It has been successfully used in a surprisingly wide variety of applications, however theoretical understanding of its convergence properties remains largely open. Better theoretical understanding of mathematical algorithms is fundamentally important for both practical implementation and for the creation of the next generation of algorithms. The aim of this proposal is to improve theoretical understanding for AA, and to develop robust and efficient variants with improved convergence properties, both in general settings and for specific nonlinear PDEs. The main theoretical components are (1) the analysis of a variant using principal component analysis; (2) the design and analysis of robust adaptive damping and algorithmic depth strategies for noncontractive operators; (3) the analysis of the superlinear convergence of accelerated Newton iterations for degenerate problems. The proposed work will include theory and practical application of AA to several difficult nonlinear PDEs from fluid mechanics and optics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
许多用于描述和预测物理、生物、化学和金融系统行为的数学模型导致了问题系数依赖于未知解的方程组。这些问题被称为非线性问题,它们通过生成一系列逐次逼近来迭代求解。对于许多这样的问题,即使是最先进的解决方案方法也可能很慢,可能会失败,并且对于潜在问题数据的更改可能不是很健壮。这个项目旨在开发更快和更可靠的迭代求解技术,使用重新组合以前近似信息的方法来创建更准确的下一次近似。理论将发展成数学展示这些方法如何改进当前的求解技术,改进的方法将在光学和流体力学中的重要实际问题产生的广泛系统上进行演示。该项目为研究生提供研究培训。有效地求解非线性方程组是整个工程和生命科学中预测物理建模所必需的高保真模拟技术的关键。自1965年以来,一种通常被称为安德森加速(AA)的外推技术已为人所知,它经常提高非线性问题迭代求解器的效率和稳健性。它已经被成功地应用于各种令人惊讶的应用中,然而对它的收敛性质的理论理解在很大程度上仍然是开放的。更好地从理论上理解数学算法对于实际实施和创建下一代算法都是至关重要的。这一建议的目的是提高对AA的理论理解,并开发出稳健而有效的变体,无论是在一般设置下还是对于特定的非线性偏微分方程组,都具有更好的收敛特性。主要的理论内容包括:(1)基于主成分分析的变型分析;(2)非压缩算子的鲁棒自适应阻尼和算法深度策略的设计与分析;(3)退化问题加速牛顿迭代的超线性收敛分析。拟议的工作将包括AA在流体力学和光学领域的几个困难的非线性PDE的理论和实际应用。该奖项反映了NSF的法定使命,并通过使用基金会的智力优势和更广泛的影响审查标准进行评估,被认为值得支持。

项目成果

期刊论文数量(11)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations
  • DOI:
    10.3934/era.2020113
  • 发表时间:
    2020-06
  • 期刊:
  • 影响因子:
    0
  • 作者:
    M. Gardner;Adam Larios;L. Rebholz;Duygu Vargun;C. Zerfas
  • 通讯作者:
    M. Gardner;Adam Larios;L. Rebholz;Duygu Vargun;C. Zerfas
An energy, momentum, and angular momentum conserving scheme for a regularization model of incompressible flow
不可压缩流正则化模型的能量、动量和角动量守恒方案
  • DOI:
    10.1515/jnma-2020-0080
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    3
  • 作者:
    Ingimarson, Sean
  • 通讯作者:
    Ingimarson, Sean
Anderson acceleration for contractive and noncontractive operators
  • DOI:
    10.1093/imanum/draa095
  • 发表时间:
    2019-09
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Sara N. Pollock;L. Rebholz
  • 通讯作者:
    Sara N. Pollock;L. Rebholz
Acceleration of nonlinear solvers for natural convection problems
  • DOI:
    10.1515/jnma-2020-0067
  • 发表时间:
    2020-04
  • 期刊:
  • 影响因子:
    3
  • 作者:
    Sara N. Pollock;L. Rebholz;Mengying Xiao
  • 通讯作者:
    Sara N. Pollock;L. Rebholz;Mengying Xiao
Anderson Acceleration of Nonlinear Solvers for the Stationary Gross-Pitaevskii Equation
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Leo Rebholz其他文献

Leo Rebholz的其他文献

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{{ truncateString('Leo Rebholz', 18)}}的其他基金

Collaborative Research: Laboratory Data Enabled Phase Field Modeling and Data Assimilation for Coupled Two-Phase Fluid Flow and Porous Media Flow
合作研究:耦合两相流体流和多孔介质流的实验室数据支持相场建模和数据同化
  • 批准号:
    2152623
  • 财政年份:
    2022
  • 资助金额:
    $ 14.53万
  • 项目类别:
    Continuing Grant
Collaborative Research: Variational Structure Preserving Methods for Incompressible Flows: Discretization, Analysis, and Parallel Solvers
合作研究:不可压缩流的变分结构保持方法:离散化、分析和并行求解器
  • 批准号:
    1522191
  • 财政年份:
    2015
  • 资助金额:
    $ 14.53万
  • 项目类别:
    Standard Grant
Eighth Annual Graduate Student Mini-conference in Computational Mathematics; Clemson, SC; February 5-6, 2016
第八届计算数学研究生小型会议;
  • 批准号:
    1547107
  • 财政年份:
    2015
  • 资助金额:
    $ 14.53万
  • 项目类别:
    Standard Grant
5th Annual Graduate Student Mini-conference in Computational Mathematics
第五届计算数学研究生小型会议
  • 批准号:
    1245607
  • 财政年份:
    2012
  • 资助金额:
    $ 14.53万
  • 项目类别:
    Standard Grant
Improved Methods for Incompressible Viscous Flow Simulation
不可压缩粘性流模拟的改进方法
  • 批准号:
    1112593
  • 财政年份:
    2011
  • 资助金额:
    $ 14.53万
  • 项目类别:
    Standard Grant
Enabling Long-Time Accuracy in Turbulent Flow Simulations
实现湍流模拟的长期精度
  • 批准号:
    0914478
  • 财政年份:
    2009
  • 资助金额:
    $ 14.53万
  • 项目类别:
    Standard Grant

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