Computational Methods for Parametrized Equations

这个项目涉及到多维参数向量非线性方程组解的计算方法的设计和研究。这种系统经常出现在科学和工程中的平衡问题中，通常关注的是在参数变化下解的变化的计算评估。本文利用在实际中通常成立的条件下，所有解的集合在状态空间和参数空间的组合中形成一个可微流形。以前已经表明，微分几何概念和技术的一致使用为分析解流形的形状和性质所需的计算方法的发展提供了一个新的和强大的工具。本项目将研究流形近似算法的进一步发展，计算结果的计算误差和计算解的灵敏度测量，多参数情况下的降基方法及其误差，以及求解隐定义流形上的普通和某些偏微分方程的方法。

This project concerns the design and study of computational methods for the solution of nonlinear systems of equations involving multi- dimensional parameter vectors. Such systems arise frequently in equilibrium problems in science and engineering where, generally, interest centers on the computational assessment of the change of the solutions under variation of the parameters. The work utilizes that under conditions, which typically hold in practice, the set of all solutions forms a differentiable manifold in the combined state and parameter space. It has been shown previously that a consistent use of differential geometric concepts and techniques provides a new and powerful tool for the development of the desired computational methods for analyzing the shape and properties of the solution manifold. This project will investigate the further development of a manifold- approximation algorithms, of the resulting computational errors and of sensitivity measures for the computed solutions, of the reduced basis method and its errors in the multi-parameter case, and of methods for solving ordinary and certain partial differential equations on implicitly defined manifolds.