广义鞍点问题的松弛维数分解预条件子

本文将Benzi等提出的松弛维数分解(Relaxed dimensionalfactorization, RDF)预条件子进一步推广到广义鞍点问题上,并称为GRDF(Generalized RDF)预条件子.该预条件子可看做是用维数分裂迭代法求解广义鞍点问题而导出的改进维数分裂(Modified dimensional split, MDS)预条件子的松弛形式, 它相比MDS预条件子更接近于系数矩阵, 因而结合Krylov子空间方法(如GMRES)有更快的收敛速度.文中分析了GRDF预处理矩阵特征值的一些性质,并用数值算例验证了新预条件子的有效性.     

关于鞍点问题的广义预处理HSS-SOR交替分裂迭代方法

本文研究了鞍点问题的迭代法. 在白中治,Golub和潘建瑜提出的预处理对称/反对称分裂（PHSS）迭代法的基础上,通过结合GSOR迭代格式,利用两个参数加速,提出了一种广义预处理HSS-SOR交替分裂迭代法,并研究了该方法的收敛性.数值结果表明本文所给方法是有效的.     

水平线性互补问题投影迭代法和模系矩阵分裂迭代法的等价性

水平线性互补问题(HLCP)是著名线性互补问题(LCP)的重要推广形式之一,投影迭代法和模系矩阵分裂迭代法是最近提出的求解HLCP两类非常有效的热点方法.本文研究表明,尽管这两类方法导出原理不同,但在一定条件下是等价的.特别地,当模系矩阵分裂迭代法中参数矩阵Ω取为特定的正对角矩阵时,投影Jacobi法、投影Gauss-Seidel法和投影SOR法分别等价于模系Jacobi迭代法、加速的模系Gauss-Seidel迭代法和加速的模系SOR迭代法.此外,对一般的正对角矩阵Ω,本文也研究了两类方法的等价性.最后,通过数值算例验证了本文的理论结果.     

鞍点问题的广义位移分裂预条件子

对于大型稀疏非Hermitian正定线性方程组,Bai等人提出了一种位移分裂预条件子(J.Comput.Math.,24(2006)539-552).本文将这种思想用到鞍点问题上并提出了一种广义位移分裂(Generalized Shift Splitting,GSS)预条件子,同时证明了该预条件子所对应分裂迭代法的无条件收敛性.最后用数值算例验证了新预条件子的有效性.     

应用迭代法求解一类非线性问题

本文应用迭代法求解一类有限维非线性问题,该方法是求解线性问题的雅可比迭代法在非线性问题上的推广,且此迭代方法具有几何收敛性质.     

求解一类隐式互补问题的加速模系矩阵分裂迭代方法

本文提出求解一类隐式互补问题的加速模系矩阵分裂迭代法.通过将隐式互补问题重新表述为一个等价的不动点方程,建立一类新的基于模系的两步矩阵分裂方法,并在一定条件下证明了方法的收敛性.数值实验表明,该方法在迭代步数上优于传统的模系矩阵分裂迭代方法.     

连续型凸动态规划的离散近似迭代法研究

为解决连续型凸动态规划的“维数灾”问题,提出了一种新的算法—离散近似迭代法.该算法的基本思路为:首先,将连续型状态变量离散化,根据网络图的构造方法将动态规划问题转化为多阶段有向赋权图;其次,运用极大代数求出起点至终点的最短路,即获得模型的一个可行解;最后,以该可行解为基础,继续迭代直到前后两个可行解非常接近.文章还证明了该算法的收敛性和线性收敛,并以一个具体例子验证了算法的有效性.     

广义鞍点问题基于PSS的约束预条件子

对于(1,1)块为非Hermitian阵的广义鞍点问题,本文给出了一种基于正定和反对称分裂(Positive definite andskew-Hermitian splitting, PSS)的约束预条件子.该预条件子的(1,1)块由求解非Hermitian正定线性方程组时的PSS迭代法所构造得到.文中分析了PSS约束预条件子的一些性质并证明了预处理迭代法的收敛性.最后用数值算例验证了该预条件子的有效性.     

连续Sylvester方程的广义正定和反Hermitian分裂迭代法及其超松弛加速

对于求解大型稀疏连续Sylvester方程,Bai提出了非常有效的Hermitian和反Hermitian分裂(HSS)迭代法.为了进一步提高求解这类方程的效率,本文建立一种广义正定和反Hermitian分裂(GPSS)迭代法,并且提出不精确GPSS(IGPSS)迭代法从而可以降低计算成本.对GPSS迭代法及其不精确变...     

投影三角分解法定价带随机波动率的美式期权

考虑数值求解Heston随机波动率美式期权定价问题,通过在空间方向采用中心差分格式离散二维偏微分算子,在时间方向利用隐式交替方向格式,将美式期权定价问题转化成求解每个时间层上的若干个线性互补问题.针对一般美式期权定价模型离散得到的线性互补问题,构造出投影三角分解法进行求解,并在理论上给出算法的收敛条件.数值实验表明,所构造的数值方法对于求解美式期权定价问题是有效的,并且优于经典的投影超松弛迭代法和算子分裂方法.     

速度追踪问题中鞍点系统的新分裂迭代

 有限元离散一类速度追踪问题后得到具有鞍点结构的线性系统,针对该鞍点系统,本文提出了一种新的分裂迭代技术.证明了新的分裂迭代方法的无条件收敛性,详细分析了新的分裂预条件子对应的预处理矩阵的谱性质.数值结果验证了对于大范围的网格参数和正则参数,新的分裂预条件子在求解有限元离散速度追踪问题得到的鞍点系统时的可行性和有效性.     

