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Establishment of New Numerical Methods for Applied Inverse and Ill-Posed Problems

应用逆问题和不适定问题的新数值方法的建立

基本信息

批准号：
16340024
负责人：
ISO Yuusuke
金额：
$ 10.3万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

The aim of this research project is mathematical analysis and numerical analysis of ill-posed problems written in partial differential equations connecting with applied inverse problems which are important in physics, medical science, and engineering. Especially, considering the future requirement in practice, it is one of our originalities that we have developed a new fast multiple-precision arithmetic environment for the sake of large scale numerical computation of the ill-posed problems with high accuracy, in addition to mathematical theory and algorithms.In the scientific computations including numerical simulations of inverse problems, approximation by floating-point arithmetic are usually used in representation and arithmetic of real numbers on digital computers. Nowadays the double precision arithmetic defined in the IEEE754 standard is the common way. This means that scientific numerical computations are carried out on the assumption that real numbers have 15 decimal digits acc … More uracy in the usual end-user environments. In the floating-point arithmetic we cannot omit rounding errors and cannot treat real numbers exactly on the digital computers. Of course we must also take discretization errors into account which appear in discretization of functional equations and partial differential equations in numerical computations. In ill-posed problems which typically appear in inverse problems, the error is fatal defect for reliable numerical computations. This is the most different point between well-posed problems which induce stable numerical schemes. Conventional numerical analysis for ill-posed problems treated only discretization errors or measurement errors, and consideration of rounding errors is not enough. The most significant points of our research is development of a new multiple-precision arithmetic in discussion on rounding errors besides the conventional numerical analysis for discretization errors and measurement errors. In the multiple-precision arithmetic environment, the new aspects have been found in high accurate discretization of functional equations, and new computational schemes have been developed and established in the project.One of the concrete results is the fast multiple-precision arithmetic environment "exflib", which was designed and implemented in the predecessor research, has been improved by co-researcher Prof. Hiroshi Fujiwara, who has succeed in implementation of special functions and in porting to supercomputers to treat scientific numerical simulations. We also apply the spectral methods, which achieve quite high accurate numerical solutions than the conventional discretization methods. Combining the multiple-precision arithmetic and the spectral methods, we have proved the proposed approach is quite effective for numerical analysis of ill-posed problems. And we give a remark on the regularization method under high accurate numerical methods, especially the relation between measurement errors, regularization parameters, and computation precisions. The remark is important in practical applied inverse problems in which we must take measurement error into account.Each problem has its own ill-posedness. Because the matter is different in each setting in inverse problems, we place mathematical analysis for inverse problems as fundamental subjects in the project and we discuss uniqueness and conditional stability of solutions. Co-researcher Professor Masahiro Yamamoto obtain sharp results in inverse scattering problems. In application of the results in mathematical and numerical analysis to practical problems, we need the fundamental research from the computational mechanics viewpoints. All co-researchers have discussed applied inverse problems in their fields. We also discuss computer aided proof and succeed in numerical verification techniques which is one of the applications of the fast multiple-precision arithmetic. Less

本研究项目的目的是数学分析和数值分析的不适定问题写在偏微分方程连接应用反问题，这是重要的物理学，医学科学和工程。特别是考虑到未来的实际需要，我们的创新之处之一是在原有的数学理论和算法的基础上，开发了一种新的快速多精度算法环境，以满足对不适定问题进行大规模高精度数值计算的需要。在数字计算机上真实的数的表示和运算中，通常使用浮点运算的近似。IEEE754标准中定义的双精度算法是目前常用的算法。这意味着科学的数值计算是在假设真实的数有15位十进制数的前提下进行的。 ...更多信息在通常的最终用户环境中的高精度。在浮点运算中，我们不能忽略舍入误差，也不能在数字计算机上精确地处理真实的数。当然，我们也必须考虑离散误差，出现在离散的函数方程和偏微分方程的数值计算。不适定问题是反问题中的一个典型问题，其误差是可靠数值计算的致命缺陷。这是诱导稳定的数值方案的适定问题之间的最大不同点。传统的不适定问题的数值分析只考虑离散误差或测量误差，对舍入误差的考虑不够。本研究的重点在于除了传统的离散化误差与量测误差的数值分析外，发展一种新的多精度算法来讨论舍入误差。在多精度运算环境中，本课题在函数方程的高精度离散化方面发现了新的方面，开发并建立了新的计算方案，具体成果之一是，共同研究员藤原弘教授对前期研究中设计并实现的快速多精度运算环境“exflib”进行了改进，他成功地实现了特殊功能，并移植到超级计算机上进行科学数值模拟。我们还应用谱方法，实现相当高的精度数值解比传统的离散方法。将多精度算法与谱方法相结合，证明了该方法对不适定问题的数值分析是十分有效的。并对高精度数值方法下的正则化方法进行了评述，特别是讨论了测量误差、正则化参数与计算精度之间的关系。这一点在实际应用中的反问题中是很重要的，因为反问题中必须考虑测量误差的影响。由于反问题在每个环境中的问题是不同的，我们把反问题的数学分析作为项目的基本主题，我们讨论解的唯一性和条件稳定性。共同研究员Masahiro Yamamoto教授在逆散射问题中获得了尖锐的结果。在将数学和数值分析的结果应用于实际问题时，需要从计算力学的角度进行基础研究。所有合作研究人员都讨论了各自领域的应用反问题。讨论了计算机辅助证明，并成功地实现了快速多精度算法的应用之一的数值验证技术。少