奇摄动线性代数方程组及其对病态方程的应用

本文首先从一个曲柄导杆机构的优化问题提出了含小参数线性代数方程组的奇摄动问题。然后利用摄动方法证明了这个问题解的存在唯一性,同时给出了解的渐近展开和误差估计。最后讨论了所得结果对求解病态方程的应用。     

Numerical analysis of a non‐singular boundary integral method: Part I. The circular case

In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we present here a method which consists in inverting, on a finite element space, a non‐singular integral operator. This operator is a geometrical perturbation of the Steklov operator, and we precisely define the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme. Furthermore, this numerical scheme does not give rise to any singular integral. The scheme can also be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single layer potential, the special quadrature formula being defined by the introduction of a neighbouring curve. In the present paper, we prove stability and we give error estimates of our numerical scheme when the Laplace problem is set on a disk. We will extend our results to any domains by using compact perturbation arguments, in a second paper. Copyright © 2001 John Wiley & Sons, Ltd.     

奇异模糊线性系统的扰动分析

当模糊线性系统的系数矩阵奇异时, 分析了模糊线性系统的右端向量和系数矩阵的扰动对模糊线性系统解的计算造成的影响,用矩阵的谱范数给出了相对误差的界.     

B-Spline Collocation Method for Nonlinear Singularly-Perturbed Two-Point Boundary-Value Problems

A B-spline collocation method is presented for nonlinear singularly-perturbed boundary-value problems with mixed boundary conditions. The quasilinearization technique is used to linearize the original nonlinear singular perturbation problem into a sequence of linear singular perturbation problems. The B-spline collocation method on piecewise uniform mesh is derived for the linear case and is used to solve each linear singular perturbation problem obtained through quasilinearization. The fitted mesh technique is employed to generate a piecewise uniform mesh, condensed in the neighborhood of the boundary layers. The convergence analysis is given and the method is shown to have second-order uniform convergence. The stability of the B-spline collocation system is discussed. Numerical experiments are conducted to demonstrate the efficiency of the method.     

非线性优化方法在大气和海洋科学数值研究中的若干应用

控制大气和海洋运动的模式是复杂的非线性模式,在考虑到线性奇异向量和线性奇异值只能描述切线性模式有效时段内小扰动发展的情况下,介绍了作者们近年来用非线性优化方法数值研究大气和海洋科学的有关工作,其中包括非线性奇异向量和非线性奇异值、条件非线性最优扰动、以及它们在数值天气和气候可预报性研究中的应用．结果表明,上述非线性优化方法在很大程度上揭示了大气和海洋运动的非线性特征;此外,对可预报性问题的新分类也做了详细介绍,即最大可预报时间、最大预报误差和最大允许初始误差A·D2这种分类的应用背景是针对数值天气预报和气候预测产品的评价;最后,讨论了数值模式敏感性分析的非线性优化方法,该方法在一定条件下可以定量识别模式误差和初始误差,量化判断数值模式的模拟能力．     

Perturbation of normally solvable linear relations in paracomplete spaces

In this paper we investigate the stability of the index, the nullity and the deficiency of normally solvable linear relations in paracomplete spaces under perturbation by strictly singular and T-strictly singular linear relations. This study led us to generalize some well-known results for operators and extend some results of small perturbation of normally solvable linear relations given by T. Alvarez (2012) [3].     

Quantitative stability analysis of optimal solutions in PDE-constrained optimization

PDE-constrained optimization problems under the influence of perturbation parameters are considered. A quantitative stability analysis for local optimal solutions is performed. The perturbation directions of greatest impact on an observed quantity are characterized using the singular value decomposition of a certain linear operator. An efficient numerical method is proposed to compute a partial singular value decomposition for discretized problems, with an emphasis on infinite-dimensional parameter and observation spaces. Numerical examples are provided.     

Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods

The nonlinear singular initial value problems including generalized Lane–Emden-type equations are investigated by combining homotopy perturbation method (HPM) and reproducing kernel Hilbert space method (RKHSM). He’s HPM is based on the use of traditional perturbation method and homotopy technique and can reduce a nonlinear problem to some linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can overcome the difficulty at the singular point of non-homogeneous, linear singular initial value problems; especially when the singularity appears on the right-hand side of this type of equations, so it can solve powerfully linear singular initial value problems. Therefore, using advantages of these two methods, more general nonlinear singular initial value problems can be solved powerfully. Some numerical examples are presented to illustrate the strength of the method.     

On piecewise-uniform meshes for upwind- and central-difference operators for solving singularly perturbed problems

Numerical methods composed of upwind-difference operators onuniform meshes are shown analytically to be defective for solvingsingularly perturbed differential equations, in the sense that,as the mesh is refined, the error in the numerical approximationincreases until the mesh parameter has decreased to the sameorder of magnitude as the singular perturbation parameter. Itis also shown that the same is true for upwind-difference operatorson piecewise uniform meshes having a transition point that isdependent solely on the singular perturbation parameter. Itis then shown, with specific analytic examples, that both upwind-and central-difference operators on specially designed piecewise-uniformmeshes give numerical methods which do not suffer from thisdefect. Conditions are also given on the structure of a piecewiseuniform mesh that are necessary if the numerical method, composedof this mesh and an upwind-difference operator, is to be convergentuniformly with respect to the singular perturbation parameter.     

