Mathematical analysis for inverse problems in mathematical scienes and establishment of numerical methods

在反问题中，人们必须根据边界上现有的数据来确定物体内部的物理性质，并从结果中找出原因，因此对各种反问题的研究在数学科学和工业等许多领域都变得重要起来。随着人们对反问题重要性的更好认识，以及计算机和观测设备的迅速改进，对反问题的数值方法和数学分析的发展提出了更高的要求。对于正问题来说，更有必要改进数值方法，使其摆脱传统的方法。然而，即使是反问题的数学研究，作为相关数值方法的基础，也还没有得到充分的研究。工业中的一个重要的反问题是风险管理：一个目标是通过对对象的内部状态进行适当的评估来实现对对象的最优控制，并且具有内在的不稳定性。在物理上可以接受的先验条件下，可以恢复稳定性，这称为条件稳定性。因此，必须选择合适的稳定方法，而不能采用常规方法。为了获得合理的数值性能，必须选择数值方法，以保证精度与原始反问题的条件稳定程度相对应。因此，必须建立条件稳定性结果。在这项研究中，我们在开发反问题的数学研究的基础上发展了数值方法。我们的数学结果是显着的各种数学知识，如复分析和偏微分方程组。此外，与该行业的联系更深，申请了一项专利。

In inverse problems, one has to determine physical properties of the interior of an object by available data on the boundary and to identify causes from results, and researches for various inverse problems become important in many fields such as mathematical sciences and industrie. The development of numerical methods as well as the mathemaical analyses for inverse problems become more requested, because the importance of the inverse problem is better recognized and computers and observation equipments are improved rapidly. It is more necessary for one to improve numerical methods which are free from the conventional manners for the forward problem. However even the mathematical researches for inverse problems on which such relevant numerical methods should be based, are not yet sufficiently done. One of important inverse problems in industry is for the risk management : one aims at the optimal control of a plant by means of suitable evaluation of the interior states of the plant, and has intrinsic instability. With physically acceptable a priori conditions, one can recover stability, which is called the conditional stability. Therefore one has to choose suitabl stabilizing methods, and should not apply conventional methods. For reasonable numerical performances, one has to choose numerical methods guaranteeing the accuracy which corresponds to the degree of conditional stability of the original inverse problem. Thus one must establish conditional stability results. In this research, we have developed numerical methods on the basis of exploited mathematical researches for inverse problems. Our mathematical results are remarkable by various mathematical knowledge such as complex analysis and partial differential equations. Moreover the link with the industry is deeper and one patent was applied.