基于Picard迭代的PN×PN-2谱元法求解定常不可压缩Navier-Stokes方程

该文给出了一种求解二维定常不可压缩Navier-Stokes方程的基于Picard线性化迭代的P_N×P_N-2谱元法.通过Picard线性化将不可压缩Navier-Stokes方程的求解转化为一系列线性的Stokes-type方程,再利用非交错网格的P_N×P_N-2谱元法计算每个迭代步的Stokes-type方程.为了消除伪压力模,压力离散比速度离散低两阶,非交错网格的应用使得方程的离散方便且不会带来相应的插值误差,从而保证了谱精度.通过此方法数值计算了有精确解的Stokes流动、Kovasznay流动和方腔顶盖驱动流,结果表明,迭代收敛非常快,误差收敛达到了谱精度收敛,并且避免了压力震荡的出现,表明了该文方法准确可靠.     

Application of amplitude–frequency formulation to nonlinear oscillation system of the motion of a rigid rod rocking back

The scope of this paper is evaluating an oscillation system with nonlinearities, using a periodic solution called amplitude–frequency formulation, such as the motion of a rigid rod rocking back. The approach proposes a choice to overcome the difficulty of computing the periodic behavior of the oscillation problems in engineering. We are to compare the solutions results of this method with the exact ones in order to validate the approach and assess the accuracy of the solutions. This method has a distinguished feature, which makes it simple to use and agree with the exact solutions for various parameters. Moreover, it is perceived that with one‐step iteration high accuracy of the solution will be achieved. We may apply the results of the solution to explain some of the practical physical problems. Copyright © 2009 John Wiley & Sons, Ltd.     

On the numerical solution of large-scale sparse discrete-time Riccati equations

We discuss the numerical solution of large-scale discrete-time algebraic Riccati equations (DAREs) as they arise, e.g., in fully discretized linear-quadratic optimal control problems for parabolic partial differential equations (PDEs). We employ variants of Newton??s method that allow to compute an approximate low-rank factor of the solution of the DARE. The principal computation in the Newton iteration is the numerical solution of a Stein (aka discrete Lyapunov) equation in each step. For this purpose, we present a low-rank Smith method as well as a low-rank alternating-direction-implicit (ADI) iteration to compute low-rank approximations to solutions of Stein equations arising in this context. Numerical results are given to verify the efficiency and accuracy of the proposed algorithms.     

Progressive genetic algorithm for solution of optimization problems with nonlinear equality and inequality constraints

A new approach, identified as progressive genetic algorithm (PGA), is proposed for the solutions of optimization problems with nonlinear equality and inequality constraints. Based on genetic algorithms (GAs) and iteration method, PGA divides the optimization process into two steps; iteration and search steps. In the iteration step, the constraints of the original problem are linearized using truncated Taylor series expansion, yielding an approximate problem with linearized constraints. In the search step, GA is applied to the problem with linearized constraints for the local optimal solution. The final solution is obtained from a progressive iterative process. Application of the proposed method to two simple examples is given to demonstrate the algorithm.     

Inexact inverse subspace iteration for generalized eigenvalue problems

n this paper, we present an inexact inverse subspace iteration method for computing a few eigenpairs of the generalized eigenvalue problem Ax=λBx. We first formulate a version of inexact inverse subspace iteration in which the approximation from one step is used as an initial approximation for the next step. We then analyze the convergence property, which relates the accuracy in the inner iteration to the convergence rate of the outer iteration. In particular, the linear convergence property of the inverse subspace iteration is preserved. Numerical examples are given to demonstrate the theoretical results.     

