Numerical and Mathematical Analysis for the reconstruction for solutions of inverse and ill-posed problems by regularization methods

我们从广义上考虑“正则化方法数值分析”，我们的目标是在本研究中提出和发展新的方法来处理逆和不适定问题。计算机断层扫描和无损检测是支撑我们现代生活的重要技术，从数学的角度来看，它们是典型的逆问题。几乎所有的逆问题在Hadamard意义上都是病态的，用标准数值方法分析它们太困难；问题的病态性意味着计算中的数值不稳定，妨碍了数值解的可靠构造。提出了正则化方法，将不适定问题简化为具有正则化项的一系列良定问题，但我们不得不满足于正则解的数值处理，这些正则解有时与精确解相差很大。为了寻求病态问题的精确、可靠的数值解，我们澄清了正则化方法的缺点，并在本项目中提出了一些分析逆问题和病态问题的新技术和方法。本研究最显著的成果是在64位计算机上以软件的形式设计并实现了新的、快速的多精度算法。该软件使我们能够对病态问题进行数值处理，而不会产生导致数值不稳定的舍入误差。在此基础上提出了一种基于谱配置方法的新算法，并通过软件进行了大量数值实验，对合适正则化参数的选择给出了深刻的评价。我们在数学和计算上提出了新的方法来重建逆问题和不适定问题的解：局部Dirichlet -Neumann映射、基于滤波理论的正则化、区间分析法等。为今后反问题的分析提供了一定的数学基础；分形波的传播分析，新的脑数学模型，弹性裂纹的敏锐分析等。少

We consider "Numerical Analysis by Regularization Methods" in wide sense, and we have aimed, in the present research, to propose and develop new methods to deal with inverse and ill-posed problems.The computer tomography and non-destructive tests are important technologies to support our present life, and they are typical inverse problems from the mathematical view points. Almost all the inverse problems are ill-posed in the sense of Hadamard, and it is too difficult to analyze them by the standard numerical methods ; ill-posedness of the problems implies numerical instability in computation and prevents reliable construction of numerical solutions. Regularization methods are proposed to reduce ill-posed problems to series of well-posed ones with the regularization terms, but we are obliged to satisfy with numerical treatments of the regularized solutions which are sometimes quite different from the exact ones. In order to seek accurate and reliable numerical solutions for the ill-pose … More d problems, we have clarified demerits of regularization methods, and we have proposed some new techniques and methods for analysis of inverse and ill-posed problems in the present project.The most remarkable results in the present research is to design and to implement new and fast multiple-precision arithmetic on 64-bits computers as a software. The software enables us numerical treatments of ill-posed problems without rounding errors which cause numerical instability. And we propose a new algorithm based on the spectral collocation methods, and we give a keen remark for the choice of the suitable regularization parameter by many numerical experiments using our software.We propose new methods to reconstruct solutions of inverse and ill-posed problems in both mathematics and in computation: localized Dirichlet -Neumann map, regularization based on the filter theory, interval analysis approach etc. And we give some mathematical foundations for the analysis of inverse problems in the near future; analysis of propagation of waves on Fractals, a new mathematical model for brain, keen analysis for cracks in elasticity etc. Less