Hilbert空间中流形上泛函极小值点的寻找方法——Ⅰ流形的恢复算子和近似曲线

Computational Methods for Finding Minimum Point of a Functional over a Manifold in Hilbert Space——Part 1. Restoration Operators and Approximate Curves of a Manifold

中文摘要:

      本文首先构造了Hilbert空间中流形的一类恢复算子,讨论了它的一些性质。然后利用恢复算子构造性地建立了流形的局部参数表示,为在Hilbert空间中寻找流形上泛函的极小值点提供构造计算方法和实现算法的依据。 另外,本文还构造了一类流形的近似曲线,利用它可以将通常的直线寻找改换成曲线寻找。为算法的实现提供一个工具。

英文摘要:

      In this paper, we first construct a class of restoration operators of a manifold in Hilbert space, anc discuss some properties of them. Secondly, the local parameter presentation of the manifold is built constructively by the restoration operators. This presentation will enable us to construct and implement the computational methods for finding minimum point of a functional over the manifold. Finally, this paper also constructs a class of approximate curves of the manifold, which providea a tool for obtaining implementoble algorithms. In these algorithms, we can apply the search along curvilinear paths rather than straight lines.