An efficient nonlinear iteration scheme for nonlinear parabolic-hyperbolic system

A nonlinear iteration method named the Picard-Newton iteration is studied for a two-dimensional nonlinear coupled parabolic-hyperbolic system. It serves as an efficient method to solve a nonlinear discrete scheme with second spatial and temporal accuracy. The nonlinear iteration scheme is constructed with a linearization-discretization approach through discretizing the linearized systems of the original nonlinear partial differential equations. It can be viewed as an improved Picard iteration, and can accelerate convergence over the standard Picard iteration. Moreover, the discretization with second-order accuracy in both spatial and temporal variants is introduced to get the Picard-Newton iteration scheme. By using the energy estimate and inductive hypothesis reasoning, the difficulties arising from the nonlinearity and the coupling of different equation types are overcome. It follows that the rigorous theoretical analysis on the approximation of the solution of the Picard-Newton iteration scheme to the solution of the original continuous problem is obtained, which is different from the traditional error estimate that usually estimates the error between the solution of the nonlinear discrete scheme and the solution of the original problem. Moreover, such approximation is independent of the iteration number. Numerical experiments verify the theoretical result, and show that the Picard-Newton iteration scheme with second-order spatial and temporal accuracy is more accurate and efficient than that of first-order temporal accuracy.     

An efficient nonlinear iteration scheme for nonlinear parabolic–hyperbolic system

A nonlinear iteration method named the Picard–Newton iteration is studied for a two-dimensional nonlinear coupled parabolic–hyperbolic system. It serves as an efficient method to solve a nonlinear discrete scheme with second spatial and temporal accuracy. The nonlinear iteration scheme is constructed with a linearization–discretization approach through discretizing the linearized systems of the original nonlinear partial differential equations. It can be viewed as an improved Picard iteration, and can accelerate convergence over the standard Picard iteration. Moreover, the discretization with second-order accuracy in both spatial and temporal variants is introduced to get the Picard–Newton iteration scheme. By using the energy estimate and inductive hypothesis reasoning, the difficulties arising from the nonlinearity and the coupling of different equation types are overcome. It follows that the rigorous theoretical analysis on the approximation of the solution of the Picard–Newton iteration scheme to the solution of the original continuous problem is obtained, which is different from the traditional error estimate that usually estimates the error between the solution of the nonlinear discrete scheme and the solution of the original problem. Moreover, such approximation is independent of the iteration number. Numerical experiments verify the theoretical result, and show that the Picard–Newton iteration scheme with second-order spatial and temporal accuracy is more accurate and efficient than that of first-order temporal accuracy.     

求解不适定问题的快速Landweber迭代法

本文从一般迭代法的级数形式出发，将一般迭代法的每一步分解为矩阵计算和求解两步，并对其中的矩阵计算部分进行了修改，在此基础上提出了快速迭代法，最后通过数值实验验证了我们的算法不仅提高了计算速度，同时也大大减少了计算量，是一种效率很高的算法。     

Variational iteration method for solving cubic nonlinear Schrödinger equation

The variational iteration method is applied to solve the cubic nonlinear Schrödinger (CNLS) equation in one and two space variables. In both cases, we will reduce the CNLS equation to a coupled system of nonlinear equations. Numerical experiments are made to verify the efficiency of the method. Comparison with the theoretical solution shows that the variational iteration method is of high accuracy.     

Interval methods for fixed-point problems

Interval analysis is applied to the fixed-point problem x=?(x) for continuous ?:S→S, where the space S is constructed from Cartesian products of the set R of real numbers, with componentwise definitions of arithmetic operations, ordering, and the product topology. With the aid of an interval inclusion φ:IS → IS in the interval space IS corresponding to S, interval iteration is used to establish the existence or nonexistence of a fixed point x^? of ? in the initial interval X₀. Each step of the interval iteration provides lower and upper bounds for fixed points of ? in the initial interval, from which approximate values and guaranteed error bounds can be obtained directly. In addition to interval iteration, operator equation and dissection methods are considered briefly.

The theory of interval iteration applies directly when only finite subsets of S, IS are used, so this method is adaptable immediately to actual computation. A numerical example is given of the use of interval iteration for the computational solution of a nonlinear integral equation of radiative transfer. It is shown that numerical results with acceptable, guaranteed accuracy can be obtained with a modest amount of computation for an extended range of the parameter involved.     

Broyden method for inverse non-symmetric Sturm-Liouville problems

In this paper, we propose a derivative-free method for recovering symmetric and non-symmetric potential functions of inverse Sturm-Liouville problems from the knowledge of eigenvalues. A class of boundary value methods obtained as an extension of Numerov’s method is the major tool for approximating the eigenvalues in each Broyden iteration step. Numerical examples demonstrate that the method is able to reduce the number of iteration steps, in particular for non-symmetric potentials, without accuracy loss.     

