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

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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