非线性迭代学习控制问题的延拓修正牛顿法

对于非线性迭代学习控制问题,提出基于延拓法和修正Newton法的具有全局收敛性的迭代学习控制新方法.由于一般的Newton型迭代学习控制律都是局部收敛的,在实际应用中有很大局限性.为拓宽收敛范围,该方法将延拓法引入迭代学习控制问题,提出基于同伦延拓的新的Newton型迭代学习控制律,使得初始控制可以较为任意的选择.新的迭代学习控制算法将求解过程分成N个子问题,每个子问题由换列修正Newton法利用简单的递推公式解出.本文给出算法收敛的充分条件,证明了算法的全局收敛性.该算法对于非线性系统迭代学习控制具有全局收敛和计算简单的优点.     

一种求解非线性方程组的混合优化算法及程序实现

非线性方程组求解是工程实践与理论研究中的一个典型问题。传统的方法主要有梯度法、Newton迭代法等。该文综合修正Newton法与梯度法的各自优势,对非线性方程组的求解问题提出了一种混合方法并用C语言编码实现该算法。将两种方法相结合,使其相互取长补短,在迭代初始值不太好的情况下也能保证收敛性,同时加快收敛速度,数值结果表明该算法是有效的。     

基于牛顿法的并行优化算法

针对非线性数值优化问题,提出一种在分布式环境下的基于牛顿法的并行算法。引入松弛变量,将不等式约束转换为等式约束,利用广义拉格朗日乘子将约束优化问题转换为无约束子优化问题。为了并行地求解这些子优化问题,将Newton迭代法中的Hessian矩阵进行适当的分裂,采用简单迭代法求解Newton法中的线性方程组。在理论上对该算法进行了收敛性分析。在HP rx2600集群上进行的数值实验结果表明并行效率达90%以上。     

求解非线性方程组的一种并行算法

提出了一种在分布式环境下求解非线性方程组的并行算法,该算法将Newton迭代法中的Jacobi矩阵进行适当的分裂,使得Newton迭代法具有很好的并行性。并在理论上进行了收敛性分析。在HP rx2600集群上进行的数值实验结果表明并行效率达70%以上。     

B样条曲面方向投影问题的几何计算方法

B样条曲面方向投影问题可以通过求解方程组的方法来解决.由于方程组所有根中往往只有一个或甚至没有根与待求解的最近点对应,因而绝大多数的求根计算量是不必要的.为此讨论了B样条曲面的方向投影问题,提出一种简单且高效稳定的几何计算方法.该方法充分利用了B样条函数的凸包性,同时结合B样条函数稳定可靠的分裂算法给出了相应的几何剪枝方法.与传统的求解非线性方程组的计算方法相比,文中方法可以剪除绝大部分非线性方程组对应的根,且不需要Newton迭代,可以应用于平面/B样条曲面间的求交测试问题及B样条曲面包围盒的计算问题.实例结果表明,该方法具有比传统的相关方法更高的计算效率和更好的稳定性.     

求解非线性回归问题的Newton算法

针对大规模非线性回归问题,提出基于静态储备池的Newton算法.利用储备池搭建高维特征空间,将原始问题转化成与储备池维数相关的线性支持向量回归问题,并应用Newton算法求解.鲁棒损失函数的应用可抑制异常点对预测结果的干扰.通过与SVR(Support Vector Regression)及储备池Tikhonov正则化方法比较,验证了所提方法的快速性、较高的预测精度和较好的鲁棒性.     

基于信赖域Newton 算法的ELM网络

针对极端学习机(ELM)网络伪逆输出权值计算方法的运算复杂度制约其训练速度问题,提出一种基于信赖域Newton算法的新型ELM网络(TRON-ELM),并采用信赖域Newton算法求解ELM网络的输出权值.该算法首先构造一个ELM网络代价函数的Newton方程,并将其作为一个无约束优化问题,采用共轭梯度法求解,避免了求代价函数Hessian矩阵逆的运算,提高了训练速度,信赖域条件的存在保证了算法的整体收敛性.仿真实验结果验证了所提出方法的有效性.     

