噪声数据数值求导的高效稳定化方法及其应用研究

本项目旨在现有研究工作基础上，主要研究整数阶与分数阶数值求导的高精度Lanczos方法和基于双曲方程正反演的高效稳定化算法，以及其在分数阶扩散方程反问题、图像处理、光谱反应分析中的应用。利用有限差分与有限元方法、不适定问题的正则化方法、数值积分等现代数值计算方法，构造出高效稳定的数值求导算法；基于现代偏微分方程理论与方法、数学物理反问题的理论等研究所得算法的条件稳定性和收敛性、正则化参数选取准则等问题。同时，积极将所得研究成果应用到分数阶扩散方程反问题、图像边缘检测、光谱响应分析等应用领域，从而促进这些科学技术研究领域的发展。本项目将获得若干高精度稳定的数值求导算法及其稳定性与收敛性结果，并获得解决若干分数阶扩散方程反问题、图像边缘检测等问题的高效算法。本项目的实施将丰富不适定问题的理论与计算方法，并服务于数学专业研究生和数学本科生的人才培养。

This project aims to study high precision Lanczos methods and stabilization algorithms based on hyperbolic equation inversion for the integer order and the fractional order numerical differentiation, and also to study their applications in the fractional diffusion equation inverse problem, the application of image processing, the spectral analysis of the reaction. By using the finite difference and finite element method, regularization methods of ill posed problems, and numerical integration methods, stable and efficient algorithms f are constructed or numerical differentiation problems. Problems of conditional stability and convergence, and choosing criterion of regularization parameters are studied based on theories and methods of modern partial differential equations and inverse problems in mathematical physics. At the same time, the research results will be applied to the fractional diffusion equation inverse problem, image edge detection, spectral response analysis and other fields of application, thereby promoting the development of these scientific and technological research fields. This project will get some high precision stable numerical differentiation algorithms with their stability and convergence results and efficient algorithms to solve problems of inverse problems for the fractional diffusion equation and image edge detection. The implementation of this project will enrich the theoretical and computational methods of ill posed problems, and serve for the cultivation of postgraduates and undergraduates.