Iterative Methods for Nonlinear Equations and Optimization

计算模拟在社会中无处不在。例如，汽车、飞机和手机不仅需要高质量的模拟器，而且需要高质量的模拟器来运行。在这个项目中，PI将研究非线性解算器，这是许多模拟器的核心组件。他将考虑理论不完整或缺失的问题类别和解决方案。对这些问题和求解器的理论理解将有助于提高求解器和使用这些求解器的模拟器的性能和可靠性。PI将把他的发现应用于材料设计基础的物理和化学问题。本课题解决了数值分析理论中一些尚未解决的算法问题。这些问题具有重要的应用。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，这项工作尤其及时。本课题的进展将有助于更好地理解如何将现有的求解器与蒙特卡罗模型相结合。安德森加速度的研究将探索一类无雅可比矩阵的解，这类解在雅可比矩阵或雅可比向量积不可用的情况下是非常重要的。Anderson已经广泛应用于电子结构计算，而电子结构计算又是计算化学软件和材料设计的核心组件。虽然关于安德森加速度的使用和操作的文献很多，包括它不能收敛的情况，但理论很少。这部分项目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.