线性互补问题模系矩阵分裂迭代方法的快速实现与应用

项目名称： 线性互补问题模系矩阵分裂迭代方法的快速实现与应用

项目编号： No.11301141

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 张丽丽

作者单位： 河南财经政法大学

项目金额： 22万元

中文摘要： 线性互补问题广泛地产生于经济、金融、控制和力学等诸多领域，如市场供需平衡、期权定价、交通网络平衡、自由边界问题、弹性接触问题和障碍问题等。因此，研究线性互补问题的高效数值解法及其具体实现具有重要的理论意义和很高的应用价值。模系矩阵分裂迭代方法是最近提出的用于求解线性互补问题的一种迭代方法。它是将线性互补问题等价转化为易于求解的线性方程组，在实际应用中易于实现且非常有效。本项目的目标是基于模系矩阵分裂迭代方法，研究求解线性互补问题高效数值算法的构造、理论分析和具体实现。为了提高求解问题的规模、减少求解时间，我们拟构造的数值算法有两类：一类是以模系矩阵分裂迭代方法为光滑子，构造多重网格算法；另一类是模系矩阵分裂迭代方法的异步并行实现。而且，我们还将分析这些算法的收敛性，并将其应用于大坝渗流和期权定价等实际问题的求解。

中文关键词： 线性互补问题；迭代方法；收敛性；并行算法；多重网格

英文摘要： Linear complementarity problems may arise in many areas of economic, financial, control and mechanics, including market equilibrium, option pricing, traffic network equilibrium, free boundary problems, elastic contact problems and obstacle problems and so on. Hence, to study effective numerical methods and their actual implementations for linear complementarity problems is of important theoretical meaning and high applied value. The modulus-based matrix splitting iterative method has recently been proposed for solving large sparse linear complementarity problems. In this method, the linear complementarity problem is reformulated equivalently into a system of fixed-point equations, which makes it practical and effective in actual applications. In this project, we will mainly study the constructions, the theory analyses and the actual implementations of effective numerical algorithms based on the modulus-based matrix splitting iterative method for linear complementarity problems. In order to solve the larger scale problems and decrease the computing time, we will construct two class of numerical algorithms for linear complementarity problems: one is the multigrid algorithms by using the modulus-based matrix splitting iterative methods as smoothers and another is the asynchronous parallel implementations of modulus

英文关键词： linear complementarity problem；iterative method；convergence；parallel algorithm；multigrid