A New Noninterior Predictor–Corrector Method for the P0 LCP

In this paper a new predictor-corrector noninterior method for LCP is presented, in which the predictor step is generated by the Levenberg-Marquadt method, which is new in the predictor-corrector-type methods, and the corrector step is generated as in [3]. The method has the following merits: (i) any cluster point of the iteration sequence is a solution of the P₀ LCP; (ii) if the generalized Jacobian is nonsingular at a solution point, then the whole sequence converges to the (unique) solution of the P₀ LCP superlinearly; (iii) for the P₀ LCP, if an accumulation point of the iteration sequence satisfies the strict complementary condition, then the whole sequence converges to this accumulation point superlinearly. Preliminary numerical experiments are reported to show the efficiency of the algorithm.     

Numerical simulation of the modified regularized long wave equation by He's variational iteration method

The modified regularized long wave (MRLW) equation, with some initial conditions, is solved numerically by variational iteration method. This method is useful for obtaining numerical solutions with high degree of accuracy. The variational iteration solution for the MRLW equation converges to its exact solution. Moreover, the conservation laws properties of the MRLW equation are also studied. Finally, interaction of two and three solitary waves is shown. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Stability and Accuracy of Inexact Interior Point Methods for Convex Quadratic Programming

We consider primal–dual interior point methods where the linear system arising at each iteration is formulated in the reduced (augmented) form and solved approximately. Focusing on the iterates close to a solution, we analyze the accuracy of the so-called inexact step, i.e., the step that solves the unreduced system, when combining the effects of both different levels of accuracy in the inexact computation and different processes for retrieving the step after block elimination. Our analysis is general and includes as special cases sources of inexactness due either to roundoff and computational errors or to the iterative solution of the augmented system using typical procedures. In the roundoff case, we recover and extend some known results.     

Periodic solution for strongly nonlinear vibration systems by He's variational iteration method

In this paper, we use variational iteration method for strongly nonlinear oscillators. This method is a combination of the traditional variational iteration and variational method. Some examples are given to illustrate the effectiveness and convenience of the method. The obtained results are valid for the whole solution domain with high accuracy. The method can be easily extended to other nonlinear oscillations and hence widely applicable in engineering and science. Copyright © 2009 John Wiley & Sons, Ltd.     

Solving Schrödinger equation by using modified variational iteration and homotopy analysis methods

In this paper, a nonlinear Schr ö dinger equation is solved by using the variational iteration method (VIM), modified variational iteration method (MVIM) and homotopy analysis method (HAM) numerically. For each method, the approximate solution of this equation is calculated based on a recursive relation which its components are computed easily. The existence and uniqueness of the solution and the convergence of the proposed methods are proved. A numerical example is studied to demonstrate the accuracy of the given algorithms     

Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt–Poincare method

In the paper, J.H. He's modified Lindstedt–Poincare method is applied to the nonlinear oscillators with discontinuities. Only one iteration leads to high accuracy of the solution.     

Reachability of Optimal Convergence Rate Estimates for High-Order Numerical Convex Optimization Methods

Doklady Mathematics - The Monteiro–Svaiter accelerated hybrid proximal extragradient method (2013) with one step of Newton’s method used at every iteration for the approximate solution...     

A domain decomposition method using efficient interface-acting preconditioners

The conjugate gradient boundary iteration (CGBI) is a domain decomposition method for symmetric elliptic problems on domains with large aspect ratio. High efficiency is reached by the construction of preconditioners that are acting only on the subdomain interfaces. The theoretical derivation of the method and some numerical results revealing a convergence rate of 0.04-0.1 per iteration step are given in this article. For the solution of the local subdomain problems, both finite element (FE) and spectral Chebyshev methods are considered.

     

非线性问题的参数迭代求解法

提出一种对非线性问题的参数迭代求解法,算例表明,其一次迭代解便有很好的精度.