A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations

In this paper we introduce a new preconditioner for linear systems of saddle point type arising from the numerical solution of the Navier-Stokes equations. Our approach is based on a dimensional splitting of the problem along the components of the velocity field, resulting in a convergent fixed-point iteration. The basic iteration is accelerated by a Krylov subspace method like restarted GMRES. The corresponding preconditioner requires at each iteration the solution of a set of discrete scalar elliptic equations, one for each component of the velocity field. Numerical experiments illustrating the convergence behavior for different finite element discretizations of Stokes and Oseen problems are included.     

Nonoverlapping Domain Decomposition Method with Mixed Element for Elliptic Problems

1.IntroductionDomaindecompositionasanewmethodofcomputationalmathematics,waJsdevel-opedsincethedevelopmentofparallelcomputersandmultiprocessorsupercomputers-Usingdomaindecompositionwecandecreasethescaleoftheproblemandimplementthesub-problemsonparallelcomputer.Fromatechnicalpointofviewmostofdo-maindecompositionmethodsconsideredsofarhavebeendealingwithfiniteelementmethods.In[1,2]ZhangandHuanghavegivenakindofnonoverlappingdomaindecompositionprocedurewithpiecewiselinearfiniteelementapproximation.…     

Multilevel methods for mixed finite elements in three dimensions

In this paper we consider second order scalar elliptic boundary value problems posed over three–dimensional domains and their discretization by means of mixed Raviart–Thomas finite elements [18]. This leads to saddle point problems featuring a discrete flux vector field as additional unknown. Following Ewing and Wang [26], the proposed solution procedure is based on splitting the flux into divergence free components and a remainder. It leads to a variational problem involving solenoidal Raviart–Thomas vector fields. A fast iterative solution method for this problem is presented. It exploits the representation of divergence free vector fields as s of the –conforming finite element functions introduced by Nédélec [43]. We show that a nodal multilevel splitting of these finite element spaces gives rise to an optimal preconditioner for the solenoidal variational problem: Duality techniques in quotient spaces and modern algebraic multigrid theory [50, 10, 31] are the main tools for the proof. Received November 4, 1996 / Revised version received February 2, 1998     

On exact and approximate boundary controllabilities for the heat equation: A numerical approach

The present article is concerned with the numerical implementation of the Hilbert uniqueness method for solving exact and approximate boundary controllability problems for the heat equation. Using convex duality, we reduce the solution of the boundary control problems to the solution of identification problems for the initial data of an adjoint heat equation. To solve these identification problems, we use a combination of finite difference methods for the time discretization, finite element methods for the space discretization, and of conjugate gradient and operator splitting methods for the iterative solution of the discrete control problems. We apply then the above methodology to the solution of exact and approximate boundary controllability test problems in two space dimensions. The numerical results validate the methods discussed in this article and clearly show the computational advantage of using second-order accurate time discretization methods to approximate the control problems.     

An algorithm using the finite volume element method and its splitting extrapolation

This paper is to present a new efficient algorithm by using the finite volume element method and its splitting extrapolation. This method combines the local conservation property of the finite volume element method and the advantages of splitting extrapolation, such as a high order of accuracy, a high degree of parallelism, less computational complexity and more flexibility than a Richardson extrapolation. Because the splitting extrapolation formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid, a large scale multidimensional problem is turned into a set of smaller discrete subproblems. Additionally, this method is efficient for solving interface problems if we regard the interfaces of the problems as the interfaces of the initial domain decomposition.     

Adaptive computation of smallest eigenvalues of self‐adjoint elliptic partial differential equations

We consider a new adaptive finite element (AFEM) algorithm for self‐adjoint elliptic PDE eigenvalue problems. In contrast to other approaches we incorporate the inexact solutions of the resulting finite‐dimensional algebraic eigenvalue problems into the adaptation process. In this way we can balance the costs of the adaptive refinement of the mesh with the costs for the iterative eigenvalue method. We present error estimates that incorporate the discretization errors, approximation errors in the eigenvalue solver and roundoff errors, and use these for the adaptation process. We show that it is also possible to restrict to very few iterations of a Krylov subspace solver for the eigenvalue problem on coarse meshes. Several examples are presented to show that this new approach achieves much better complexity than the previous AFEM approaches which assume that the algebraic eigenvalue problem is solved to full accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

A hybrid projective method for solving system of equilibrium problems with demicontractive mappings applicable in image restoration problems

In this paper, we propose a general iterative scheme based on CQ projection method for finding a common solution of system of equilibrium problems and the fixed point set of a finite family of demicontractive mappings. We also prove strong convergence of the scheme to a common element of the two above-described sets. We then give a numerical example to justify our main result. An example is given in an infinite dimensional space for supporting our main result. Moreover, we apply our main result to solve the unconstrained image restoration problems with a finite family of blurring operators. Our results extend and improve some existing results in the literature.     

半导体器件瞬态模拟的对称正定混合元方法

提出具有对称正定特性的混合元格式求解非稳态半导体器件瞬态模拟问题。提出一个最小二乘混合元方法、一个新的具有分裂和对称正定性质的混合元格式和一个解经典混合元方程的对称正定失窃工格式求解电场位势和电场强度方程;提出一个最小二乘混合元格式求解关于电子与空穴浓度的非稳态对流扩散方程,浓度函数和流函数被同时求解;采用标准的有限元方法求解热传导方程。建立了误差分析理论。     

一类求解鞍点问题的广义不精确Uzawa方法

本文提出了一类求解大型稀疏鞍点问题的新的广义不精确Uzawa算法.该方法不仅可以包含 前人的方法, 而且可以拓展出很多新方法. 理论分析给出该方法收敛的条件, 并详细的分析了其收敛性质和参数矩阵的选取方法. 通过对有限元离散的Stokes问题的数值实验表明, 新方法是行之有效的, 其收敛速度明显优于原来的算法.