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

许多数学模型用于描述和预测物理、生物、化学和金融系统的行为，导致方程组的问题系数取决于未知的解。这些被称为非线性问题，并且通过生成一系列连续近似来迭代求解。 对于许多这样的问题，即使是最先进的解决方法也可能很慢，可能失败，并且可能对于底层问题数据的变化不鲁棒。该项目旨在开发更快，更可靠的迭代求解技术，使用重新组合以前近似的信息以创建更准确的下一个近似的方法。 理论将发展到数学上显示这些方法如何改善当前的解决方案技术，并将在光学和流体力学中的重要实际问题所产生的广泛系统上证明改进的方法。本项目为研究生提供研究训练的机会。非线性方程组的有效解对于整个工程和生命科学领域的预测物理建模所需的高保真模拟技术至关重要。 自1965年以来，通常称为安德森加速（AA）的外推技术已经被公知，以经常提高用于非线性问题的迭代求解器的效率和鲁棒性。它已成功地用于各种各样的应用程序，但其收敛性的理论理解仍然很大程度上是开放的。更好地理解数学算法的理论对于实际实现和下一代算法的创建都非常重要。本建议的目的是提高理论的理解AA，并开发强大的和有效的变种，改善收敛性能，无论是在一般设置和特定的非线性偏微分方程。主要的理论组成部分是（1）使用主成分分析的变体的分析;（2）非压缩算子的鲁棒自适应阻尼和算法深度策略的设计和分析;（3）退化问题的加速牛顿迭代的超线性收敛性分析。 拟议的工作将包括理论和实际应用AA的几个困难的非线性偏微分方程从流体力学和optics.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.