拟Newton法在高阶矩阵中的应用——求解最大特征值及特征向量

将求解高阶矩阵的最大特征值及其对应的特征向量问题转化为高阶非线性方程组的求解问题。在此基础上,提出了求解矩阵最大特征值及其对应特征向量的拟Newton法,给出求解矩阵最大特征值及其单位化向量重新整理后的Broyden方法公式、BFS方法公式、DFP方法公式及其对应的Broyden算法,BFS算法,DFP算法。以层次分析法中高阶判断矩阵为例验证了该方法的可行性,说明了该方法相对收敛速度快的优势。     

Riccati方程子矩阵约束对称解的非精确Newton-MCG算法

采用修正共轭梯度法(MCG算法)求由Newton算法每一步迭代计算导出的线性矩阵方程的近似子矩阵约束(SMC)对称解或者近似SMC对称最小二乘解,建立求离散时间代数Riccati矩阵方程SMC对称解的非精确Newton-MCG算法.该算法仅要求Riccati矩阵方程有SMC对称解,不要求它的SMC对称解唯一,也不要求导出的线性矩阵方程有相应的SMC对称解.数值算例表明,非精确Newton-MCG算法是有效的.     

两类相场方程的紧致指数时间差分法设计

本文针对相场方程提出稳定的高阶紧致指数时间差分算法.该算法具有完全显式的特性,从而避免了求解线性或非线性方程组.算法使用精确指数时间差分和多步法近似以保证精确性;通过线性算子分裂控制刚性非线性项以增强稳定性;同时引入有限差分格式的紧致表示大大降低了指数时间差分法的存储需求和计算量.算法的精确性和高效性通过CahnHilliard方程和Willmore问题相场模型的大规模三维模拟进行了验证.     

ILU预处理Newton-Krylov方法的潮流计算

由于电力系统修正方程组具有高维、稀疏的特点,本文提出将预处理Krylov子空间方法应用于潮流修正方程组的求解,形成预处理Newton-Krylov的潮流计算方法。结合ILU预处理方法,比较了最常用的3类Newton-Krylov方法求解潮流方程的计算效果。通过对 IEEE30、IEEE118、IEEE300 和3个Poland大规模电力系统进行潮流计算,结果表明：3类Newton-Krylov方法是电力系统潮流计算的有效方法,呈现出良好的收敛特性和计算效率。     

Uniformly Convergent Iterative Methods for Discontinuous Galerkin Discretizations

We present iterative and preconditioning techniques for the solution of the linear systems resulting from several discontinuous Galerkin (DG) Interior Penalty (IP) discretizations of elliptic problems. We analyze the convergence properties of these algorithms for both symmetric and non-symmetric IP schemes. The iterative methods are based on a “natural” decomposition of the first order DG finite element space as a direct sum of the Crouzeix-Raviart non-conforming finite element space and a subspace that contains functions discontinuous at interior faces. We also present numerical examples confirming the theoretical results.     

Parallel Krylov Methods for Econometric Model Simulation

This paper investigates parallel solution methods to simulate large-scalemacroeconometric models with forward-looking variables. The method chosen isthe Newton-Krylov algorithm, and we concentrate on a parallel solution to thesparse linear system arising in the Newton algorithm. We empirically analyzethe scalability of the GMRES method, which belongs to the class of so-calledKrylov subspace methods. The results obtained using an implementation of thePETSc 2.0 software library on an IBM SP2 show a near linear scalability forthe problem tested.     

Multigrid method for stability problems

The problem of calculating the stability of steady state solutions of differential equations is treated. Leading eigenvalues (i.e., having maximal real part) of large matrices that arise from discretization are to be calculated. An efficient multigrid method for solving these problems is presented. The method begins by obtaining an initial approximation for the dominant subspace on a coarse level using a damped Jacobi relaxation. This proceeds until enough accuracy for the dominant subspace has been obtained. The resulting grid functions are then used as an initial approximation for appropriate eigenvalue problems. These problems are solved first on coarse levels, followed by refinement until a desired accuracy for the eigenvalues has been achieved. The method employs local relaxation on all levels together with a global change on the coarsest level only, which is designed to separate the different eigenfunctions as well as to update their corresponding eigenvalues. Coarsening is done using the FAS formulation in a nonstandard way in which the right-hand side of the coarse grid equations involves unknown parameters to be solved for on the coarse grid. This in particular leads to a new multigrid method for calculating the eigenvalues of symmetric problems. Numerical experiments with a model problem that are presented demonstrate the effectiveness of the method proposed. Using an FMG algorithm a solution to the level of discretization errors is obtained in just a few work units (less than 10), where a work unit is the work involved in one Jacobi relaxation on the finest level.     

