Numerical Analysis for Ill-posed Problems Related with Engineering

在目前的研究中，我们主要把逆问题作为病态问题来处理。我们用Hadamard意义上的非适定问题来表示“不适定问题”；目标微分方程在Hadamard意义上不是适定的，其解与给定数据之间几乎没有连续性。病态性意味着问题数值分析中数值解的不稳定性，意味着用通常的数值方法无法进行可靠的数值计算。反问题，如无损检测，是近年来工程中比较流行的问题，但几乎所有的反问题在上述意义上都是病态的。通常的数值方法几乎不可能给出这一问题的数值解，我们需要一些新的思想来处理它。本文从这一观点出发，给出了一些研究结果。在上述情况下，我们将自己局限于不适定问题数值分析的理论方法。我们的研究人员之一Masahiro Yamamoto博士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。它在弱意义上是数值解的稳定性，我们把反问题的唯一性研究作为研究的主要课题之一。久保正弘博士给出了双曲型反问题唯一性的一个重要结果。此外。由于其独特性，本研究的其他研究者研究应用了Tikhonov正则化方法、toaka博士提出的滤波方法和iso博士提出的多精度技术，每种方法都被认为是病态问题数值分析的一种合适的新方法。

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.