奇异线性方程组和具有特定结构的非线性问题的研究与应用

项目名称： 奇异线性方程组和具有特定结构的非线性问题的研究与应用

项目编号： No.11471122

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 陈果良

作者单位： 华东师范大学

项目金额： 60万元

中文摘要： 大型稀疏奇异线性方程组和具有特殊结构的非线性问题，在计算流体力学、控制系统、神经网络、非线性波动、量子力学等领域中存在着广泛应用。由于奇异线性方程组的不稳定性、解的不唯一性，使得求解此问题时面临实际困难；同时由于特殊结构非线性问题的复杂性，其理论分析与数值算法有待进一步研究。因此成为近年来数值代数方向和非线性分析的新兴课题。 本项目拟解决如下两个重要问题：（1）给出求解大型稀疏奇异线性方程组的高效迭代算法及有效的奇异预条件子，同时针对几类多参数定常迭代算法给出最优迭代参数。（2）针对具有特殊结构的非线性方程组和非线性特征值的数值求解问题，建立非线性方程的可解性条件和收敛理论体系，给出非线性特征值的高精度快速求解算法。本项目将更加系统和深入的研究上述两类问题的求解理论与算法分析，并为解决相关实际科学计算问题提供科学依据。

中文关键词： 线性方程组；非线性方程组；迭代算法；数值方法；收敛性分析

英文摘要： The problems of large, sparse, singular linear equations and nonlinear questions with special structure arise in a number of scientific computing and engineering applications, such as computational fluid dynamics, control system, neural network, nonlinear wave, quantum mechanics, and so on. Due to the instability and the uniqueness solution of the singular linear equations, it has pratical difficulties while solving the problems. At the same time, because of the complexity of the special structured nonlinear questions, the theoretical analysis and numerical algorithms need further research. Therefore, they have become the emerging issues of numerical algebra and nonlinear analysis in recent years. This project is proposed to solve the following two important questions:(1)The efficient iterative algorithms of solving large, sparse, singular linear equations will be given, and the effective preconditioner will also be given. Moreover, we will give the optimal iterative parameters for several multi-parameter constant iterative algorithms.(2)For the problems of solving special structured nonlinear equations and nonlinear eigenvalues, we will establish the solvability conditions and convergent theory system, and give the high precision and fast algorithms of solving nonlinear eigenvalues.This project will be more systematic and indepth studied about the solving theory and algorithm analysis of the above two types of problems.And it will provided scientific basis for the related practical scientific computing problems.

英文关键词： Linear equations;Nonlinear equations;Iterative algorithms;Numerical methods;Convergence analysis