INVERSE PROBLEMS OF SCATTERING : MATHEMATICS,NUMERICAL ANALYSIS AND GRAPHICS

关于薛定谔算子的逆问题，isozaki进行了深入的研究，得到了非常重要的结果。矶崎从Faddeev的思想出发，明确了薛定谔算符的势与散射振幅的关系。Isozaki所显示的这个关系给出了薛定谔算子逆问题解的一个表示。他被邀请参加了几次国际会议，并得到了很高的评价。对于我们研究的另一个核心，即数值分析，我们引入了Silicon Graphic Co.Ltd.的计算机。， challenge Indigo 2和Indy。在Sakane老师的带领下，在研究生（特别是与Senda）的合作下，我们研究了获得Greobner基的算法。不用说，Greobner基在与多项式环有关的各种问题中起着重要的作用。但是为了明确地得到给定具体问题的格里奥布纳基，它的计算量很大，而且需要花费很多时间。因此，将算法改进为高效算法是非常重要的。我们成功地提出了一种新的算法，称为GRASIS，它使我们能够比以前更快地实现Greobner基的计算。作为我们发明的新算法GRASIS的一个应用，我们发现了新的爱因斯坦度量。我们必须承认，在开始对逆散射问题进行数值分析之前，我们的研究已经结束了。我们将继续反问题的数值研究。

Concerning the inverse problems for Schrodinger operatores, H.Isozaki studied deeply and got very important results. Starting from the idea of Faddeev, Isozaki made clear the relationship between the potential of Schrodinger operators and the scattering amplitude. This relationship shown by Isozaki gives a representation of the solution of the inverse problem for the Schrodinger operators. He was invited to several international conferences and was given a high evaluation.Concerning the another core of our research, that is, numerical analysis, we introduced computers of Silicon Graphic Co.Ltd., Chalenge Indigo 2 and Indy. By the leadership of Sakane, with cooperation of graduate students (especially with Senda), we studied algorism for getting Greobner basis. Needless to say, Greobner basis play an essential role in various problems related to polynomial rings. But in order to get Greobner basis explicitely for given concrete problems, it becomes a huge computation and it takes a lot of time. So, it is very important to improve algorithm into a efficient ones.We succeeded to make a new algorism, named GRASIS,which enable us to acheive computation of Greobner basis much faster than before. As an application of our invention of new algorism GRASIS,we found new Einstein metrics.We have to confess that the term of our research has ended before we set about numerical analysis of inverse scattering problems. We shall continue the numerical study of inverse problems.