Computational Methods for Parametrized Equations

本研究的目的是继续研究非线性、参数相关方程的有效计算方法。这些方程包含一个状态变量，通常是无限维的，一个有限维的参数向量。在适当的条件下，得到了状态空间和参数空间的解。这类问题出现在物理和工程中许多实际重要的平衡研究中。例如，该方程可以描述机械结构的平衡位置，在这种情况下，状态变量表征，例如，变形，而参数向量包含有关载荷点，载荷方向，材料特性，几何数据等信息。在这种情况下，正如在这种类型的大多数应用中一样，兴趣很少集中在计算固定参数选择的几个解上，而是必须解决通常称为灵敏度问题的问题。这涉及到参数变化时解的变化问题。特别是，需要确定解的性质发生变化时的参数值，例如，当稳定性丧失时。因此，本质上，灵敏度问题要求对解流形的形状和性质进行计算研究。解流形存在的假设假定问题已适当展开。在大多数应用中，这是一个非常自然的假设，对于参数化方程的数值解具有相当大的优势。到目前为止，数值分析文献很少关注多参数非线性方程的微分几何方面。但是，对微分几何概念和结果的一致利用可以为统一分析参数化方程解的性质提供有效的计算方法的发展的有力工具。

The purpose of this investigation is to continue work on the development of effective computational methods for nonlinear, parameter dependent equations. The equations involve a state variable, which is usually infinite dimensional, a finite dimensional parameter vector. Under appropriate conditions, the solutions of the state space and the parameter space. Problems of this type arise in numerous, practically important equilibrium studies in physics and engineering. For example, the equation may describe the equilibrium positions of a mechanical structure, in which case the state variable characterizes, say, the deformations while the parameter vector incorporates information about the load points, load directions, material properties, geometrical data, etc. In this, as in most applications of this type, interest centers rarely on the computation of a few solutions for fixed choices of the parameters, but instead one has to address, what is often called, the sensitivity problem. This concerns the question of the change of the solutions under variation of the parameters. In particular, one needs to determine those values of the parameters where the character of the solution changes, for instance, where stability is lost. Thus, in essence, the sensitivity problem requires a computational study of the shape and properties of the solution manifold. The assumption of the existence of the solution manifold presumes that the problem has been suitably unfolded. This turns out to be a very natural assumption in most applications and to have considerable advantages for the numerical solution of parametrized equation. Up to now, the numerical analysis literature has paid little attention to this differential-geometric aspect of multi-parameter nonlinear equations. But a consistent utilization of differential-geometric concepts and results can provide a powerful tool in the development of effective computational methods for a unified analysis of the solution properties of the parametrized equation.