求解跨音速势流的自适应并行多重网格算法

本文用并行Ｓｃｈｗａｒｚ方法求解了轴向大扰动、径向小扰动的跨音速势流方程，并用自适应多重网格算法作为整体修正。数值计算表明：自适应并行多重网格算法可使计算效率大为提高。     

多重网格技术对SIMPLER算法的加速性能研究

将多重网格技术引入SIMPLER算法以加快其收敛速度，从而节约计算时间。通过计算不同雷诺数下的二维方腔顶盖驱动流，研究了多重网格方法中的V循环、W循环对SIMPLER算法的加速效果，并讨论了网格层数对加速性能的影响。研究结果表明，在不同雷诺数下，多重网格方法均可以起到良好的加速效果；在相同雷诺数和精度要求下，W循环方式的外迭代次数少于V循环方式的外迭代次数，而且网格层数对多重网格加速性能的影响并不显著。     

多重网格格子Boltzmann方法的并行算法

针对复杂流动数值模拟中的格子Boltzmann方法存在计算网格量大、收敛速度慢的缺点,提出了基于三维几何边界的多重笛卡儿网格并行生成算法,并基于该网格生成方法提出了多重网格并行格子Boltzmann方法（LBM）。该方法结合不同尺度网格间的耦合计算,有效减少了计算网格量,提高了收敛速度;而且测试结果也表明该并行算法具有良好的可扩展性。     

大尺度图像编辑的泊松方程并行多重网格求解算法

随着获取设备的发展,大尺度、高分辫率数字图像已逐步进入人们的生活,大尺度图像的梯度域编辑显得更为重要,求解大规模未知数的泊松方程是大尺度图像梯度域编辑的关键。传统多重网格算法的迭代、约束和插值操作单独进行,内存和外存间通讯量大,算法效率低,为此提出了一种面向大尺度图像梯度域编辑的并行多重网格求解泊松方程的算法。该算法利用多重网格的迭代、约束和插值过程的内存数据访问局部性和更新相关性,构造滑动工作窗口,使迭代、约束和插值操作并行运行,提高了多重网格算法求解泊松方程的计算效率。全景图拼接实验表明,所提算法的运行效率高于超松弛迭代、高斯塞德尔迭代和传统多重网格算法。     

多重网格方法求解两类Helmholtz方程

详细给出了多重网格方法的实现过程,借助正定Helmholtz方程及不定Helmholtz方程的求解来探讨多重网格方法的特性。对多重网格V环、W环以及F环三种不同迭代格式的收敛效果进行了对比。通过正定Helmholtz方程的求解,发现多重网格的确有很高的计算效率。对于不定Helmholtz方程,随着波数的增加,利用多重网格方法得到结果不收敛,原因出在细网格光滑和粗网格矫正过程。如何针对此问题对多重网格进行有效改进还有待进一步研究。     

一种非线性系数的粒子群优化算法

针对粒子群优化算法（Particle Swarm Optimization-PSO）存在算法初期容易陷入局部极值、进化后期收敛速度慢和精度低的缺点,提出了一种用非线性函数调整惯性权重和加速系数的粒子群优化算法（nfPSO）。nfPSO通过一个与当前迭代次数相关的非线性函数控制惯性权重和加速系数,从而提高了算法的收敛速度与精度。通过与两个算法对三个基准测试函数的仿真实验结果对比,说明了nfPSO算法具有良好的收敛速度与精度。     

Cahn-Hilliard方程多重网格求解器收敛性分析

Cahn-Hilliard(CH)方程是相场模型中的一个基本的非线性方程，通常使用数值方法进行分析。在对CH方程进行数值离散后会得到一个非线性的方程组，全逼近格式(Full Approximation Storage, FAS)是求解这类非线性方程组的一个高效多重网格迭代格式。目前众多的求解CH方程主要关注数值格式的收敛性，而没有论证求解器的可靠性。文中给出了求解CH方程离散得到的非线性方程组的多重网格算法的收敛性证明，从理论上保证了计算过程的可靠性。针对CH方程的时间二阶全离散差分数值格式，利用快速子空间下降(Fast Subspace Descent, FASD)框架给出其FAS格式多重网格求解器的收敛常数估计。为了完成这一目标，首先将原本的差分问题转化为完全等价的有限元问题，再论证有限元问题来自一个凸泛函能量形式的极小化，然后验证能量形式及空间分解满足FASD框架假设，最终得到原多重网格算法的收敛系数估计。结果显示，在非线性情形下，CH方程中的参数ε对网格尺度添加了限制，太小的参数会导致数值计算过程不收敛。最后通过数值实验验证了收敛系数与方程参数及网格尺度的依赖关系。     

