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

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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