Mathematical Sciences: Optimization Problems in Function Spaces

这个项目将研究大规模优化问题和由无限维问题的离散化产生的非线性方程的算法的设计和分析。目标包括发展逐点拟牛顿方法的统一理论，研究在传统和逐点情况下无限维的对称一阶方法，分析一类新的方法，这些方法使用梯度有限差分计算中的步骤作为工具来避免简单光滑函数和小幅度、高频、噪声项，以及无限维问题的全局收敛算法。基础研究将应用于辐射传输、微波器件的计算机辅助设计和竞争系统优化的数值方法等问题。

This project will study the design and analysis of algorithms for the solution of large scale optimization problems and nonlinear equations that arise from discretizations of infinite dimensional problems. The goals include development of a unifying theory for pointwise quasi-Newton methods, study of the symmetric rank one method in infinite dimensions in both the conventional and pointwise contexts, analysis of a new class of methods that use the step in finite difference computation of gradients as a tool to avoid local minima for problems that are sums of simple smooth functions and small amplitude, high frequency, noise terms, and globally convergent algorithms for infinite dimensional problems. The basic research will be applied to problems in radiative transfer, computer sided design of microwave devices, and numerical methods for optimization of competitive systems.