一类求解鞍点问题的推广的GSOR迭代法

为了快速有效地求解大型稀疏鞍点问题,在SOR-like迭代算法的基础上,通过引入新的待定参数对原有迭代算法进行加速的思想,构造了一种解鞍点问题的具有多个待定参数的一般加速超松弛迭代算法,并给出了该算法收敛性的条件。数值例子表明：通过参数值的选择,新算法比SOR-like和GSOR算法都具有更快的收敛速度和更小的迭代次数,选择了合适的参数值后,可以大大提高算法的收敛效率。     

结构化多重网格粘性流场数值模拟

流体力学控制方程的数值求解过程中,当网格加密或者粘性效应强的时候,流场收敛非常缓慢.为了解决计算的效率问题,在结构网格的基础上采用多重网格技术,模拟了二维RAE2822超临界翼型的亚音速绕流及三维M6机翼跨音速流场,仿真结果表明,采用多重网格方法在二维,三维粘性流场的计算结果都与实验结果吻合良好,与不采用多重网格方法比较,在求解中获得了相当满意的加速收敛效果.还比较了两种不同循环方式:V循环,W循环的加速效率,为多重网格的工程应用奠下基础.     

三维对流扩散方程的多重网格算法研究

研究了三维对流扩散方程基于有限差分法的多重网格算法。差分格式采用一般网格步长下的二阶中心差分格式和四阶紧致差分格式,建立了与两种格式相适应的部分半粗化的多重网格算法,构造了相应的限制算子和插值算子,并与传统的等距网格下的完全粗化的多重网格算法进行了比较。数值研究结果表明,对于各向异性问题,一般网格步长下的部分半粗化多重网格算法比等距网格下的完全粗化多重网格算法具有个更高的精度和更好的收敛效率。     

A multilevel hybrid Newton–Krylov–Schwarz method for the Bidomain model of electrocardiology

A multilevel hybrid Newton–Krylov–Schwarz (NKS) method is constructed and studied numerically for implicit time discretizations of the Bidomain reaction–diffusion system in three dimensions. This model describes the bioelectrical activity of the heart by coupling two degenerate parabolic equations with a stiff system of ordinary differential equations. The NKS Bidomain solver employs an outer inexact Newton iteration to solve the nonlinear finite element system originating at each time step of the implicit discretization. The Jacobian update during the Newton iteration is solved by a Krylov method employing a multilevel hybrid overlapping Schwarz preconditioner, additive within the levels and multiplicative among the levels. Several parallel tests on Linux clusters are performed, showing that the convergence of the method is independent of the number of subdomains (scalability), the discretization parameters and the number of levels (optimality).     

A parallel multigrid FAS scheme for transputer networks

This paper discusses the parallel implementation of a multigrid full approximation scheme (FAS) for the solution of non-linear elliptic PDEs in both 2 and 3 dimensions. The method used for smoothing is Red Black Newton approximation. The purpose of this paper is to investigate whether it is possible to construct a 16 processor transputer network which permits the efficient execution of multigrid algorithms. In particular, our aim is to maintain the parallel efficiency of the underlying iterative method, whilst achieving vastly improved convergence rates due to multigrid.     

A Cascadic Multigrid Algorithm for Semilinear Indefinite Elliptic Problems

We propose a cascadic multigrid algorithm for a semilinear indefinite elliptic problem. We use a standard finite element discretization with piecewise linear finite elements. The arising nonlinear equations are solved by a cascadic organization of Newton's method with frozen derivative on a sequence of nested grids. This gives a simple version of a multigrid method without projections on coarser grids. The cascadic multigrid algorithm starts on a comparatively coarse grid where the number of unknowns is small enough to obtain an approximate solution within sufficiently high precision without substantial computational effort. On each finer grid we perform exactly one Newton step taking the approximate solution from the coarsest grid as initial guess. The linear Newton systems are solved iteratively by a Jacobi-type iteration with special parameters using the approximate solution from the previous grid as initial guess. We prove that for a sufficiently fine initial grid and for a sufficiently good start approximation the algorithm yields an approximate solution within the discretization error on the finest grid and that the method has multigrid complexity with logarithmic multiplier. Received February 1999, revised July 13, 1999     

