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

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。该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。这个项目为研究生提供了研究培训的机会。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。

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.