关于高超声速绕流问题的激波层方法

切尔诺依(ГГ.Черный)的高超声速绕流激波层的级数解法,不适用于(γ-1)/(γ+1)<<2/(γ十1)M2sin2β(γ=c_p/c_v为绝热指数,M为马赫数,β为激波倾角)的情形。本文只假定在激波邻近存在激波层,在此薄层内,气体密度很大,但不排除(γ-1)/(γ+1)<<2/(γ+1)M2sin2β的情形。     

On the Newton Interior-Point Method for Nonlinear Programming Problems

Interior-point methods have been developed largely for nonlinear programming problems. In this paper, we generalize the global Newton interior-point method introduced in Ref. 1 and we establish a global convergence theory for it, under the same assumptions as those stated in Ref. 1. The generalized algorithm gives the possibility of choosing different descent directions for a merit function so that difficulties due to small steplength for the perturbed Newton direction can be avoided. The particular choice of the perturbation enables us to interpret the generalized method as an inexact Newton method. Also, we suggest a more general criterion for backtracking, which is useful when the perturbed Newton system is not solved exactly. We include numerical experimentation on discrete optimal control problems.     

On Restarting the Arnoldi Method for Large Nonsymmetric Eigenvalue Problems

The Arnoldi method computes eigenvalues of large nonsymmetric matrices. Restarting is generally needed to reduce storage requirements and orthogonalization costs. However, restarting slows down the convergence and makes the choice of the new starting vector difficult if several eigenvalues are desired. We analyze several approaches to restarting and show why Sorensen's implicit QR approach is generally far superior to the others. Ritz vectors are combined in precisely the right way for an effective new starting vector. Also, a new method for restarting Arnoldi is presented. It is mathematically equivalent to the Sorensen approach but has additional uses.

     

On the Gradient Method for Some Nonlinear Elliptic Boundary Value Problems

The infinite-dimensional gradient method is applied to the iterative solution of quasilinear elliptic boundary value problems. Earlier results on uniformly monotone problems are extended to a general case within the scope of Hilbert space well-posedness. Linear convergence is proved in Sobolev norm. This revised version was published online in June 2006 with corrections to the Cover Date.     

On a Method for Solving Boundary Value Problems of Aerohydrodynamics

Journal of Applied and Industrial Mathematics - Under consideration is the well-known boundary value problem of aerohydrodynamics in which it is required to find the velocity distribution of fluid...     

关于亚音速绕流问题的强奇性基本解法

本文对求解亚音速流的偶极子基本解法作了新的处理,导出了一个关于求解偶极子强度的强奇性积分方程,给出了强奇性积分有效主值的定义及计算公式.由此可以导出多种整体连续分布的数值基本解法.适用于亚音速气动力计算.     

On the Convergence of Finite Element Method for Second Order Elliptic Interface Problems

The purpose of this paper is to study the convergence of finite element approximation to the exact solution of general self-adjoint elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it is difficult to achieve optimal order of convergence with classical finite element methods [Numer. Math. 1998; 79:175–202]. In this paper, an isoparametric type of discretization is used to prove optimal order error estimates in L ² and H ¹ norms when the global regularity of the solution is low. The interface is assumed to be of arbitrary shape and is smooth for our purpose. Further, for the purpose of numerical computations, we discuss the effect of numerical quadrature on finite element solution, and the related optimal order estimates are also established.     

On the Convergence Regions of Generalized Accelerated Overrelaxation Method for Linear Complementarity Problems

In this paper, we use a generalized Accelerated Overrelaxation (GAOR) method and analyze the convergence of this method for solving linear complementarity problems. Furthermore, we improve on the convergence region of this method with acknowledgement of the maximum norm. A numerical example is also given, to illustrate the efficiency of our results.     

非凸无约束优化问题的广义拟牛顿法的全局收敛性

本文对无约束优化问题提出一类新的广义拟牛顿法,并采用一类非精确线搜索证明了算法对一般非凸目标函数极小化问题的全局收敛性.     

线性互补问题模系多重网格方法的AMSOR光滑算子

为了改进求解大型稀疏线性互补问题模系多重网格方法的收敛速度和计算时间,本文采用加速模系超松弛(AMSOR)迭代方法作为光滑算子.局部傅里叶分析和数值结果表明此光滑算子能有效地改进模系多重网格方法的收敛因子、迭代次数和计算时间.     

关于摄动分支问题的奇异摄动法

本文较全面地阐述了用奇异摄动法计算摄动分支问题(1.6)在y=0,λ=0,δ=0附近的解的一致渐近展式的一般数学原理和方法,并推广了牛顿多边形原理。最后,给出了两个计算实例。     

关于吴方法在双层规划中的一个应用

双层规划及多层规划这一数学规划研究中的较新领域因其坚实的经济背景及丰富的数学内涵在二十年来的发展过程中变得越来越吸引人们的关注. 在通常解双层规划问题时往往采用数值计算的方法, 得到的解并不一定是全局最优解. 该文介绍了一个全新的解双层规划问题的方法,它与数值计算法不同, 采用的是符号计算, 依据了计算机代数与代数几何的理论. 作者通过对文献[1]中的几个双层规划问题的上机计算, 得出了与之不同的全面彻底的解答, 在比较过程中, 发现不仅所得的结果要比文献[1]中答案更进一步, 而且也证明了文章的新方法在解这一类问题时,是简明和行之有效的.     

On Finite and Strong Convergence of a Proximal Method for Equilibrium Problems

We introduce the notions of conditioning and well-posedness for equilibrium problems. Using these concepts, we obtain finite and strong convergence results for the proximal method that improve, develop, and unify several theorems in optimization and nonlinear analysis.     

On a Hybrid Extragradient-Viscosity Method for Monotone Operators and Fixed Point Problems

The purpose of this article is to give a more general scheme for approximating a common element of the fixed-point set of a certain mapping and the set of solutions of a variational inequality problem. This scheme is inspired by the recent work of Maingé [A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim. 47, 1499–1515 (2008)]. We also show that some assumption imposed in his result can be relaxed. Moreover, our scheme is a genuine generalization of Maingé's result because there is a class of mappings to which our scheme is applicable, but which is beyond the scope of his result.     

基于非拟牛顿方法无约束最优化问题的全局收敛性

In this paper, the non-quasi-Newton's family with inexact line search applied to unconstrained optimization problems is studied. A new update formula for non-quasi-Newton's family is proposed. It is proved that the constituted algorithm with either Wolfe-type or Armijotype line search converges globally and Q-superlinearly if the function to be minimized has Lipschitz continuous gradient.     

Smoothing Method for Minimax Problems

In this paper, we propose a smoothing method for minimax problem. The method is based on the exponential penalty function of Kort and Bertsekas for constrained optimization. Under suitable condition, the method is globally convergent. Preliminary numerical experiments indicate the promising of the algorithm.     

求解动力学问题的普遍方法

本文阐明了标准化的Routh方程[1]的应用,给出了求解系统动力学问题的约束反力及运动状态变化的普遍方法,并给出了相应的矩阵方程。     

Convergence of the BFGS-SQP Method for Degenerate Problems

In this paper, we propose a BFGS (Broyden–Fletcher–Goldfarb–Shanno)-SQP (sequential quadratic programming) method for nonlinear inequality constrained optimization. At each step, the method generates a direction by solving a quadratic programming subproblem. A good feature of this subproblem is that it is always consistent. Moreover, we propose a practical update formula for the quasi-Newton matrix. Under mild conditions, we prove the global and superlinear convergence of the method. We also present some numerical results.