Fast Optimization Methods and Application to Data Science and Nonlinear Partial Differential Equations

该项目结合了优化方法和非线性多重网格方法的一些最新发展，以提供一种新的技术来提高实际应用的计算效率。我们的快速优化方法的成功集成将打开一个广泛的新的应用领域，从偏微分方程的数值解到大规模机器学习的优化方法。Facebook和GitHub等社交媒体将用于传播应用和计算数学的基础知识，并向学术界和工业界更广泛的受众推广研究，以及提高公众对计算数学如何帮助推进其他物理和数据科学研究的认识。本项目将为研究生提供培训机会，重点研究一种特殊的非线性多重网格方法，即快速子空间下降（FASD）方法，用于解决偏微分方程数值解和数据科学问题等各种应用中的优化问题。例如，待研究的非线性多重网格方法可以解决工程应用中的挑战性问题，包括相场模型中的梯度流，数学生物学中的Poisson-Boltzmann方程，以及数据科学中的凸复合优化问题。加速度是现代优化理论中最富有成效的思想之一。该框架为新旧优化方法的设计和分析带来了更多的见解和数学工具，特别是加速梯度下降方法。该项目的另一个重要方面是为一大类优化方法提供严格的理论基础。该奖项反映了NSF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。

This projects incorporates several recent developments in optimization methods and nonlinear multigrid methods to provide a new technique to improve the computational efficiency of practical applications. Successful integration of our fast optimization methods will open a wide new area of applications ranging from numerical solution of partial differential equations to optimization methods for large-scale machine learning. Social media such as Facebook and GitHub will be used to disseminate basics on applied and computational mathematics and promote the research to a wider audience in both academia and industry, as well as increase the public awareness of how computational mathematics help the advancement of research in other physical and data sciences. This project will provide training opportunities for graduate students.The project focuses on a particular nonlinear multigrid method, the fast subspace descent (FASD) method, for solving optimization problems arising from various applications such as numerical solution of partial differential equations and data science problems. For example, the nonlinear multigrid methods to be studied can address the challenging problems in engineering applications including gradient flow in phase field models, Poisson-Boltzmann equation in math biology, and convex composite optimization problems in data science. Acceleration has been one of the most productive ideas in modern optimization theory. This framework brings more insight and mathematical tools for the design and analysis of old and new optimization methods, especially the accelerated gradient descent methods. Another important aspect of this project will be the rigorous theoretical foundation for a large class of optimization methods.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.