Numerical Analysis for Ill-posed Problems Related with Engineering

在目前的研究中，我们主要把反问题作为我们的目标不适定问题。所谓“不适定问题”，是指Hadamard意义下的不适定问题;所研究的微分方程是Hadamard意义下的不适定问题，其解对于给定的数据几乎没有连续性。在数值分析中，这种不适定性意味着数值解的不稳定性，也意味着用常规的数值方法进行可靠的数值计算是不可能的。G.非破坏性检测等方法是近年来工程中比较流行的方法，但它们几乎都是上述意义下的不适定问题。用通常的数值方法几乎不可能给出很好的数值解，我们需要一些新的思想来处理它们。本文从这一观点出发，给出了一些结果，在上述情况下，我们仅限于对不适定问题进行数值分析的理论方法。Masahiro Yamamoto博士是我们研究的研究者之一，他提出了不适定问题的广义适定性，并从泛函分析中证明了目标问题解的唯一性意味着其稳定性。它是指数值解在弱意义下的稳定性，我们把反问题的唯一性研究作为研究的主要内容之一。久保昌弘博士对双曲型反问题的唯一性给出了一个非常重要的结果.另外。针对这一唯一性，本文研究了Tikhonov正则化方法、Toxie博士提出的滤波方法和Iso博士提出的多精度技术的应用，每种方法都被认为是不适定问题数值分析中一种合适的新方法。

We treat mainly, in the present research, inverse problems as our aimed ill-posed problems. We mean "ill-posed problems" by the not well-posed problems in the sense of Hadamard ; the aimed differential equations are not well-posed in the sense of Hadamard and solutions have almost no continuity in connection with the given data. The ill-posedness implies instability of numerical solutions in numerical analysis for our problems, and it means impossibility of reliable numerical computation by usual numerical methods applied to the problems.Inverse Problems , e. g. non-destructive test, are popular in the recent engineering, but almost all of them are ill-posed in the above sense. It is almost impossible to give good numerical solutions for the problem by usual numerical techniques, and we need some new idea to treat them. We give some results from this view point in the present research.Under the circumastance stated above, we restrict ourselves for theoretical approach for numerical analysis for ill-posed problems. Dr. Masahiro Yamamoto, who is one of the investigators of our research, proposed generalized well-posedness for ill-posed problems, and he shows the uniqueness of the solution for an aimed problem implies its stability from the functional analysis. It means stability of numerical solution in a weak sense, and we put the research of uniqueness in inverse problems as one of the main topics in the research. Dr.Masahiro Kubo gives a very important result for the uniqueness in the hyperbolic inverse problems. In addition. to the uniqueness, other investigators of the present research study application of Tikhonov regularization methods, the filter methods proposed by Dr.Tosaka and multi-precision techniques proposed 5y Dr.Iso, and each method is regarded as a suitable new method in numerical analysis for ill-posed problems.