Parallel Two-Grid Semismooth Newton-Krylov-Schwarz Method for Nonlinear Complementarity Problems

We develop scalable parallel domain decomposition algorithms for nonlinear complementarity problems including, for example, obstacle problems and free boundary value problems. Semismooth Newton is a popular approach for such problems, however, the method is not suitable for large scale calculations because the number of Newton iterations is not scalable with respect to the grid size; i.e., when the grid is refined, the number of Newton iterations often increases drastically. In this paper, we introduce a family of Newton-Krylov-Schwarz methods based on a smoothed grid sequencing method, a semismooth inexact Newton method, and a two-grid restricted overlapping Schwarz preconditioner. We show numerically that such an approach is totally scalable in the sense that the number of Newton iterations and the number of linear iterations are both nearly independent of the grid size and the number of processors. In addition, the method is not sensitive to the sharp discontinuity often associated with obstacle problems. We present numerical results for several large scale calculations obtained on machines with hundreds of processors.     

A class of parallel two-level nonlinear Schwarz preconditioned inexact Newton algorithms

We propose and test a new class of two-level nonlinear additive Schwarz preconditioned inexact Newton algorithms (ASPIN). The two-level ASPIN combines a local nonlinear additive Schwarz preconditioner and a global linear coarse preconditioner. This approach is more attractive than the two-level method introduced in [X.-C. Cai, D.E. Keyes, L. Marcinkowski, Nonlinear additive Schwarz preconditioners and applications in computational fluid dynamics, Int. J. Numer. Methods Fluids, 40 (2002), 1463-1470], which is nonlinear on both levels. Since the coarse part of the global function evaluation requires only the solution of a linear coarse system rather than a nonlinear coarse system derived from the discretization of original partial differential equations, the overall computational cost is reduced considerably. Our parallel numerical results based on an incompressible lid-driven flow problem show that the new two-level ASPIN is quite scalable with respect to the number of processors and the fine mesh size when the coarse mesh size is fine enough, and in addition the convergence is not sensitive to the Reynolds numbers.     

铂电阻测温的非线性补偿算法分析

针对铂电阻测温时存在的本质非线性特征,分析了产生非线性误差的主要原因,阐明了反向分度函数法、牛顿迭代法、查表法等几种主要铂电阻非线性补偿算法的原理与特点,并利用基于单片机的温度测控系统的实测数据作为样本,在Matlab仿真环境下,按不同的曲线拟合阶次和方式等条件,对这些算法的补偿误差进行了对比分析。结果表明：采用分段最小二乘曲线拟合法既简单又能有效地减少补偿误差;在相同条件下,牛顿迭代法的补偿精度高于反向分度函数法和查表法。     

Incomplete Jacobian Newton method for nonlinear equations

This paper presents a new modified Newton method for nonlinear equations. This method uses a part of elements of the Jacobian matrix to obtain the next iteration point and is refereed to as the incomplete Jacobian Newton (IJN) method. The IJN method may be fit for solving large scale nonlinear equations with dense Jacobian. The conditions of linear, superlinear and quadratic convergence of the IJN method are given and the local convergence results are analyzed and proved. Some special IJN algorithms are designed and numerical experiments are given. The results show that the IJN method is promising.     

Multigrid method for nonlinear poroelasticity equations

In this study, a nonlinear multigrid method is applied for solving the system of incompressible poroelasticity equations considering nonlinear hydraulic conductivity. For the unsteady problem, an additional artificial term is utilized to stabilize the solutions when the equations are discretized on collocated grids. We employ two nonlinear multigrid methods, i.e. the “full approximation scheme” and “Newton multigrid” for solving the corresponding system of equations arising after discretization. For the steady case, both homogeneous and heterogeneous cases are solved and two different smoothers are examined to search for an efficient multigrid method. Numerical results show a good convergence performance for all the strategies.     

分布式存储环境下的加性Schwarz方法

引言 区域分裂方法起源于古老的schwarz交替方法[l].八十年代末期,法国数学家P.L.LionS提出了schwarz交替方法的投影解释[2一4],使得人们对schwarz交替方法有了全新的认识,为其进一步发展奠定了理论基础.由于并行计算环境的逐渐成熟以及预处理技术的兴起和大规模科学计算的需要,由严格串行的scliwarz交替方法发展了多种可完全并行的