Computational Methods for Parametrized Equations

这个项目涉及计算方法的设计和研究 对于涉及多个非线性方程组的解， 维参数向量 这种系统经常出现在 在科学和工程中的平衡问题，一般来说， 兴趣集中在计算评估的变化， 参数变化下的解。 这项工作利用了这一点 在通常在实践中成立的条件下， 解在组合状态下形成可微流形， 参数空间 以前已经表明， 微分几何的概念和技术提供了一个新的， 用于开发所需计算方法的强大工具 用于分析解流形的形状和性质。 这 该项目将研究多方面的进一步发展- 近似算法，由此产生的计算误差和 简化基的计算解的灵敏度测量 方法及其在多参数情况下的误差，以及 求解常微分方程和某些偏微分方程 隐式定义的流形。

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.