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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：
2011519
负责人：
Sara Pollock
金额：
$ 17.54万
依托单位：
University of Florida
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

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2011519&HistoricalAwards=false
关键词：
Collaborative Research Advancing Theoretical Understanding

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 which 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 opportunities 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问题系数依赖于未知解的方程组。这些被称为非线性问题，它们是通过产生一系列连续的近似来迭代地解决的。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，并且对于底层问题数据的变化可能不够健壮。该项目旨在开发更快，更可靠的迭代解决技术，使用从以前的近似中重新组合信息的方法来创建更准确的下一个近似。理论将发展，以数学方式显示这些方法如何改进当前的解决技术，改进的方法将在光学和流体力学中产生的重要实际问题的广泛系统中进行演示。本项目为研究生提供研究训练机会。非线性方程组的有效解对于整个工程和生命科学中预测物理建模所需的高保真仿真技术至关重要。自1965年以来，一种通常被称为安德森加速（AA）的外推技术已经被人们所知，它经常提高非线性问题迭代求解器的效率和鲁棒性。它已经成功地应用于各种各样的应用中，然而，对其收敛性质的理论理解仍然很大程度上是开放的。对数学算法更好的理论理解对于实际实现和下一代算法的创建都是至关重要的。本提案的目的是提高对AA的理论理解，并开发具有改进收敛特性的鲁棒和高效变体，无论是在一般设置还是特定的非线性偏微分方程。主要理论成分有：(1)用主成分分析法对变量进行分析；(2)非收缩算子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性分析。提出的工作将包括理论和实际应用的几个困难的非线性偏微分方程从流体力学和光学。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。