Symmetric and non-symmetric waves in the osmosis K(2, 2) equation

We give an improved qualitative method to solve the osmosis K(2, 2) equation. This method combines several characteristics of other methods. Using this method, the existence of symmetric and non-symmetric wave solutions of the osmosis K(2, 2) equation is studied. Besides abundant symmetric forms such as smooth wave solutions, peaked waves, cusped waves, looped waves, stumpons and fractal-like waves, this equation also admits non-symmetric ones including breaking kink wave solutions, breaking anti-kink wave solutions and rampons. As regards this equation most of those solutions, either symmetric or non-symmetric solutions, have not appeared in the literature. We also study the limiting behavior of all periodic solutions as the parameters tend to some special values.     

Modified HSS iteration methods for a class of complex symmetric linear systems

In this paper, we introduce and analyze a modification of the Hermitian and skew-Hermitian splitting iteration method for solving a broad class of complex symmetric linear systems. We show that the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method is unconditionally convergent. Each iteration of this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices. These two systems can be solved inexactly. We consider acceleration of the MHSS iteration by Krylov subspace methods. Numerical experiments on a few model problems are used to illustrate the performance of the new method.     

Incomplete factorization methods for three-dimensional non-symmetric problems

Various incomplete factorization methods (ILU) are described for the solution of non-symmetric linear systems arising from systems of partial differential equations in three dimensions. Pivot stabilization techniques are investigated for problems which are strongly non-diagonally dominant. Test results are presented for model problems and matrices generated from a simulation of enhanced oil recovery by steam injection.     

The effect of orderings on sparse approximate inverse preconditioners for non-symmetric problems

We experimentally study how reordering techniques affect the rate of convergence of preconditioned Krylov subspace methods for non-symmetric sparse linear systems, where the preconditioner is a sparse approximate inverse. In addition, we show how the reordering reduces the number of entries in the approximate inverse and thus, the amount of storage and computation required for a given accuracy. These properties are illustrated with several numerical experiments taken from the discretization of PDEs by a finite element method and from a standard matrix collection.     

Dual reciprocity boundary element method in Laplace domain applied to anisotropic dynamic crack problems

In this paper the dual reciprocity boundary element method in the Laplace domain for anisotropic dynamic fracture mechanic problems is presented. Crack problems are analyzed using the subregion technique. The dynamic stress intensity factors are computed using traction singular quarter-point elements positioned at the tip of the crack. Numerical inversion from the Laplace domain to the time domain is achieved by the Durbin method. Numerical examples of dynamic stress intensity factor evaluation are considered for symmetric and non-symmetric problems. The influence of the number of Laplace parameters and internal points in the solution is investigated.     

Preconditioned Krylov subspace methods for solving radiative transfer problems with scattering and reflection

Two Krylov subspace methods, the GMRES and the BiCGSTAB, are analyzed for solving the linear systems arising from the mixed finite element discretization of the discrete ordinates radiative transfer equation. To increase their convergence rate and stability, the Jacobi and block Jacobi methods are used as preconditioners for both Krylov subspace methods. Numerical experiments, designed to test the effectiveness of the (preconditioned) GMRES and the BiCGSTAB, are performed on various radiative transfer problems: (i) transparent, (ii) absorption dominant, (iii) scattering dominant, and (iv) with specular reflection. It is observed that the BiCGSTAB is superior to the GMRES, with lower iteration counts, solving times, and memory consumption. In particular, the BiCGSTAB preconditioned by the block Jacobi method performed best amongst the set of other solvers. To better understand the discrete systems for radiative problems (i) to (iv), an eigenvalue spectrum analysis has also been performed. It revealed that the linear system conditioning deteriorates for scattering media problems in comparison to absorbing or transparent media problems. This conditioning further deteriorates when reflection is involved.