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

本研究项目的目的是对病态问题的数学分析和数值分析，这些病态问题写在偏微分方程中，与应用逆问题相联系，在物理、医学和工程中很重要。特别是考虑到未来的实际需要，除了数学理论和算法之外，我们还开发了一种新的快速多精度算法环境，以便对病态问题进行高精度的大规模数值计算。在包括反问题数值模拟在内的科学计算中，数字计算机上实数的表示和运算通常采用浮点近似。目前常用的方法是IEEE754标准中定义的双精度算法。这意味着科学的数值计算是在假设实数有15位十进制数字的前提下进行的……在通常的最终用户环境中更准确。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实数。当然，我们也必须考虑数值计算中泛函方程和偏微分方程离散化中出现的离散化误差。在逆问题中常见的病态问题中，误差是可靠数值计算的致命缺陷。这是良定问题与可导出稳定数值格式的问题之间最大的不同之处。传统的不适定问题数值分析只处理离散化误差或测量误差，而对舍入误差的考虑不够。本研究的重点是在传统的离散误差和测量误差的数值分析之外，开发了一种新的多精度算法来讨论舍入误差。在多精度算法环境下，发现了函数方程高精度离散化的新方面，开发并建立了新的计算格式。其中一个具体的成果是在前人的研究中设计和实现的快速多精度算术环境“exflib”，由共同研究员Hiroshi Fujiwara教授改进，他成功地实现了特殊功能并移植到超级计算机上处理科学数值模拟。我们还采用了谱方法，与传统的离散化方法相比，谱方法得到了精度很高的数值解。将多精度算法与谱方法相结合，证明了该方法对病态问题的数值分析是非常有效的。讨论了高精度数值方法下的正则化方法，特别是测量误差、正则化参数与计算精度之间的关系。这一点在实际应用的反问题中是很重要的，在这些问题中我们必须考虑测量误差。每个问题都有自己的问题。由于反问题在每一种情况下的问题是不同的，所以我们将反问题的数学分析作为项目的基础课题，并讨论解的唯一性和条件稳定性。共同研究员Masahiro Yamamoto教授在逆散射问题上获得了尖锐的结果。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。所有共同研究人员都讨论了各自领域的应用逆问题。我们还讨论了计算机辅助证明和成功的数值验证技术，这是快速多精度算法的应用之一。少

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