Iterative Methods for Nonlinear Equations and Optimization

计算模拟在社会中是普遍存在的。例如，汽车、飞机和移动的电话不仅需要高质量的模拟器来进行设计，还需要高质量的模拟器来进行操作。在这个项目中，PI将研究非线性求解器，这是许多模拟器的核心组件。他将考虑类的问题和解决方案的理论是不完整或失踪。 对这些问题和求解器的理论理解将导致求解器和使用这些求解器的模拟器的性能和可靠性的改进。PI将把他的发现应用于材料设计的物理和化学问题。本计画针对数值分析理论中一些尚未解决的算法问题。这些问题有重要的应用。在目标函数或残差中嵌入蒙特卡罗方法的优化和非线性求解器的工作是一个非常新的研究方向。鉴于蒙特卡罗方法在大规模并行环境中的鲁棒性和弹性，这项工作特别及时。本主题的进展将有助于更好地理解如何将现有的求解器耦合到Monte Carlo模型。对安德森加速度的研究将探索一类无雅可比矩阵的求解器，这类求解器在雅可比矩阵或雅可比向量积不可用时非常重要。安德森已经被广泛用于电子结构计算，而电子结构计算又是计算化学软件和材料设计的核心组成部分。虽然有广泛的文献使用和操作的安德森加速，包括情况下，它无法收敛，有很少的理论。 本项目这一部分的目的是发展这一理论。

Computation simulation is pervasive in society. Automobiles, aircraft, and mobile phones, for example, need high-quality simulators not only for their design, but also for their operation. In this project the PI will investigate nonlinear solvers, which are core components of many simulators. He will consider classes of problems and solvers for which theory is incomplete or missing. A theoretical understanding of these problems and solvers will lead to improved performance and reliability of the solvers and the simulators which use those solvers. The PI will apply his findings to problems in physics and chemistry basic to the design of materials. This project addresses some unresolved algorithmic questions in numerical analysis theory. These problems have important applications. The work on optimization and nonlinear solvers with embedded Monte Carlo methods in the objective function or residual is a very new line of research. The work is particularly timely in view of Monte Carlo methods robustness and resiliency in a massively parallel environment. Progress in this topic will lead to better understanding of how to couple existing solvers to Monte Carlo models. The research on Anderson acceleration will explore a class of Jacobian-free solvers which are very important when Jacobians or Jacobian vector products are unavailable. Anderson is already widely used in electronic structure computations which, in turn, are core components in computational chemistry software and the design of materials. While there is wide literature on the use and operation of Anderson acceleration, including cases where it fails to converge, there is very little theory. The object of this part of the project is to develop that theory.