Piecewise homotopy perturbation method for solving linear and nonlinear weakly singular VIE of second kind

The purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular Volterra integral equation of the second kind, especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem. For this purpose, we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like. Through modifying homotopy perturbation method, an algorithm is successfully established to solve the linear and nonlinear weakly singular Volterra integral equation of the second kind. And the convergence of the algorithm is proved strictly. Comparisons are made between our method and other methods, and the results reveal that the modified homotopy perturbation is effective.     

A numerical method for singularly perturbed weakly coupled system of two second order ordinary differential equations with discontinuous source term

In this paper, a numerical method based on finite difference scheme and Shishkin mesh for singularly perturbed two second order weakly coupled system of ordinary differential equations with discontinuous source term is presented. An error estimate is derived to show that the method is uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented to illustrate the theoretical results.     

Convergence results for general linear methods on singular perturbation problems

Many numerical methods used to solve Ordinary Differential Equations, or Differential Algebraic Equations can be written as general linear methods. The B-convergence results for general linear methods are for algebraically stable methods, and therefore useless for nearly A-stable methods. The purpose of this paper is to show convergence for singular perturbation problems for the class of general linear methods without assuming A-stability.     

Convergence of linear multistep methods for two-parameter singular perturbation problems

1. IntroductionIt is well known that the IVPs of ODEs in singular perturbation form may be consideredas a special class of stab problems. But it is regrettable that they can't be satisfactorily covered by B-theory (of. [2-8]) because of their very special structure. In the recede ten yeaxslmany importallt and illteresting results on the convergence of linear multistep methods,Runge-Kutta methods, Rosenbrock methods and general linear methods applied to oneparameter singular perturbation pr…     

Lower bounds of error estimates for singular perturbation problems with Jacobi-Spectral approximations

With weighted orthogonal Jacobi polynomials, we study spectral approximations for singular perturbation problems on an interval. The singular parameters of the model are included in the basis functions, and then its stiff matrix is diagonal. Considering the estimations for weighted orthogonal coefficients, a special technique is proposed to investigate the a posteriori error estimates. In view of the difficulty of a posteriori error estimates for spectral approximations, we employ a truncation projection to study lower bounds for the models. Specially, we present the lower bounds of a posteriori error estimates with two different weighted norms in details.     

Morley-Wang-Xu element methods with penalty for a fourth order elliptic singular perturbation problem

Two Morley-Wang-Xu element methods with penalty for the fourth order elliptic singular perturbation problem are proposed in this paper, including the interior penalty Morley-Wang-Xu element method and the super penalty Morley-Wang-Xu element method. The key idea in designing these two methods is combining the Morley-Wang-Xu element and penalty formulation for the Laplace operator. Robust a priori error estimates are derived under minimal regularity assumptions on the exact solution by means of some established a posteriori error estimates. Finally, we present some numerical results to demonstrate the theoretical estimates.     

A mixed bvp for flow in strongly anisotropic media

Initial-boundary value problems for a class of linear parabolic equations are considered. The anisotropy of the medium is characterised by a small parameter. The solution structure is analysed by singular perturbation methods which include the construction of outer solutions and boundary and initial layer terms. The analysis is justified by convergence results     

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction-diffusion problems

The numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems. The quasi-optimal order error estimates are proved in the ε-weighted H¹-norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε-weighted H¹-norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.     

Invariance of deficiency indices under perturbation for discrete Hamiltonian systems

This paper deals with discrete Hamiltonian systems with one singular endpoint. Using Hermitian linear relation generalized by linear Hamiltonian system, the invariance of the minimal and maximal deficiency indices under bounded perturbation for discrete Hamiltonian systems is built. This parallels the well-known results for linear Hamiltonian differential systems obtained by F.V. Atkinson.     

在激发介质中单扩散回卷波组织中心的运动

应用奇异摄动方法,讨论了在单扩散型激发介质中无扭曲的三维回卷波组织中心的运动规律.确定了无扭曲的组织中心的曲率关系和线性律,这些结果具有明显的物理意义,而且与实验符合得很好.     

Hyperbolic-parabolic singular perturbation for quasilinear equations of Kirchhoff type

We consider a hyperbolic-parabolic singular perturbation problem for a quasilinear equation of Kirchhoff type, and obtain parameter-dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of Kirchhoff type and the corresponding quasilinear parabolic equation. For this purpose we show time decay estimates for hyperbolic-parabolic singular perturbation problem for linear equations with a time-dependent coefficient.