Computational Methods for Parametrized Equations

本项目涉及涉及多维参数向量的非线性方程组的求解的计算方法的设计和研究。这类系统经常出现在科学和工程中的平衡问题中，通常，人们的兴趣集中在对参数变化时解的变化的计算评估上。这项工作利用了在实际中通常存在的条件下，所有解的集合在组合状态和参数空间中形成可微流形。以前已经证明，微分几何概念和技术的一致使用为发展所需的分析解流形的形状和性质的计算方法提供了一种新的强有力的工具。这个项目将研究流形近似算法的进一步发展，计算解的计算误差和灵敏度度量，多参数情况下的约化基法及其误差，以及求解隐式定义流形上的常微分方程组和某些偏微分方程组的方法。

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.