结构矩阵线性互补问题的模系矩阵分裂迭代方法

项目名称： 结构矩阵线性互补问题的模系矩阵分裂迭代方法

项目编号： No.11501300

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 徐玮玮

作者单位： 南京信息工程大学

项目金额： 18万元

中文摘要： 结构矩阵线性互补问题来源于科学与工程计算的许多领域, 如在凸二次优化问题中寻找纳什均衡点,运动的刚体单边约束,不平等的最优控制问题,流体力学中的自由边界问题等等问题中都有广泛的应用。如何有效地求解结构矩阵线性互补问题开始成为计算数学界的一个研究热点。本项目主要研究结构矩阵线性互补问题的模系矩阵分裂迭代方法。内容包括: 研究结构系统矩阵不同类型的分裂；基于这些分裂建立模系矩阵分裂迭代方法；讨论模系矩阵分裂迭代方法的收敛性质以及迭代公式中参数选取的问题。对模系矩阵分裂迭代方法做扰动分析的研究,研究系统系数矩阵发生小的扰动或小的误差对数值解会产生怎样的影响，即数值解会产生怎样的误差。同时，对模系矩阵分裂迭代算法的稳定性和敏感性方面进行分析。本项目旨在促进结构矩阵线性互补问题的模系矩阵分裂迭代方法研究, 为求解结构矩阵线性互补问题提供有效的方法和理论，有着一定的理论和实际意义。

中文关键词： 结构矩阵；线性互补问题；矩阵分裂；不动点方程

英文摘要： Linear complementarity problems of structured matrices come from science and engineering computing in many fields, such as finding Nash equilibrium points in convex optimization problem of a rigid body in two, unilateral constrained motion of the electrical network, the ideal diode, widely applied to inequality and optimal control problems in fluid mechanics the free boundary problem. How to effectively solve linear complementarity problems of structured matrices began to become a hot research topic in computational mathematics circles. This project is mainly on the research of modulus-based matrix splitting iterative methods for linear complementarity problems of structured matrices. The contents include: various types of splits based on the system matrix, the establishment of a series of modulus matrix splitting iterative methods and discussing their convergence properties and how to select the parameters of the problem. Doing some perturbation theory analysis of the new iterative method, studying the system coefficient matrix to small perturbations or small errors of the numerical solution will produce what kind of effect, namely the numerical solution will produce what kind of error. At the same time, stability and sensitivity of the new iterative algorithm are also analyzed and measured. This research project aims to promote modulus matrix splitting iterative methods of the linear complementarity problem of structured matrices, and provides the effective methods and theories for solving linear complementarity problems of structured matrices. Therefore, it is of important theoretical and practical significance.

英文关键词： structured matrices; linear complementarity problem; matrix splitting ;fixed-point equation