INVERSE PROBLEMS OF SCATTERING : MATHEMATICS,NUMERICAL ANALYSIS AND GRAPHICS

关于薛定谔算子的反问题，矶崎健三进行了深入的研究，得到了重要的结果。Isozaki从Faddeev的思想出发，阐明了薛定谔算符的势与散射振幅之间的关系。Isozaki所示的这种关系给出了薛定谔算子逆问题的解的表示。他被邀请参加了几次国际会议，并得到了很高的评价。关于我们研究的另一个核心，即数值分析，我们引进了Silicon Graphic有限公司的计算机，Chalenge靛蓝2和Indy。在坂根的领导下，与研究生（特别是与仙台）合作，我们研究了获得Greobner基础的算法。不用说，Greobner基在多项式环的各种问题中起着至关重要的作用。但为了针对给定的具体问题显式地获得Greobner基，这将成为一个巨大的计算并且需要大量的时间。因此，改进算法成为一种有效的算法是非常重要的，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，在我们开始逆散射问题的数值分析之前，我们的研究已经结束了。我们将继续进行反问题的数值研究.

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.