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

数学领域反问题的数学分析和数值方法的建立

基本信息

批准号：
15340027
负责人：
YAMAMOTO Masahiro
金额：
$ 8.9万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

项目摘要

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.

在反问题中，人们必须通过边界上的可用数据来确定物体内部的物理性质，并从结果中找出原因，各种反问题的研究在数学科学和工业等许多领域变得重要。随着人们对反问题重要性的认识日益加深，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，更有必要对摆脱传统方法的数值方法进行改进。然而，即使是作为相关数值方法基础的反问题的数学研究，也还不够充分。风险管理是工业上一个重要的逆问题，它通过对工厂内部状态的适当评价来达到对工厂的最优控制，具有内在的不稳定性。在物理上可接受的先验条件下，人们可以恢复稳定性，这被称为条件稳定性。因此，必须选择合适的稳定方法，而不应采用常规方法。为了获得合理的数值性能，必须选择与原反问题的条件稳定程度相对应的精度保证的数值方法。因此，必须建立条件稳定性结果。在此研究中，我们在反问题数学研究的基础上发展了数值方法。我们的数学成果是显著的各种数学知识，如复分析和偏微分方程。与行业联系更紧密，申请专利1项。